# An Extensive Categorisation of Infinite Paradoxes

In­fini­ties are one of the most com­plex and con­found­ing top­ics in math­e­mat­ics and they lead to an ab­surd num­ber of para­doxes. How­ever, many of the para­doxes turn out to be vari­a­tions on the same theme once you dig into what is ac­tu­ally hap­pen­ing. I will provide in­for­mal hints on how sur­real num­bers could help us solve some of these para­doxes, al­though the fo­cus on this post is pri­mar­ily cat­e­gori­sa­tion, so please don’t mis­take these for for­mal proofs. I’m also aware that sim­ply not­ing that a for­mal­i­sa­tion pro­vides a satis­fac­tory solu­tion doesn’t philo­soph­i­cally jus­tify its use, but this is also not the fo­cus of this post. I plan to up­date this post to in­clude new in­finite para­doxes that I learn about, in­clud­ing ones that are posted in the com­ments.

In­fini­tar­ian Paral­y­sis: Sup­pose there are an in­finite num­ber of peo­ple and they are happy so there is in­finite util­ity. A man punches 100 peo­ple and de­stroys 1000 util­ity. He then ar­gues that he hasn’t done any­thing wrong as there was an in­finite amount util­ity be­fore and that there is still an in­finite util­ity af­ter. What is wrong with this ar­gu­ment?

If we use car­di­nal num­bers, then we can’t make such a dis­tinc­tion. How­ever, if we use sur­real num­bers, then n+1 is differ­ent from n ev­ery when n is in­finite.

Para­dox of the Gods: “A man walks a mile from a point α. But there is an in­finity of gods each of whom, un­known to the oth­ers, in­tends to ob­struct him. One of them will raise a bar­rier to stop his fur­ther ad­vance if he reaches the half-mile point, a sec­ond if he reaches the quar­ter-mile point, a third if he goes one-eighth of a mile, and so on ad in­fini­tum. So he can­not even get started, be­cause how­ever short a dis­tance he trav­els he will already have been stopped by a bar­rier. But in that case no bar­rier will rise, so that there is noth­ing to stop him set­ting off. He has been forced to stay where he is by the mere un­fulfilled in­ten­tions of the gods”

Let n be a sur­real num­ber rep­re­sent­ing the num­ber of Gods. The man can move: 1/​2^n dis­tance be­fore he is stopped.

Two En­velopes Para­dox: Sup­pose that there are two sealed en­velopes with one hav­ing twice as much money in it as the other such that you can’t see how much is in ei­ther. Hav­ing picked one, should you switch?

If your cur­rent en­velope has x, then there is a 50% chance of re­ceiv­ing 2x and a 50% chance of re­ceiv­ing x/​2, which then cre­ates an ex­pected value of 5x/​4. But then ac­cord­ing to this logic we should also want to switch again, but then we’d get back to our origi­nal en­velope.

This para­dox is un­der­stood to be due to treat­ing a con­di­tional prob­a­bil­ity as an un­con­di­tional prob­a­bil­ity. The ex­pected value calcu­la­tion should be 1/​2[EV(B|A>B) + EV(B|A<B)] = 1/​2[EV(A/​2|A>B) + EV(2A|A<B)] = 14 EV(A|A>B) + EV(A|A<B)

How­ever, if we have a uniform dis­tri­bu­tion over the pos­i­tive re­als then ar­guably EV(A|A>B) = EV(A|A<B) in which case the para­dox re­oc­curs. Such a prior is typ­i­cally called an im­proper prior as it has an in­finite in­te­gral so it can’t be nor­mal­ised. How­ever, even if we can’t prac­ti­cally work with such a prior, these pri­ors are still rele­vant as it can pro­duce a use­ful pos­te­rior prior. For this rea­son, it seems worth­while un­der­stand­ing how this para­dox is pro­duced in this case

Part 2:

Sup­pose we uniformly con­sider all num­bers be­tween 1/​n and n for a sur­real n and we dou­ble this to gen­er­ate a sec­ond num­ber be­tween 2/​n and 2n. We can then flip a coin to de­ter­mine the or­der. This now cre­ates a situ­a­tion where EV(A|A>B) = n+1/​n and EV(A|A<B) = (n+1/​n)/​2. So these ex­pected val­ues are un­equal and we avoid the para­dox.

Sphere of Suffer­ing: Sup­pose we have an in­finite uni­verse with peo­ple in a 3-di­men­sional grid. In world 1, ev­ery­one is ini­tially suffer­ing, ex­cept for a per­son at the ori­gin and by time t the hap­piness spreads out to peo­ple within a dis­tance of t from the ori­gin. In world 2, ev­ery­one is ini­tially happy, ex­cept for a per­son at the ori­gin who is suffer­ing and this spreads out. Which world is bet­ter? In the first world, no mat­ter how much time passes, more peo­ple will be suffer­ing than happy, but ev­ery­one be­comes happy af­ter some finite time and re­mains that way for the rest of time.

With sur­real num­bers, we can use l to rep­re­sent how far the grid ex­tents in any par­tic­u­lar di­rec­tion and q to rep­re­sent how many time steps there are. We should then be able to figure out whether q is long enough that the ma­jor­ity of in­di­vi­d­ual time slices are happy or suffer­ing.

Sup­pose we have an in­finite set of finite num­bers (for ex­am­ple the nat­u­ral num­bers or the re­als), for any finite num­ber x, there ex­ists an­other finite num­ber 2x. Non-for­mally, we can say that each finite num­ber is in the first half; in­deed the first 1/​x for any in­te­ger x. Similarly, for any num­ber x, there’s always an­other num­ber x+n for ar­bi­trary n.

The canon­i­cal ex­am­ple is the Hilbert Ho­tel. If we have an in­finite num­ber of rooms la­bel­led 1, 2, 3… all of which are full, we fit an ex­tra per­son in by shift­ing ev­ery­one up one room num­ber. Similarly, we can fit an in­finite num­ber of ad­di­tional peo­ple in the ho­tel by send­ing each per­son in room x to room 2x to free up all of the odd rooms. This kind of trick only works be­cause any finite-in­dexed per­son isn’t in the last n rooms or in the last-half of rooms.

If we model the sizes of sets with car­di­nal num­bers, then all countable sets have the same size. In con­trast, if we were to say that there were n rooms where n is a sur­real num­ber, then n would be the last room and any­one in a room > n/​2 would be in the last half of rooms. So by mod­el­ling these situ­a­tions in this man­ner pre­vents last half para­doxes, but ar­guably in­sist­ing on the ex­is­tence of rooms with in­finite in­dexes changes the prob­lem. After all, in the origi­nal Hilbert Ho­tel, all rooms have a finite nat­u­ral num­ber, but the to­tal num­ber of rooms are in­finite.

Gal­ileo’s Para­dox: Similar to the Hilbert Ho­tel, but notes that ev­ery nat­u­ral num­ber has a square, but the that only some are squares, so the num­ber of squares must be both at least the num­ber of in­te­gers and less than the num­ber of in­te­gers.

Ba­con’s Puz­zle: Sup­pose that there is an in­finitely many peo­ple in a line each with ei­ther a black hat or a white hat. Each per­son can see the hats of ev­ery­one in front of them, but they can’t see their own hat. Sup­pose that ev­ery­one was able to get to­gether be­fore­hand to co-or­di­nate on a strat­egy. How would they be able to en­sure that only a finite num­ber of peo­ple would guess the colour of their hat in­cor­rectly? This pos­si­bil­ity seems para­dox­i­cal as each per­son should only have a 50% chance of guess­ing their hat cor­rectly.

We can di­vide each in­finite se­quence into equiv­alence classes where two se­quences are equiv­a­lent if they stop differ­ing af­ter a finite num­ber of places. Let’s sup­pose that we choose one rep­re­sen­ta­tive from each equiv­alence class (this re­quires the ax­iom of choice). If each per­son guesses their hat as per this rep­re­sen­ta­tive, af­ter a finite num­ber of places, each per­son’s guess will be cor­rect.

This para­dox is a re­sult of the rep­re­sen­ta­tive never last differ­ing from the ac­tu­ally cho­sen se­quence in the last half of the line due to this in­dex hav­ing to be finite.

Trumped: Don­ald Trump is re­peat­edly offered two days in heaven for one day in hell. Since heaven is as good as hell is bad, Trump de­cides that this is a good deal and ac­cepts. After com­plet­ing his first day in hell, God comes back and offers him the deal again. Each day he ac­cepts, but the re­sult is that Trump is never let into heaven.

Let n be a sur­real num­ber rep­re­sent­ing how many fu­ture time steps there are. Trump should ac­cept the deal as long as the day num­ber is less than n/​3

St Peters­berg Para­dox: A coin is tossed un­til it comes up heads and if it comes up heads on the nth toss, then you win 2^n dol­lars and the game ends. What is the fair price for play­ing this game? The ex­pected value is in­finite, but if you pay in­finity, then it is im­pos­si­ble for you to win money.

If we model this with sur­re­als, then sim­ply stat­ing that there is po­ten­tially an in­finite num­ber of tosses is un­defined. Let’s sup­pose first that there is a sur­real num­ber n that bounds the max­i­mum num­ber of tosses. Then your ex­pected value is $n and if we were to pay$n/​2, then we’d ex­pect to make $n/​2 profit. Now, set­ting a limit on the num­ber of tosses, even an in­finite limit is prob­a­bly against the spirit of the prob­lem, but the point re­mains, that pay­ing in­finity to play this game isn’t un­rea­son­able as it first sounds and the game effec­tively re­duces to Pas­cal’s Wager. Trou­ble in St. Peters­berg: Sup­pose we have a coin and we toss it un­til we re­ceive a tails and then stop. We are offered the fol­low­ing deals: • Offer 1: Lose$1 if it never lands on tails, gain $3 if it lands on tails on toss 1 • Offer 2: Lose$4 if if lands on tails on toss 1, gain $9 if it lands on tails on toss 2 • Offer 2: Lose$10 if if lands on tails on toss 2, gain $13 if it lands on tails on toss 3 And so on, when we calcu­late the losses by dou­bling and adding two and the gains by dou­bling and adding 3. Each deal has a pos­i­tive ex­pected value, so we should ac­cept all the deals, but then we ex­pect to lose in each pos­si­bil­ity. In the finite case, there is a large amount of gain in the fi­nal offer and this isn’t coun­tered by a loss in a sub­se­quence bet, which is how the EV ends up be­ing pos­i­tive de­spite be­ing nega­tive in ev­ery other case. The para­dox oc­curs for in­fini­ties be­cause there is no last n with an un­miti­gated gain, but if we say that we are offered n deals where n is a sur­real num­ber, then deal n will have this prop­erty. Dice Room Mur­ders: Sup­pose that a se­rial kil­ler takes a man hostage. They then roll a ten-sided dice and re­lease them if it comes up 10, kil­ling them oth­er­wise. If they were kil­led, then the se­rial kil­ler re­peats tak­ing twice as many hostages each time un­til they are re­leased. It seems that each per­son taken hostage should have a 110 chance of sur­viv­ing, but there are always more peo­ple who sur­vive than die. There’s an in­finites­i­mal chance that the dice never comes up a 10. In­clud­ing this pos­si­bil­ity, ap­prox­i­mately 50% of peo­ple should sur­vive, ex­clud­ing it the prob­a­bil­ity is 110. This is a re­sult of a bi­ased sam­ple. To see a finite ver­sion, imag­ine that there are two peo­ple. One tosses a coin. If it’s not heads, the other tosses a coin too. The per­son-flip in­stances will be 5050 heads/​tails, but if you av­er­age over peo­ple be­fore av­er­ag­ing over pos­si­bil­ities, you’ll ex­pect most of the flips to be heads. How you in­ter­pret this de­pends on your per­spec­tive on SSA and SIA. In any case, noth­ing to do with in­fini­ties. Ross-Lit­tle­wood Para­dox: Sup­pose we have a line and at each time pe­riod we add 10 balls to the end and then re­move the first ball re­main­ing. How many balls re­main at the end if we perform this an in­finite num­ber of times? Our first in­tu­ition would be to an­swer in­finite as 9 balls are added in each time pe­riod. On the other hand, the nth ball added is re­moved at time 10n, so ar­guably all balls are re­moved. Clearly, we can see that this para­dox is a re­sult of all balls be­ing in the first 110. Soc­cer Teams: Sup­pose that there are two soc­cer teams. Sup­pose one team has play­ers with abil­ities …-3,-2,-1,0,1,2,3… Now lets sup­pose a sec­ond team started off equally as good, but all the play­ers trained un­til they raised their abil­ity lev­els by one. It seems that the sec­ond team should be bet­ter as each player has strictly im­proved, but it also seems like it should be equally good as both teams have the same dis­tri­bu­tion. With sur­real num­bers, sup­pose that the teams origi­nally have play­ers be­tween -n and n in abil­ity. After the play­ers have trained, their abil­ities end up be­ing be­tween -n+1 and n+1. So the dis­tri­bu­tion ends up be­ing differ­ent af­ter all. Pos­i­tive Soc­cer Teams: Imag­ine that the skill level of your soc­cer team is 1,2,3,4… You im­prove ev­ery­one’s score by two which should im­prove the team, how­ever your skill lev­els are now 3,4… Since your skill lev­els are now a sub­set of the origi­nal it could be ar­gued that the origi­nal team is bet­ter since it has two ad­di­tion (weakly pos­i­tive) play­ers as­sist­ing. Can God Pick an In­te­ger at Ran­dom? - Sup­pose that there are an in­finite amount of planets la­bel­led 1,2,3… and God has cho­sen ex­actly one. If you bet that God didn’t pick that planet, you lose$2 if God ac­tu­ally chose it and you re­ceive $1/​2^n. On each planet, there is a 1/​∞ chance of it be­ing cho­sen and a (∞-1)/​∞ chance of it not be­ing cho­sen. So there is an in­finites­i­mal ex­pected loss and a finite ex­pected gain, given a pos­i­tive ex­pected value. This sug­gests we should bet on each planet, but then we lose$2 on one planet win less than \$1 from all the other planets.

Let sur­real n be the num­ber of planets. We ex­pect to win on each finite val­ued planet, but for sur­real-val­ued planets, our ex­pected gain be­comes not only in­finites­i­mal, but smaller than our ex­pected loss.

Banach-Tarski Para­dox: This para­dox in­volves de­scribing a way in which a ball can be di­vided up into five sets that can then be re­assem­bled into two iden­ti­cal balls. How does this make any sense? This might not ap­pear like an in­finite para­dox as first, but this be­comes ap­par­ent once you dig into the de­tails.

(Other var­i­ants in­clude the Haus­dorff para­dox, Sier­pin­ski-Mazurk­iewicz Para­dox and Von Neu­mann Para­dox)

Let’s first ex­plain how the proof of this para­dox works. We di­vide up the sphere by us­ing the free group of rank 2 to cre­ate equiv­alence classes of points. If you don’t know group the­ory, you can sim­ply think of this as com­bi­na­tions of where an el­e­ment is not al­lowed to be next to its in­verse, plus the spe­cial el­e­ment 1. We can then think of this purely in terms of these se­quences.

Let S rep­re­sent all such se­quences and S[a] rep­re­sent a se­quence start­ing with a. Then .

Fur­ther

In other words, not­ing that there is a bi­jec­tion be­tween all se­quences and all se­quences start­ing with any sym­bol s, we can write

Break­ing down similarly, we get:

In other words, al­most pre­cisely two copies of it­self apart from 1.

How­ever, in­stead of just con­sid­er­ing the se­quences of in­finite length, it might be helpful to as­sign them a sur­real length n. Then S[a] con­sists of “a” plus a string of n-1 char­ac­ters. So S[aa] isn’t ac­tu­ally con­gru­ent to S[a] as the former has n-2 ad­di­tion char­ac­ters and the later n-1. This time in­stead of the para­dox rely­ing on there be­ing that no finite num­bers are in the last half, it’s rely­ing on there be­ing no finite length strings that can’t have ei­ther “a” or “b” prepended in front of them which is prac­ti­cally equiv­a­lent if we think of these strings as rep­re­sent­ing num­bers in qua­ter­nary.

The Headache: Imag­ine that peo­ple live for 80 years. In one world each per­son has a headache for the first month of their life and are happy the rest, in the other, each per­son is happy for the first month, but has a headache af­ter that. Fur­ther as­sume that the pop­u­la­tion triples at the end of each month. Which world is bet­ter? In the first, peo­ple live the ma­jor­ity of their life headache free, but in the sec­ond, the ma­jor­ity of peo­ple at any time are headache free.

If we say that the world runs for t timesteps where t is a sur­real num­ber, then the peo­ple in the last timesteps don’t get to live all of their lives, so it’s bet­ter to choose the world where peo­ple only have a headache for the first month.

The Magic Dart­board: Imag­ine that we have a dart­board where each point is col­ored ei­ther black or white. It is pos­si­ble to con­struct a dart­board where all but mea­sure 0 of each ver­ti­cal line is black and all but mea­sure 0 of each hori­zon­tal line is white. This means that that we should ex­pect any par­tic­u­lar point to be black with prob­a­bil­ity 0 and white with prob­a­bil­ity 0, but it has to be some color.

One way of con­struct­ing this situ­a­tion is to first start with a bi­jec­tion f from [0, 1] onto the countable or­di­nals which is known to ex­ist. We let the black points be those ones where f(x) < f(y). So given any f(y) there are only a countable num­ber of or­di­nals less than it, so only a countable num­ber of x that are black. This means that the mea­sure of black points in that line must be 0, and by sym­me­try we can get the same re­sult for any hori­zon­tal line.

We only know that there will be a countable num­ber of black x for each hori­zon­tal line be­cause f(x) will always be in the first 1/​n of the or­di­nals for ar­bi­trary n. If on the other hand we al­lowed f(x) to be say in the last half of countable or­di­nals, then for that x we would get the ma­jor­ity of points be­ing black. This is dis­tinct from the other para­doxes in this sec­tion as for this ar­gu­ment to be cor­rect, this the­o­rem would have to be wrong. If we were to bite this bul­let, it would sug­gest any other proof us­ing similar tech­niques might also be wrong. I haven’t in­ves­ti­gated enough to con­clude whether this would be a rea­son­able thing to do, but it could have all kinds of con­se­quences.

The canon­i­cal ex­am­ple is Thom­son’s Lamp. Sup­pose we have a lamp that is turned on at t=-1, off at t=-1/​2, on at t=1/​4, ect. At t=0, will the lamp be on or off?

With sur­real num­bers, this ques­tion will de­pend on whether the num­ber of times that the switch is pressed is rep­re­sented by an odd or even Om­rific num­ber, which will de­pend in a rel­a­tively com­plex man­ner on how we define the re­als.

Grandi’s Series: What is the sum of 1-1+1-1...?

Us­ing sur­real num­bers, we can as­sign a length n to the se­ries as merely say­ing that it is in­finite lacks re­s­olute. The sum then de­pends on whether n is even or odd.

This is one class of para­doxes that sur­real num­bers don’t help with as sur­re­als don’t have a largest finite num­ber or a small­est in­fnity.

Satan’s Ap­ple: Satan has cut a deli­cious ap­ple into in­finitely many pieces. Eve can take as many pieces as she likes, but if she takes in­finitely many pieces she will be kicked out of par­adise and this will out­weigh the ap­ple. For any finite num­ber i, it seems like she should take that ap­ple piece, but then she will end up tak­ing in­finitely many pieces.

I find this para­dox the most trou­bling for at­tempts to for­mal­ise ac­tual in­fini­ties. If we ac­tu­ally had in­finitely many pieces, then we should be able to paint all finitely num­bered pieces red and all in­finitely num­bered pieces blue, but any finite num­ber plus one is finite and any in­finite num­ber minus one is in­finite, so it doesn’t seem like we can have a red and blue piece next to each other. But then, what does the bound­ary look like.

Th­ese “para­doxes” may point to in­ter­est­ing math­e­mat­i­cal phe­nomenon, but are so eas­ily re­solved that they hardly de­serve to be called para­doxes.

Gabriel’s Horn: Con­sider ro­tat­ing 1/​x around the x-axis. This can be proven to have finite vol­ume, but in­finite sur­face area. So it can’t con­tain enough paint to paint its sur­face.

It only can’t paint its sur­face if we as­sume a fixed finite thick­ness of paint. As x ap­proaches in­finity the size of the cross-sec­tion of the horn ap­proaches 0, so past a cer­tain point, this would make the paint thicker than the horn.

Miscellaneous

Ber­trand Para­dox: Sup­pose we have an equilat­eral tri­an­gle in­scribed in a cir­cle. If we choose a chord at ran­dom, what is the prob­a­bil­ity that the length of chord is longer than a side of the tri­an­gle.

There are at least three differ­ent meth­ods that give differ­ent re­sults:

• Pick­ing two ran­dom end points gives a prob­a­bil­ity of 13

• Pick­ing a ran­dom ra­dius then a ran­dom point on that ra­dius gives prob­a­bil­ity of 12

• Pick­ing a ran­dom point and us­ing it as a mid­point gives 14

Now some of these method will count di­ame­ters mul­ti­ple times, but even af­ter these are ex­cluded, we still ob­tain the same prob­a­bil­ities.

We need to bite the bul­let here and say that all of these prob­a­bil­ities are valid. It’s just that we can’t just choose a “ran­dom chord” with­out spec­i­fy­ing this more pre­cisely. In other words, there isn’t just a sin­gle set of chords, but mul­ti­ple that can be defined in differ­ent ways.

Zeno’s Para­doxes: There are tech­ni­cally mul­ti­ple para­doxes, but let’s go with the Dich­tomy Para­dox. Be­fore you can walk a dis­tance, you must go half way. But be­fore you can get halfway, you must get a quar­ter-way and be­fore that an eight of the way. So mov­ing a finite dis­tance re­quires an in­finite num­ber of tasks to be com­plete which is im­pos­si­ble.

It’s gen­er­ally con­sider un­con­tro­ver­sial and bor­ing these days that in­finite se­quences can con­verge. But more in­ter­est­ing, this para­dox seems to be a re­sult of claiming an in­finite amount of time in­ter­vals to di­verge, whilst al­low­ing an in­finite num­ber of space in­ter­vals to con­verge, which is a ma­jor in­con­sis­tency.

Skolem’s Para­dox: Any countable ax­iomi­sa­tion of set the­ory has a countable model ac­cord­ing to the Löwen­heim–Skolem the­o­rem, but Can­tor’s The­o­rem proves that there must be an un­countable set. It seems like this con­fu­sion arises from mix­ing up whether we want to know if there ex­ists a set that con­tains un­countably many el­e­ments or if the set con­tains un­countably many el­e­ments in the model (the cor­re­spond­ing defi­ni­tion of mem­ber­ship in the model only refers to el­e­ments in the model). So at a high level, there doesn’t seem to be very much in­ter­est­ing here, but I haven’t dug enough into the philo­soph­i­cal dis­cus­sion to ver­ify that it isn’t ac­tu­ally rele­vant.

This post was writ­ten with the sup­port of the EA Hotel

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• Al­most noth­ing in this post is cor­rect. This post dis­plays not just a mi­suse of and failure to un­der­stand sur­real num­bers, but a failure to un­der­stand car­di­nals, or­di­nals, free groups, lots of other things, and just how to think about such mat­ters gen­er­ally, much as in our last ex­change. The fact that (as I write this) this is sit­ting at +32 is an em­bar­rass­ment to this web­site. You re­ally, re­ally, need to go back and re­learn all of this from scratch, be­cause go­ing by this post you don’t have the slight­est clue what you’re talk­ing about. I would en­courage ev­ery­one else to stop up­vot­ing this crap.

This whole post is just throw­ing words around and mak­ing as­ser­tions that as­sume things gen­er­al­ize in a par­tic­u­lar naïve way that you ex­pect. Well, they don’t, and cer­tainly not ob­vi­ously.

Really, the whole idea here is wrong. The fact that some­thing does not ex­tend to in­fini­ties or in­finites­i­mals is not some­how a para­dox. Many things don’t ex­tend. There’s noth­ing wrong with that. Some things, of course, do ex­tend if you do things prop­erly. Some things ex­tend in more than one way, with none of them be­ing more nat­u­ral than the oth­ers! But if some­thing doesnt ex­tend, it doesn’t ex­tend. That’s not a para­dox.

Similarly, the fact that some­thing has un­ex­pected re­sults is not a para­dox. The right solu­tion for some of these is just to ac­tu­ally for­mal­ize them and ac­cept the re­sults. No fur­ther “re­s­olu­tion” is re­quired.

In the hopes of mak­ing my point ab­solutely clear, I am go­ing to take these one by one. ~(As per the bul­lshit asym­me­try prin­ci­ple, I’m afraid my re­sponse will be much longer than the origi­nal post.)~ (OK, I guess that turned out not to be true.) Those that in­volve philo­soph­i­cal prob­lems in ad­di­tion to just math­e­mat­i­cal prob­lems I will skip on my first pass, if you don’t mind (well, some of them, any­way; and I may have slightly mis­judged some of the ones I skipped, be­cause, well, I skipped them—point is I’m skip­ping some, it hardly mat­ters, the rest are enough to demon­strate the point, but maybe I will get back to the skipped ones later). Note that I’m go­ing to fo­cus on prob­lems in­volv­ing in­fini­ties some­how; if there are prob­lems not in­volv­ing in­fini­ties I’ll likely miss them.

In­fini­tar­ian paral­y­sis: Skip­ping for now due to philo­soph­i­cal prob­lems in ad­di­tion to math­e­mat­i­cal ones.

Para­dox of the gods: You haven’t stated your setup here for­mally, but if I try to for­mal­ize it (us­ing real num­bers as is prob­a­bly ap­pro­pri­ate here) I come to the con­clu­sion that yes, the man can­not leave the start­ing point. Is this a “para­dox”? No, it’s just what you get if you ac­tu­ally for­mal­ize this. The con­tinuum is coun­ter­in­tu­itive! It doesn’t quite fit our usual no­tions of causal­ity! Think about differ­en­tial equa­tions for a mo­ment—is it a “para­dox” that some differ­en­tial equa­tions have nonunique solu­tions, even though it seems that a par­ti­cle’s po­si­tion, ve­loc­ity, and re­la­tion be­tween the two ought to “cause” its fu­ture tra­jec­tory? No! This is the same sort of thing; con­tin­u­ous time and con­tin­u­ous space do not work like dis­crete time and dis­crete space.

But in ad­di­tion to your “re­s­olu­tion” be­ing un­nec­es­sary, it’s also non­sen­si­cal. You’re tak­ing the num­ber of gods as a sur­real num­ber. That’s non­sense. Sur­real num­bers are not for count­ing how many of some­thing there are. Are you try­ing to map car­di­nals to sur­re­als? I mean, yeah, you could define such a map, it’s easy to do with AC, but is it mean­ingful? Not re­ally. You do not count num­bers of things with sur­re­als, as you seem to be sug­gest­ing.

Of course, there’s more than one way to mea­sure the size of an in­finite set, not just car­di­nal­ity. Since you trans­late the num­ber into a sur­real, per­haps you meant the set of gods to be re­verse well-or­dered, so that you can talk about its re­verse or­der type, as an or­di­nal, and take that as a sur­real? That would go a lit­tle way to mak­ing this less non­sen­si­cal, but, well, you never said any such thing.

Of course, your solu­tion seems to in­volve im­plic­itly chang­ing the set­ting to have sur­real-val­ued time and space, but that makes sense—it does make sense to try to make such “para­doxes” make more sense by ex­tend­ing the do­main you’re talk­ing about. You might want to make more of an ex­plicit note of it, though. Any­way, let’s get back to non­sense.

So let’s say we ac­cept this re­verse-well-or­der­ing hy­poth­e­sis. Does your “re­s­olu­tion” fol­low? Does it even make sense? No to both! First, your “re­s­olu­tion” isn’t so much a de­duc­tion as a new as­sump­tion—that these re­verse-well-or­dered gods are placed at po­si­tions 1/​2^α for or­di­nals α. I mean, I guess that’s a sen­si­ble ex­ten­sion of the setup, but… let’s note here that you ac­tu­ally are chang­ing the setup sig­nifi­cantly at this point; the origi­nal setup pretty clearly had ω gods, not more. But, OK, that’s fine—you’re gen­er­al­iz­ing, from the case of ω to the case of more. You should be more ex­plicit that you’re do­ing that, but I guess that’s not wrong.

But your con­clu­sion still is wrong. Why? Sev­eral rea­sons. Let’s fo­cus on the case of ω-many gods, that the origi­nal setup de­scribes. You say that the man is stopped at 1/​2^ω. Ques­tion: Why? Is 1/​2^ω the min­i­mum of the set {1/​2^n : n ∈ N } in­side the sur­re­als? Well, ob­vi­ously not, be­cause that set ob­vi­ously has no small­est el­e­ment.

But is it the in­fi­mum (or equiv­a­lently limit), then, in­side the sur­re­als, if not the min­i­mum? Ac­tu­ally, let’s put that ques­tion aside for now and note that the an­swer to this ques­tion is ac­tu­ally ir­rele­vant! Be­cause if you ac­cept the logic that the in­fi­mum (or equiv­a­lently limit) con­trols, then, guess what, you already have your re­s­olu­tion to the para­dox back in the real num­bers, where there’s it’s an in­fi­mum and it’s 0. So all the rest of this is ir­rele­vant.

But let’s go on—is it the in­fi­mum (or equiv­a­lently limit)? No! It’s not! Be­cause there is no in­fi­mum! A sub­sets of the sur­re­als with no min­i­mum also has no in­fi­mum, always, un­con­di­tion­ally! The sur­real num­bers are not at all like the real num­bers. You ba­si­cally can’t do limits there, as we’ve already dis­cussed. So there’s noth­ing par­tic­u­larly dis­t­in­guish­ing about the point 1/​2^ω, no par­tic­u­lar rea­son why that’s where the man would stop. (There’s no god there! We’re talk­ing about the case of ω gods, not ω+1 gods.)

We haven’t even asked the ques­tion of what you mean by 2^s for a sur­real s. I’m go­ing to as­sume, since you’re talk­ing about sur­re­als and didn’t spec­ify oth­er­wise, that you mean exp(s log 2), us­ing the usual sur­real ex­po­nen­tial. But, since you’re only con­cerned with the case where s is an or­di­nal, maybe you ac­tu­ally meant tak­ing 2^s us­ing or­di­nal ex­po­nen­ti­a­tion, and then tak­ing the re­cip­ro­cal as a sur­real. Th­ese are differ­ent, I hope you re­al­ize that!

What about if we use {left set|right set} in­tead of limits and in­fima? Well, there’s now even less rea­son to be­lieve that such a point has any rele­vance to this prob­lem, but let’s make a note of what we get. What is {|1, 12, 14, …}? Well, it’s 0, duh. OK, what if we ex­clude that by ask­ing for {0|1, 12, 14, …} in­stead? That’s 1/​ω. This isn’t 1/​2^ω; it’s larger—well, un­less you meant “use or­di­nal ex­po­nen­ti­a­tion and then in­vert”, in which case it is in­deed equal and you need to be a hell of a lot clearer but it’s all still ir­rele­vant to any­thing. (Us­ing or­di­nal ex­po­nen­ti­a­tion, 2^ω = ω; while us­ing the sur­real ex­po­nen­tial, 2^ω = ω^(ω log 2) > ω.)

(What if we use sign-se­quence limits, FWIW? That’ll still get us 1/​ω. You re­ally shouldn’t use those though.)

Any­way, in short, your re­s­olu­tion makes no sense. Mov­ing on...

Two en­velopes para­dox: OK, I’m ig­nor­ing all the parts that don’t have to do with sur­re­als, in­clud­ing the use of an im­proper prior (aka not a prob­a­bil­ity dis­tri­bu­tion); I’m just go­ing to ex­am­ine the use of sur­re­als.

Please. Ex­plain. How, on earth, does one put a uniform dis­tri­bu­tion on an in­ter­val of sur­real num­bers?

So, if we look at the in­ter­val from 0 to 1, say, then the prob­a­bil­ity of pick­ing a num­ber be­tween a and b, for a<b, is b-a? For sur­real a and b?

So, first off, that’s not a prob­a­bil­ity. Prob­a­bil­ities are real, for very good rea­son. This is ex­plic­itly a de­ci­sion-the­ory con­text, so don’t tell me that doesn’t ap­ply!

But OK. Let’s ac­cept the premise that you’re us­ing a sur­real-val­ued prob­a­bil­ity mea­sure in­stead of a real one. Ex­cept, wait, how is that go­ing to work? How is countable ad­di­tivity go­ing to work, for in­stance? We’ve already es­tab­lished that in­finite sums do not (in gen­eral) work in the sur­re­als! (See ear­lier dis­cus­sion.) But OK, we can ig­nore that—hell, Sav­age’s the­o­rem doesn’t guaran­tee countable ad­di­tivity, so let’s just ac­cept finite ad­di­tivity. There is the ques­tion of just how you’re go­ing to define this in gen­er­al­ity—it takes quite a bit of work to ex­tend Jor­dan “mea­sure” into Lebesgue mea­sure, you know—but you’re ba­si­cally just us­ing in­ter­vals so I’ll ac­cept we can just treat that part naïvely.

But now you’re tak­ing ex­pected val­ues! Of a sur­real-val­ued prob­a­bil­ity dis­tri­bu­tion over the sur­re­als! So ba­si­cally you’re hav­ing to in­te­grate a sur­real-val­ued func­tion over the sur­re­als. As I’ve men­tioned be­fore, there is no known the­ory of this, no known gen­eral way to define this. I sup­pose since you’re just deal­ing with step func­tions we can treat this naïvely, but ugh. Noth­ing you’re do­ing is re­ally defined. This is pure “just go with it, OK?” This one is less bad than the pre­vi­ous one, this one con­tains things one can po­ten­tially just go with, but you don’t seem to re­al­ize that the things you’re do­ing aren’t ac­tu­ally defined, that this is naïve heuris­tic rea­son­ing rather than ac­tual prop­erly-founded math­e­mat­ics.

Sphere of suffer­ing: Skip­ping for now due to philo­soph­i­cal prob­lems in ad­di­tion to math­e­mat­i­cal ones.

Hilbert Ho­tel: So, first off, there’s no para­dox here. This sort of ba­sic car­di­nal ar­ith­metic of countable sets is well-un­der­stood. Yes, it’s coun­ter­in­tu­itive. That’s not a para­dox.

But let’s ex­am­ine your re­s­olu­tion, be­cause, again, it makes no sense. First, you talk about there be­ing n rooms, where n is a sur­real num­ber. Again: You can­not mea­sure sizes of sets with sur­real num­bers! That is mean­ingless!

But let’s be gen­er­ous and sup­pose you’re talk­ing about well-or­dered sets, and you’re mea­sur­ing their size with or­di­nals, since those em­bed in the sur­re­als. As you note, this is chang­ing the prob­lem, but let’s go with it any­way. Guess what—you’ve still de­scribed it wrong! If you have ω rooms, there is no last room. The last room isn’t room ω, that’d be if you had ω+1 rooms. Hav­ing ω rooms is the origi­nal Hilbert Ho­tel with no mod­ifi­ca­tion.

I’m as­sum­ing when you say n/​2 you mean that in the sur­real sense. OK. Let’s go back to the origi­nal prob­lem and say n=ω. Then n/​2 is ω/​2, which is still big­ger than any nat­u­ral num­ber, so there’s still no­body in the “last half” of rooms! What if n=ω+1, in­stead? Then ω/​2+1/​2 is still big­ger than any nat­u­ral num­ber, so your “last half” con­sists only of ω+1 -- it’s not of the same car­di­nal­ity as your “first half”. Is that what you in­tended?

But ul­ti­mately… even ig­nor­ing all these prob­lems… I don’t un­der­stand how any of this is sup­posed to “re­solve” any para­doxes. It re­solves it by mak­ing it im­pos­si­ble to add more peo­ple? Um, OK. I don’t see why we should want that.

But it doesn’t even suc­ceed at that! Be­cause if you have [Dedekind-]in­finitely many, then for adding finitely many, you have that ini­tial ω, so you can just perform your al­ter­a­tions on that and leave the rest alone. You haven’t pre­vented the Hilbert Ho­tel “para­dox” at all! And for dou­bling, well, as­sum­ing well-or­der­ing (be­cause you’re mea­sur­ing sizes with or­di­nals, maybe?? or be­cause we’re as­sum­ing choice) well, you can par­ti­tion things into copies of ω and go from there.

Gal­ileo’s para­dox: Skip­ping this one as I have noth­ing more to add on this sub­ject, re­ally.

Ba­con’s puz­zle: This one, hav­ing noth­ing to do with sur­re­als, is com­pletely cor­rect! It’s not new, but it’s cor­rect, and it’s neat to know about, so that’s good. (Although I have to won­der: Why is it on this one you ac­cept con­ven­tional math­e­mat­ics of the in­finite, in­stead of ob­ject­ing that it’s a “para­dox” and try­ing to shoe­horn in sur­re­als?)

Trumped and the St. Peters­burg ones: Skip­ping for now due to philo­soph­i­cal prob­lems in ad­di­tion to math­e­mat­i­cal ones

Dice-room mur­ders: An in­finites­i­mal chance the die never comes up 10? No, there’s a 0 chance. That’s how prob­a­bil­ity the­ory works. Again, prob­a­bil­ity is real-val­ued for very good rea­sons, and re­als don’t have in­finites­i­mals. If you want to in­tro­duce prob­a­bil­ities val­ued over some other codomain, you’re go­ing to have to spec­ify what and ex­plain how it’s go­ing to work. “In­finites­i­mal” is not very spe­cific.

The rest as you say has noth­ing to do with in­fini­ties and seems cor­rect so I’ll ig­nore it.

Ross-Lit­tle­wood para­dox: Er… you haven’t re­solved this one at all? The con­ven­tional an­swer, FWIW, is that you should take the limit of the sets, not the limit of the car­di­nal­ities, so that none are left, and this demon­strates the dis­con­ti­nu­ity of car­di­nal­ity. But, um, you just haven’t an­swered this one? I mean I guess that’s not wrong as such...

Soc­cer teams: Your re­s­olu­tion bears lit­tle re­sem­blance to the origi­nal prob­lem. You ini­tially pos­tu­lated that the set of abil­ities was Z, then in your re­s­olu­tion you said it was an in­ter­val in the sur­re­als. Z is not an in­ter­val in the sur­re­als. In fact, no set is an in­ter­val in the sur­re­als; be­tween any two given sur­re­als there is a whole proper class of sur­re­als. Per­haps you meant in the om­nific in­te­gers? Sorry, Z isn’t an in­ter­val in there ei­ther. Per­haps you meant in some­thing of your own in­ven­tion? Well, you didn’t de­scribe it. Ul­ti­mately it’s ir­rele­vant—be­cause the fact is that, yes, if you add 1 to each el­e­ment of Z, you get Z. No al­ter­nate way of de­scribing it will change that.

Pos­i­tive soc­cer teams: You, uh, once again didn’t sup­ply a re­s­olu­tion? In any case this whole prob­lem is ill-defined since you didn’t ac­tu­ally spec­ify any way to mea­sure which of two teams is bet­ter. Although, if we just as­sume there is some way, then pre­sum­ably we want it to be a pre­order (since teams can be tied), and then it seems pretty clear that the two teams should be tied (be­cause each should be no greater than the other for the two rea­sons you gave). (Ac­tu­ally it’s not too hard to come up with an ac­tual pre­order here that does what you want, and then you can ver­ify that, yup, the two teams are tied in it.) This hap­pens a lot with in­fini­ties—things that are or­ders in the finite case be­come pre­orders. Just some­thing you have to learn to live with, once again.

Can God pick an in­te­ger at ran­dom?: This is… not how prob­a­bil­ity works. There is no uniform prob­a­bil­ity dis­tri­bu­tion on the nat­u­ral num­bers, by countable ad­di­tivity. Or, in short, no, God can­not pick an in­te­ger at ran­dom. You then go on to talk about non­sen­si­cal 1/​∞ chances. In short, the only para­dox here is due to a non­sen­si­cal setup.

But then you go and give it a non­sen­si­cal re­s­olu­tion, too. So, first off, once again, you can’t count things with sur­re­als. I will once again gen­er­ously as­sume that you in­tended there to be a well-or­dered set of planets and are count­ing with or­di­nals rather than sur­re­als.

It doesn’t mat­ter. Not only do you then fail to re­ject the non­sen­si­cal setup, you do the most non­sen­si­cal thing yet: You ex­plic­itly mix sur­real num­bers with ex­tended real num­bers, and at­tempt to com­pare the two. What. Are you im­plic­itly think­ing of ∞ as ω here? Be­cause you sure didn’t say any­thing like that! Se­ri­ously, these don’t go to­gether.

I am tempted to do the for­mal ma­nipu­la­tions to see if there is any way one might come to your con­clu­sions by such mean­ingless for­mal ma­nipu­la­tion, but I’ll just give you the benefit of the doubt there, be­cause I don’t want to give my­self a headache do­ing mean­ingless for­mal ma­nipu­la­tions in­volv­ing two differ­ent num­ber sys­tems that can’t be mean­ingfully com­bined.

Banach-Tarski para­dox: This starts out as a de­cent ex­pla­na­tion of Banach-Tarski; it’s miss­ing some im­por­tant de­tails, but what­ever. But then you start talk­ing about se­quences of in­finite length. (Some­thing that wasn’t there be­fore—you act as if this was already there, but it wasn’t.) Which once again you mean­inglessly as­sign a sur­real length. I’ll once again as­sume you meant an or­di­nal length in­stead. Ex­cept that doesn’t help much be­cause this whole thing is mean­ingless—you can’t take in­finite prod­ucts in groups.

Or maybe you can, in this case, since we’re re­ally work­ing in F_2 em­bed­ded in SO(3), rather than just in F_2? So you could take the limit in SO(3), if it ex­ists. (SO(3) is com­pact, so there will cer­tainly be at least one limit point, but I don’t see any ob­vi­ous rea­son it’d be unique.)

Ex­cept the way you talk about it, you talk as if these in­finite se­quences are still in our free group. Which, no. That is not how free groups work. They con­tain finite words only.

Maybe you’re in­tend­ing this to be in some sort of “free topolog­i­cal group”, which does con­tain in­finite and trans­finite words? Yeah, there’s no such thing in any non­triv­ial man­ner. Be­cause if you have any el­e­ment g, then you can ob­serve that g(ggg...) = ggg..., and there­fore (be­cause this is a group) ggg...=1. Well, OK, that’s not a full ar­gu­ment, I’ll ad­mit. But, that’s just a quick ex­am­ple of how this doesn’t work, I hope you don’t mind. Point is: You haven’t defined this new set­ting you’re work­ing in, and if you try, you’ll find it makes no sense. But it sure as hell ain’t the free group F_2.

I also have no idea what you’re say­ing this does to the Banach-Tarski para­dox. Hon­estly, it doesn’t mat­ter, be­cause the logic be­hind Banach-Tarski re­mains the same re­gard­less.

The magic dart­board: No, a bi­jec­tion be­tween the countable or­di­nals and [0,1] is not known to ex­ist. That’s only true if you as­sume the con­tinuum hy­poth­e­sis. Are you as­sum­ing the con­tinuum hy­poth­e­sis? You didn’t men­tion any such thing.

You then give a com­pletely wrong and non­sen­si­cal ar­gu­ment as to why this con­struc­tion has the de­sired “magic dart­board” prop­erty, in which you talk about cer­tain or­di­nals be­ing in the “first 1/​n” of the countable or­di­nals, or the “last half” of the countable or­di­nals. This is com­pletely mean­ingless. There is no first 1/​n, or last half, of the countable or­di­nals. If you had some mean­ing in mind, you’re go­ing to have to ex­plain it. And if you mean go­ing into the sur­re­als and com­par­ing them against ω_1/​n, then, un­sur­pris­ingly, the en­tire countable or­di­nals will always fall in your first 1/​n. The con­struc­tion does yield a magic dart­board, but you’re com­pletely wrong as to why.

Thom­son’s lamp: Your re­s­olu­tion here is non­sense. Now, our presses our oc­cur­ring in a well-or­dered se­quence, so it’s most ap­pro­pri­ate to re­gard the num­ber of presses as an or­di­nal. In which case, the num­ber of presses is ω. It’s not a ques­tion—that’s what it is. It doesn’t de­pend on how we define the re­als, WTF? The re­als are the re­als (un­less you’re go­ing to start do­ing con­struc­tive math­e­mat­ics, in which case the things you wrote will pre­sum­ably be wrong in many more ways). It might de­pend on how you define the prob­lem, but you were pretty ex­plicit about what the press timings are. Any­way, ω is even as an om­nific in­te­ger, but does that mean we should con­sider the lamp to be on? I see no rea­son to con­clude this. The lamp’s state has no well-defined limit, af­ter all. This is once again naïvely ex­tend­ing some­thing from the finite to the in­finite with­out check­ing whether it ac­tu­ally ex­tends (it doesn’t).

Really, the ba­sic mis­take here is as­sum­ing there must be an an­swer. As I said, the lamp’s state has no limit, so there re­ally just isn’t any well-defined an­swer to this prob­lem.

Grandi’s se­ries: You once again as­sign a vari­able sur­real length (which still makes no sense) to some­thing which has a very definite length, namely ω. In any case, Grandi’s se­ries has no limit. You say it de­pends on whether the length is even or odd. Sup­pose we in­ter­pret that as “even or odd as an om­nific in­te­ger” (i.e. hav­ing even or odd finite part). OK. So you’re say­ing that Grandi’s se­ries sums to 0, then, since ω is even as an om­nific in­te­ger? It doesn’t mat­ter; the se­ries has no limit, and if you tried to ex­tend it trans­finitely, you’d get stuck at ω when there’s already no limit there.

I mean, I sup­pose you could define a new no­tion of what it means to sum a di­ver­gent (pos­si­bly trans­finite se­ries), and ap­ply it to Grandi’s se­ries (pos­si­bly ex­tended trans­finitely) as an ex­am­ple, but you haven’t done that. You’ve just said what the limit “is”. It isn’t. More naïve ex­ten­sion and for­mal ma­nipu­la­tion in place of ac­tual math­e­mat­i­cal rea­son­ing.

Satan’s ap­ple: Skip­ping, you didn’t men­tion sur­re­als and the para­dox is en­tirely philo­soph­i­cal rather than math­e­mat­i­cal (you also ad­mit­ted con­fu­sion on this one rather than giv­ing a fake re­s­olu­tion, so good for you)

Gabriel’s horn: Yup, you de­scribed this one cor­rectly at least!

Ber­trand para­dox: You al­most had this, but still snuck in an in­cor­rect state­ment re­veal­ing a se­ri­ous con­cep­tual er­ror. There aren’t mul­ti­ple sets of chords; there are mul­ti­ple prob­a­bil­ity dis­tri­bu­tions on the set of chords. Really, it’s not that all the prob­a­bil­ities are valid, it’s just that it de­pends on how you pick, but I was giv­ing you the benefit of the doubt on that one un­til you added that bit about mul­ti­ple sets of chords.

Zeno’s para­doxes: We can ar­gue all we like about the “real” re­s­olu­tion here philo­soph­i­cally but what­ever, you seem to grasp the math­e­mat­ics of it at least, so let’s move on

Skolem’s para­dox: You’ve mostly summed this one up cor­rectly. I must nit­pick and point out that mem­ber­ship in the model is not nec­es­sar­ily the same as mem­ber­ship out­side the model even for those sets that are in the model—some­thing which you might re­al­ize but your ex­pla­na­tion doesn’t make clear—but this is a small er­ror com­pared to the gi­ant con­cep­tual er­rors that fill most of what you’ve writ­ten here.

Whew. OK. I will maybe get back to the ones I skipped, but prob­a­bly not be­cause this is enough to demon­strate my point. This post is hor­ribly wrong nearly in its en­tirely, shot through with se­ri­ous con­cep­tual er­rors. You re­ally need to re­learn this stuff from scratch, be­cause al­most noth­ing you’re say­ing makes sense. I urge ev­ery­one else to ig­nore this post and not take any­thing it says as re­li­able.

• My pri­mary re­sponse to this com­ment will take the form of a post, but I should add that I wrote: “I will provide in­for­mal hints on how sur­real num­bers could help us solve some of these para­doxes, al­though the fo­cus on this post is pri­mar­ily cat­e­gori­sa­tion, so please don’t mis­take these for for­mal proofs”.

Your com­ment seems to com­pletely ig­nore this stipu­la­tion. Take for ex­am­ple this:

“Of course, your solu­tion seems to in­volve im­plic­itly chang­ing the set­ting to have sur­real-val­ued time and space… You might want to make more of an ex­plicit note of it, though”

Yes, there’s a lot of philo­soph­i­cal ground­work that would need to be done to jus­tify the sur­real ap­proach. That’s why I said that it was only an in­for­mal hint.

I’m go­ing to as­sume, since you’re talk­ing about sur­re­als and didn’t spec­ify oth­er­wise, that you mean exp(s log 2), us­ing the usual sur­real exponential

Yes, I ac­tu­ally did look up that there was a way of defin­ing 2^s where s is a sur­real num­ber.

Let’s ac­cept the premise that you’re us­ing a sur­real-val­ued prob­a­bil­ity mea­sure in­stead of a real one

I wrote a sum­mary of a pa­per by Chen and Ru­bio that pro­vides the start of a sur­real de­ci­sion the­ory. This isn’t a com­plete prob­a­bil­ity the­ory as it only sup­ports finite ad­di­tivity in­stead of countable ad­di­tivity, but it sug­gests that this ap­proach might be vi­able.

I could keep go­ing, but I think I’ve made my point that you’re eval­u­at­ing these in­for­mal com­ments as though I’d claimed they were a for­mal proof. This post was already long enough and took enough time to write as is.

I will ad­mit that I could have been clearer that many of these re­marks were spec­u­la­tive, in the sense of be­ing ar­gu­ments that I be­lieved were worth work­ing to­wards for­mal­is­ing, even if all of the math­e­mat­i­cal ma­chin­ery doesn’t nec­es­sar­ily ex­ist at this time. My point is that jus­tify­ing the use of sur­re­als num­bers doesn’t nec­es­sar­ily in­volve solv­ing ev­ery para­dox; it should also be per­sua­sive to solve a good num­ber of them and then to demon­strate that there is good rea­son to be­lieve that we may be able to solve the rest in the fu­ture. In this sense, in­for­mal ar­gu­ments aren’t val­ue­less.

• My pri­mary re­sponse to this com­ment will take the form of a post, but I should add that I wrote: “I will provide in­for­mal hints on how sur­real num­bers could help us solve some of these para­doxes, al­though the fo­cus on this post is pri­mar­ily cat­e­gori­sa­tion, so please don’t mis­take these for for­mal proofs”.

You’re right; I did miss that, thanks. It was per­haps un­fair of me then to pick on such gaps in for­mal­ism. Un­for­tu­nately, this is only enough to res­cue a small por­tion in the post. Ig­nor­ing the ones I skipped—maybe it would be worth my time to get back to those af­ter all—I think the only one po­ten­tially res­cued that way is the en­velope prob­lem. (I’m still skep­ti­cal that it is—I haven’t looked at it in enough de­tail to say—but I’ll grant you that it could be.)

(Edit: After recheck­ing, I guess I’d count Grandi’s se­ries and Thom­son’s lamp here too, but only barely, in the sense that—af­ter giv­ing you quite a bit of benefit of the doubt—yeah I guess you could define things that way but I see ab­solutely no rea­son why one would want to and I se­ri­ously doubt you gain any­thing from do­ing so. (I was about to in­clude god pick­ing a ran­dom in­te­ger here, too, but on recheck­ing again, no, that one still has se­ri­ous other prob­lems even if I give you more lee­way than I ini­tially did. Like, if you try to iden­tify ∞ with a spe­cific sur­real, say ω, there’s no sur­real you can iden­tify it with that will make your con­clu­sion cor­rect.))

The rest of the ones I pointed out as wrong (in­volv­ing sur­re­als, any­way) all con­tain more sub­stan­tial er­rors. In some cases this be­comes ev­i­dent af­ter do­ing the work and at­tempt­ing to for­mal­ize your hints; in other cases they’re ev­i­dent im­me­di­ately, and clearly do not work even in­for­mally.

The magic dart­board is a good ex­am­ple of the lat­ter—you’ve sim­ply given an in­cor­rect proof of why the magic dart­board con­struc­tion works. In it you talk about ω_1 hav­ing a first half and a sec­ond half. You don’t need to do any deep think­ing about sur­re­als to see the prob­lem here—that’s just not what ω_1 looks like, at all. If you do fol­low the hint, and com­pare the el­e­ments of ω_1 to (ω_1)/​2 in the sur­re­als, then, as already noted, you find ev­ery­thing falls in the first half, which is not very helpful. (Again: This is the sort of thing that causes me to say, I sus­pect you need to re­learn or­di­nals and prob­a­bly other things, not just sur­re­als. If you ac­tu­ally un­der­stand or­di­nals, you should not have any trou­ble prov­ing that the magic dart­board acts as claimed, with­out any need to go into the sur­re­als and perform di­vi­sion.)

Mean­while the para­dox of the gods is, as I’ve already laid out in de­tail, an ex­am­ple of the former. It sounds like a nice in­for­mal an­swer that could pos­si­bly be for­mal­ized, sure; but if you try to ac­tu­ally fol­low the hint and do that—switch­ing to sur­real time and space as needed, of course—it still makes no sense for the rea­sons I’ve de­scribed above. Be­cause, e.g., ω is a limit or­di­nal and not a suc­ces­sor or­di­nal (this is a re­peated mis­take through­out the post, ig­nor­ing the ex­is­tence of limit or­di­nals), be­cause in the sur­re­als there are no in­fima of sets (that aren’t min­ima), be­cause the fact that a sur­real ex­po­nen­tial ex­ists doesn’t mean that it acts like you want it to (alge­braically it does ev­ery­thing you might want, but this prob­lem isn’t about alge­braic prop­er­ties) or that there’s any­thing spe­cial about the points it picks out.

In ad­di­tion, some of the things one is ex­pected to just go with would re­quire not just more ex­pla­na­tion to for­mal­ize (like sur­real in­te­gra­tion) but to even make even in­for­mal sense of (like what struc­ture you are putting on a set, or what you are em­bed­ding it in, that would make a sur­real an ap­pro­pri­ate mea­sure of its size).

In short, your hints are not hints to­wards an already-ex­ist­ing solu­tion (or at least, not one that any­one other than you would ac­cept); they’re anal­ogy-driven spec­u­la­tion as to what a solu­tion could look like. Ob­vi­ously there’s noth­ing wrong with anal­ogy-driven spec­u­la­tion! I could definitely go on about some anal­ogy-driven spec­u­la­tion of mine in­volv­ing sur­re­als! But, firstly, that’s not what you pre­sented it as; sec­ondly, in most of your cases it’s ac­tu­ally fairly easy (with a bit of rele­vant knowl­edge) to fol­low the bread­crumb trail and see that in fact it goes nowhere, as I did in my re­ply; and, thirdly, you’re pur­port­ing to “solve” things that aren’t ac­tu­ally prob­lems in the first place. The sec­ond be­ing the most im­por­tant here, to be clear.

(And I think the ones I skipped demon­strate even more math­e­mat­i­cal prob­lems that I didn’t get to, but, well, I haven’t got­ten to those.)

FWIW, I’d say sur­real de­ci­sion the­ory is a bad idea, be­cause, well, Sav­age’s the­o­rem—that’s a lot of my philo­soph­i­cal ob­jec­tions right there. But I should get to the ac­tual math­e­mat­i­cal prob­lems some­time; the philo­soph­i­cal ob­jec­tions, while im­por­tant, are, I ex­pect, not as in­ter­est­ing to you.

Ba­si­cally, the post treats the sur­re­als as a sort of de­vice for au­to­mat­i­cally mak­ing the in­finite be­have like the finite. They’re not. Yes, their struc­ture as an or­dered field (or­dered ex­po­nen­tial field, even) means that their alge­braic be­hav­ior re­sem­bles such fa­mil­iar finite set­tings as the real num­bers, in con­trast to the quite differ­ent ar­ith­metic of (say) the or­di­nal or car­di­nal num­bers (one might even in­clude here the ex­tended real line, with its mostly-all-ab­sorb­ing ∞). But the things you’re try­ing to do here of­ten in­volve more than ar­ith­metic or alge­bra, and then the analo­gies quickly fall apart. (Again, I’d see our pre­vi­ous ex­change here for ex­am­ples.)

• OK, time for the sec­ond half, where I get to the er­rors in the ones I ini­tially skipped. And yes, I’m go­ing to as­sert some philo­soph­i­cal po­si­tions which (for what­ever rea­son) aren’t well-ac­cepted on this site, but there’s still plenty of math­e­mat­i­cal er­rors to go around even once you ig­nore any philosph­i­cal prob­lems. And yeah, I’m still go­ing to point out miss­ing for­mal­ism, but I will try to fo­cus on the more sub­stan­tive er­rors, of which there are plenty.

So, let’s get those philo­soph­i­cal prob­lems out of the way first, and quickly re­view util­ity func­tions and util­i­tar­i­anism, be­cause this ap­plies to a bunch of what you dis­cuss here. Like, this whole post takes a very naive view of the idea of “util­ity”, and this needs some break­ing down. Apolo­gies if you already know all of what I’m about to say, but I think given the con­text it bears re­peat­ing.

So: There are two differ­ent things meant by “util­ity func­tion”. The first is de­ci­sion-the­o­retic; an agent’s util­ity func­tion is a func­tion whose ex­pected value it at­tempts to max­i­mize. The sec­ond is the one used by util­i­tar­i­anism, which in­volves (at pre­sent, poorly-defined) “E-util­ity” func­tions, which are not util­ity func­tions in the de­ci­sion-the­o­retic sense, that are then some­how ag­gre­gated (maybe by ad­di­tion? who knows?) into a de­ci­sion-the­o­retic util­ity func­tion. Yes, this ter­minol­ogy is ter­ribly con­fus­ing. But these are two sep­a­rate things and need to be kept sep­a­rate.

Ba­si­cally, any agent that satis­fies ap­pro­pri­ate ra­tio­nal­ity con­di­tions has a util­ity func­tion in the de­ci­sion-the­o­retic sense (ob­vi­ously such ideal­ized agents don’t ac­tu­ally ex­ist, but it’s still a use­ful ab­strac­tion). So you could say, roughly speak­ing, any ra­tio­nal con­se­quen­tial­ist has a de­ci­sion-the­o­retic util­ity func­tion. Whereas E-util­ity is speci­fi­cally a util­i­tar­ian no­tion, rather than a gen­eral con­se­quen­tal­ist or purely de­scrip­tive no­tion like de­ci­sion-the­o­retic util­ity (it’s also not at all clear how to define it).

Any­way, if you want sur­real E-util­ity func­tions… well, I think that’s still prob­a­bly pretty dumb for rea­sons I’ll get to, but since E-util­ity is so poorly defined that’s not ob­vi­ously wrong. But let’s talk about de­ci­sion-the­o­retic util­ity func­tions. Th­ese need to be real-val­ued for very good rea­sons.

Be­cause, well, why use util­ity func­tions at all? What makes us think that a ra­tio­nal agent’s prefer­ences can be de­scribed in terms of a util­ity func­tion in the first place? Well, there’s an an­swer to that: Sav­age’s the­o­rem. I’ve already de­scribed this above—it gives ra­tio­nal­ity con­di­tions, phrased di­rectly in terms of an agent’s prefer­ences, that to­gether suffice to guaran­tee that said prefer­ences can be de­scribed by a util­ity func­tion. And yes, it’s real-val­ued.

(And, OK, it’s real-val­ued be­cause Sav­age in­cludes an Archimedean as­sump­tion, but, well—do you think that’s a bad as­sump­tion? Let me re­peat here a naive ar­gu­ment against in­finite and in­finites­i­mal util­ities I’ve seen be­fore on this site (I for­get due to who; I think Eliezer maybe?). Sup­pose we go with a naive treat­ment of in­finites­i­mal util­ities, and A has in­finites­i­mal util­ity com­pared to B. Then since any ac­tion you take at all has some pos­i­tive (real, non-in­finites­i­mal) prob­a­bil­ity of bring­ing about B, even sit­ting in your room wav­ing your hand back and forth in the air, A sim­ply has no effect on your de­ci­sion mak­ing; all con­sid­er­a­tions of B, even stupid ones, com­pletely wash it out. Which means that A’s in­finites­i­mal util­ity does not, in fact, have any place in a de­ci­sion-the­o­retic util­ity func­tion. Do you re­ally want to throw out that Archimedean as­sump­tion? Also if you do throw it out, I don’t think that ac­tu­ally gets you non-real-val­ued util­ities, I think it just, y’know, doesn’t get you util­ities. The agent’s prefer­ences can’t nec­es­sar­ily be de­scribed with a util­ity func­tion of any sort. Ad­mit­tedly I could be wrong about that last part; I haven’t checked.)

In short, your philo­soph­i­cal mis­take here is of a kind with your math­e­mat­i­cal mis­takes—in both cases, you’re start­ing from a sys­tem of num­bers (sur­re­als) and try­ing awk­wardly to fit it to the prob­lem, even when it blatantly does not fit, does not have the prop­er­ties that are re­quired; rather than see­ing what re­quire­ments the prob­lem ac­tu­ally calls for and find­ing some­thing that meets those needs. As I’ve pointed out mul­ti­ple times by now, you’re try­ing to make use of prop­er­ties that the sur­real num­bers just don’t have. Work for­ward from the re­quire­ments, don’t try to force into them things that don’t meet them!

By the way, Sav­age’s the­o­rem also shows that util­ity func­tions must be bounded. That util­ity func­tions must be bounded does not, for what­ever rea­son, seem to be a well-ac­cepted po­si­tion on this site, but, well, it’s cor­rect so I’m go­ing to con­tinue as­sert­ing it, in­clud­ing here. :P Now it’s true that the VNM the­o­rem doesn’t prove this, but that’s due to a defi­ciency in the VNM the­o­rem’s as­sump­tions, and with that gap fixed it does. I don’t want to be­la­bor this point here, so I’ll just re­fer you to this pre­vi­ous dis­cus­sion.

(Also the VNM the­o­rem is just a worse foun­da­tion gen­er­ally be­cause it as­sumes real-val­ued prob­a­bil­ities to be­gin with, but that’s a sep­a­rate mat­ter. Though maybe here it’s not—since you can’t claim to avoid the bound­ed­ness re­quire­ment by say­ing you’re jus­tify­ing the use of util­ities with VNM rather than Sav­age, since you seem to want to al­low sur­real-val­ued prob­a­bil­ities!)

Any­way, so, yes, util­ities should be real-val­ued (and bounded) or else you have no good rea­son to use them—to use sur­real-val­ued util­ities is to start from the as­sump­tion that you should use util­ities (a big as­sump­tion! why would one ever as­sume such a thing?) when it should be a con­clu­sion (a con­clu­sion of the­o­rems that say it must be real-val­ued).

Ah, but could in­fini­ties or in­finites­i­mals ap­pear in an E-util­ity func­tion, that the util­i­tar­i­ans use? I’ve been ig­nor­ing those, af­ter all. But, since they’re get­ting ag­gre­gated into a de­ci­sion-the­o­retic util­ity func­tion, which is real-val­ued (or maybe it’s not quite a de­ci­sion-the­o­retic util­ity func­tion, but it should still be real-val­ued by the naive ar­gu­ment above), un­less this ag­gre­ga­tion func­tion can mag­nify an in­finites­i­mal into a non-in­finites­i­mal, the same prob­lem will arise, the in­finites­i­mals will still have no rele­vance, and thus should never have been in­cluded.

(Yeah, I sup­pose in what you write you con­sider “sum­ming over an in­finite num­ber of peo­ple”. But: 1. such in­finite sums with in­finites­i­mals don’t ac­tu­ally work math­e­mat­i­cally, for rea­sons I’ve already cov­ered, and 2. you can’t ac­tu­ally have an in­finite num­ber of peo­ple, so it’s all moot any­way.)

Yikes, all that and I haven’t even got­ten to ex­am­in­ing in de­tail the par­tic­u­lar math­e­mat­i­cal prob­lems in the re­main­ing ones! You know what, I’ll end this here and split that com­ment out into a third post. Point is, now in these re­main­ing ones, when I want to point out philo­soph­i­cal prob­lems, I can just point back to this com­ment rather than re­peat­ing all this again.

• OK, time to ac­tu­ally now get into what’s wrong with the ones I skipped ini­tially. Already wrote the in­tro above so not re­peat­ing that. Time to just go.

In­fini­tar­ian paral­y­sis: So, philo­soph­i­cal prob­lems to start: As an ac­tual de­ci­sion the­ory prob­lem this is all moot since you can’t ac­tu­ally have an in­finite num­ber of peo­ple. I.e. it’s not clear why this is a prob­lem at all. Se­condly, naive as­sump­tion of util­i­tar­ian ag­gre­ga­tion as men­tioned above, etc, not go­ing over this again. Enough of this, let’s move on.

So what are the math­e­mat­i­cal prob­lems here? Well, you haven’t said a lot here, but here’s what it’s look like to me. I think you’ve writ­ten one thing here that is es­sen­tially cor­rect, which is that, if you did have some sys­tem of sur­real val­ued-util­ities, it would in­deed likely make the dis­tinc­tion you want.

But, once again, that’s a big “if”, and not just for philo­soph­i­cal rea­sons but for the math­e­mat­i­cal rea­sons I’ve already brought up so many times right now—you can’t do in­finite sums in the sur­re­als like you want, for rea­sons I’ve already cov­ered. So there’s a rea­son I in­cluded the word “likely” above, be­cause if you did find an ap­pro­pri­ate way of do­ing such a sum, I can’t even nec­es­sar­ily guaran­tee that it would be­have like you want (yes, finite sums should, but in­finite sums re­quire defi­ni­tion, and who knows if they’ll ac­tu­ally be com­pat­i­ble with finite sums like they should be?).

But the re­ally jar­ring thing here, the thing that re­ally ex­poses a se­ri­ous er­ror in your thought (well, OK, that does so to a greater ex­tent), is not in your pro­posed solu­tion—it’s in what you con­trast it with. Car­di­nal val­ued-util­ities? Noth­ing about that makes sense! That’s not a re­motely well-defined al­ter­na­tive you can con­trast with! And the thing that bugs me about this er­ror is that it’s just so un­forced—I mean, man, you could have said “ex­tended re­als” rather than car­di­nals, and made es­sen­tially the same point while mak­ing at least some sense! This is just demon­strat­ing once again that not only do you not un­der­stand sur­re­als, you do not un­der­stand car­di­nals or or­di­nals ei­ther.

(Well, I sup­pose tech­ni­cally there’s the pos­si­bil­ity that you do but ex­pect your au­di­ence doesn’t and are talk­ing down to them, but since you’re writ­ing here on Less Wrong, I’m go­ing to as­sume that’s not the case.)

Se­ri­ously, car­di­nals and util­ities do not go to­gether. I mean, car­di­nals and real num­bers do not go to­gether. Like sur­re­als and util­ities don’t go to­gether ei­ther, but at least the sur­re­als in­clude the re­als! At least you can at­tempt to treat it naively in spe­cial cases, as you’ve done in a num­ber of these ex­am­ples, even if the re­sult prob­a­bly isn’t mean­ingful! Car­di­nals you can’t even do that.

And once again, there’s no rea­son any­one who un­der­stood car­di­nals would even want car­di­nal-val­ued util­ities. That’s just not what car­di­nals are for! Car­di­nals are for count­ing how many there are of some­thing. Utility calcu­la­tions are not a “how many” prob­lem.

Sphere of suffer­ing: Once again we have in­finitely many peo­ple (so this whole prob­lem is again a non-prob­lem) and once again we have some sort of naive util­ity ag­gre­ga­tion over those in­finitely many peo­ple with all the math­e­mat­i­cal prob­lems that brings (only now it’s over time-slices as well?). Enough of this, mov­ing on.

Hon­estly I don’t have much new to say about the bad math­e­mat­ics here, much of it is the same sort of mis­takes as you made in the ones I cov­ered in my ini­tial com­ment. To cover those ones briefly:

1. Sur­real num­bers do not mea­sure how far a grid ex­tends (similar to ex­am­ples I’ve already cov­ered)

2. There’s not a ques­tion of how far the grid ex­tends, al­low­ing it to be a trans­finite vari­able l is just chang­ing the prob­lem (similar to ex­am­ples I’ve already cov­ered)

3. Sur­real num­bers also do not mea­sure num­ber of time steps, you want or­di­nals for that (similar to ex­am­ples I’ve already cov­ered)

4. Re­peat #2 but for the time steps (similar to ex­am­ples I’ve already cov­ered)

But OK. The one new thing here, I guess, is that now you’re talk­ing about a “ma­jor­ity” of the time slices? Yeah, that is once again not well-defined at all. Car­di­nal­ity won’t help you here, ob­vi­ously; are you putting a mea­sure on this some­how? I think you’re go­ing to have some prob­lems there.

Trumped: Same prob­lems I’ve dis­cussed be­fore. Sur­real num­bers do not count time steps, you’re chang­ing the prob­lem by in­tro­duc­ing a vari­able, util­ity ag­gre­ga­tion over an in­finite set (this time of time-slices rather than peo­ple), you know the drill.

But ac­tu­ally here you’re chang­ing the prob­lem in a differ­ent way, by sup­pos­ing that Trump knows in ad­vance the num­ber of time steps? The origi­nal prob­lem just had this as a re­peated offer. Maybe that’s a philo­soph­i­cal rather than math­e­mat­i­cal prob­lem. What­ever. It’s chang­ing the prob­lem, is the point.

And then on top of that your solu­tion doesn’t even make any sense. Let’s sup­pose you meant an or­di­nal num­ber of days rather than a sur­real num­ber of days, since that is what you’d ac­tu­ally use in this con­text. OK. Sup­pose for ex­am­ple then that the num­ber of days is ω (which is, af­ter all, the origi­nal prob­lem be­fore you changed it). So your solu­tion says that Trump should ac­cept the deal so long as the day num­ber is less than the sur­real num­ber ω/​3. Ex­cept, oops! Every or­di­nal less than ω is also less than ω/​3. Trump always ac­cepts the deal, we’re back at the origi­nal prob­lem.

I.e., even grant­ing that you can some­how make all the for­mal­ism work, this is still just wrong.

St. Peters­burg para­dox: OK, so, there’s a lot wrong here. Let me get the philo­soph­i­cal prob­lem out of the way first—the real solu­tion to the St. Peters­burg para­dox is that you must look not at ex­pected money, but at ex­pected util­ity, and util­ity func­tions must be bounded, so this prob­lem can’t arise. But let’s get to the math, be­cause, like I said, there’s a lot wrong here.

Let’s get the easy-to-de­scribe prob­lems out of the way first: You are once again us­ing sur­re­als where you should be us­ing or­di­nals; you are once again as­sum­ing some sort of the­ory of in­finite sums of sur­re­als; get­ting in­finitely many heads has zero prob­a­bil­ity, not in­finites­i­mal (prob­a­bil­ities are real-val­ued, you could try to in­tro­duce a the­ory of sur­real prob­a­bil­ities but that will have prob­lems already dis­cussed), what hap­pens in that case is ir­rele­vant; you are once again chang­ing the prob­lem by al­low­ing things to go on be­yond ω steps; and, minor point, but where on earth did the func­tion n |-> n comes from? Don’t you mean n |-> 2^n?

OK, that’s largely stuff I’ve said be­fore. But the thing that puz­zled me the most in your claimed solu­tion is the first sen­tence:

If we model this with sur­re­als, then sim­ply stat­ing that there is po­ten­tially an in­finite num­ber of tosses is un­defined.

What? I mean, yeah, sure, the sur­re­als have mul­ti­ple in­fini­ties while, say, the ex­tended non­nega­tive re­als have only one, no ques­tion there. But that sen­tence still makes no sense! It, like, seems to re­veal a fun­da­men­tal mi­s­un­der­stand­ing so great I’m hav­ing trou­ble com­pre­hend­ing it. But I will give it my best shot.

So the thing is, that—ig­nor­ing the is­sue of un­bounded util­ity and what’s the cor­rect de­ci­sion—the origi­nal setup has no am­bi­gui­ties. You can’t choose to make it differ­ent by chang­ing what sys­tem of num­bers you de­scribe it with. Now, I don’t know if you’re mak­ing the mis­take I think you’re mak­ing, be­cause who knows what mis­take you might be mak­ing, but it looks to me that you are con­fus­ing num­bers that are part of the ac­tual prob­lem speci­fi­ca­tion, with aux­iliary num­bers just used to de­scribe the prob­lem.

Like, what’s ac­tu­ally go­ing on here is that there is a set of coin flips, right? The el­e­ments of that set will be in­dexed by the nat­u­ral num­bers, and will form a (pos­si­bly im­proper, though with prob­a­bil­ity 0) ini­tial seg­ment of it—those num­bers are part of the ac­tual prob­lem speci­fi­ca­tion. The idea though that there might be in­finitely many coin flips… that’s just a de­scrip­tion. When I say “With prob­a­bil­ity 0, the set of flips will be in­finite”, that’s just an­other way of say­ing, “With prob­a­bil­ity 0, the set of flips will be N.” It doesn’t make sense to ask “Ah, but what sys­tem of num­bers are you us­ing to mea­sure its in­fini­tude?” It doesn’t mat­ter! The set I’m de­scribing is N! (And in any case I just said it was an in­finite set, al­though I sup­pose you could say I was im­plic­itly us­ing car­di­nals.)

This is, I sup­pose, an idea that’s shown up over and over in your claimed solu­tions, but since I skipped over this par­tic­u­lar one be­fore, I guess I never got it so ex­plic­itly be­fore. Again, I’m hav­ing to guess what you think, but it looks to me like you think that the num­bers are what’s pri­mary, rather than the ac­tual ob­jects the prob­lems are about, and so you can just change the num­bers sys­tem and get a differ­ent ver­sion of the same prob­lem. I mean, OK, of­ten the num­bers are pri­mary and you can do that! But some­times they’re just de­scrip­tive.

Oy. I have no idea whether I’ve cor­rectly de­scribed what your mi­s­un­der­stand­ing, but what­ever it is, it’s pretty big. Let’s just move on.

Trou­ble in St. Peters­burg: Can I first just com­plain that your num­bers don’t seem to match up with your text? 13 is not 9*2+3. I’m just go­ing to as­sume you meant 21 rather than 13, be­cause none of the other in­ter­pre­ta­tions I can come up with make sense.

Also this prob­lem once again re­lies on un­bounded util­ities, but I don’t need to go on about that. (Although if you were to some­how re­for­mu­late it with­out those—though that doesn’t seem pos­si­ble in this coin-flip for­mu­la­tion—then the prob­lem would be ba­si­cally similar to Satan’s Ap­ple. I have my own thoughts on that prob­lem, but, well, I’m not go­ing to go into it here be­cause that’s not the point.)

Any­way, let’s get to the sur­real abuse! Well, OK, again I don’t have much new to say here, it’s the same sort of sur­real abuse as you’ve made be­fore. Namely: Us­ing sur­re­als where they don’t make sense (time steps should be counted by or­di­nals); chang­ing the prob­lem by in­tro­duc­ing a trans­finite vari­able; think­ing that all or­di­nals are suc­ces­sor or­di­nals (sorry, but with n=ω, i.e. the origi­nal prob­lem, there’s still no last step).

Ul­ti­mately you don’t offer any solu­tion? What­ever. The er­rors above still stand.

The headache: More naive ag­gre­ga­tion and think­ing you can do in­finite sums and etc. Or at least so I’m gath­er­ing from your claimed solu­tion. Any­way that’s bor­ing.

The sur­real abuse here though is also bor­ing, same types as we’ve seen be­fore—us­ing sur­re­als where they make no sense but where or­di­nals would; ig­nor­ing the ex­is­tence of limit or­di­nals; and of course the afore­men­tioned in­finite sums and such.

OK. That’s all of them. I’m stop­ping there. I think the first com­ment was re­ally enough to demon­strate my point, but now I can hon­estly claim to have ad­dressed ev­ery one of your ex­am­ples. Time to go sleep now.

• This is quite a long post, so it may take some time to write a proper re­ply, but I’ll get back to you when I can. The fo­cus of this post was on gath­er­ing to­gether all the in­finite para­doxes that I could man­age. I also added some in­for­mal thoughts on how sur­real num­bers could help us con­cep­tu­al­ise the solu­tion to these prob­lems, al­though this wasn’t the main fo­cus (it was just con­ve­nient to put them in the same space).

Un­for­tu­nately, I haven’t con­tinued the se­quence since I’ve been caught up with other things (travel, AI, ap­ply­ing for jobs), but hope­fully I’ll write up some new posts soon. I’ve ac­tu­ally be­come much less op­ti­mistic about sur­real num­bers for philo­soph­i­cal rea­sons which I’ll write up soon. So my in­tent is for my next post to ex­am­ine the defi­ni­tion of in­finity and why this makes me less op­ti­mistic about this ap­proach. After that, I want to write up a bit more for­mally how the sur­real ap­proach would work, be­cause even though I’m less op­ti­mistic about this ap­proach, per­haps some­one else will dis­agree with my pes­simism. Fur­ther, I think it’s use­ful to un­der­stand how the sur­real ap­proach would try to re­solve these prob­lems, even if only to provide a solid tar­get for crit­i­cism.

• Sur­real num­bers are use­less for all of these para­doxes.

In­fini­tar­ian paral­y­sis: Us­ing sur­real-val­ued util­ities cre­ates more in­fini­tar­ian paral­y­sis than it solves, I think. You’ll never take an op­por­tu­nity to in­crease util­ity by be­cause it will always have higher ex­pected util­ity to fo­cus all of your at­ten­tion on try­ing to find ways to in­crease util­ity by , since there’s some (how­ever small) prob­a­bil­ity that such efforts would suc­ceed, so the ex­pected util­ity of fo­cus­ing your efforts on look­ing for ways to in­crease util­ity by will have ex­pected util­ity , which is higher than . I think a bet­ter solu­tion would be to note that for any per­son, a nonzero frac­tion of peo­ple are close enough to iden­ti­cal to that per­son that they will make the same de­ci­sions, so any de­ci­sion that any­one makes af­fects a nonzero frac­tion of peo­ple. Mea­sure the­ory is prob­a­bly a bet­ter frame­work than sur­real num­bers for for­mal­iz­ing what is meant by “frac­tion” here.

Para­dox of the gods: The in­tro­duc­tion of sur­real num­bers solves noth­ing. Why wouldn’t he be able to ad­vance more than miles if no gods erect any bar­ri­ers un­til he ad­vances miles for some finite ?

Two-en­velopes para­dox: it doesn’t make sense to model your un­cer­tainty over how much money is in the first en­velope with a uniform sur­real-val­ued prob­a­bil­ity dis­tri­bu­tion on for an in­finite sur­real , be­cause then the prob­a­bil­ity that there is a finite amount of money in the en­velope is in­finites­i­mal, but we’re try­ing to model the situ­a­tion in which we know there’s a finite amount of money in the en­velope and just have no idea which finite amount.

Sphere of suffer­ing: Sur­real num­bers are not the right tool for mea­sur­ing the vol­ume of Eu­clidean space or the du­ra­tion of for­ever.

Hilbert ho­tel: As you men­tioned, us­ing sur­re­als in the way you pro­pose changes the prob­lem.

Trumped, Trou­ble in St. Peters­burg, Soc­cer teams, Can God choose an in­te­ger at ran­dom?, The Headache: Us­ing sur­re­als in the way you pro­pose in each of these changes the prob­lems in ex­actly the same way it does for the Hilbert ho­tel.

St. Peters­burg para­dox: If you pay in­finity dol­lars to play the game, then you lose in­finity dol­lars with prob­a­bil­ity 1. Doesn’t sound like a great deal.

Banach-Tarski Para­dox: The free group only con­sists of se­quences of finite length.

The Magic Dart­board: First, a nit­pick: that proof re­lies on the con­tinuum hy­poth­e­sis, which is in­de­pen­dent of ZFC. Aside from that, the proof is cor­rect, which means any re­s­olu­tion along the lines you’re imag­in­ing that im­ply that no magic dart­boards ex­ist is go­ing to im­ply that the con­tinuum hy­poth­e­sis is false. Worse, the fact that for any countable or­di­nal, there are countably many smaller countable or­di­nals and un­countably many larger countable or­di­nals fol­lows from very min­i­mal math­e­mat­i­cal as­sump­tions, and is of­ten used in de­scrip­tive set the­ory with­out bring­ing in the con­tinuum hy­poth­e­sis at all, so if you start try­ing to change math to make sense of “the sec­ond half of the countable or­di­nals”, you’re go­ing to have a bad time.

Par­ity para­doxes: The lengths of the se­quences in­volved here are the or­di­nal , not a sur­real num­ber. You might ob­ject that there is also a sur­real num­ber called , but this is differ­ent from the or­di­nal . Arith­metic op­er­a­tions act differ­ently on or­di­nals than they do on the copies of those or­di­nals in the sur­real num­bers, so there’s no rea­son­able sense in which the sur­re­als con­tain the or­di­nals. Ex­am­ple: if you add an­other el­e­ment to the be­gin­ning of ei­ther se­quence (i.e. flip the switch at , or add a at the be­gin­ning of the sum, re­spec­tively), then you’ve added one thing, so the sur­real num­ber should in­crease by , but the or­der-type is un­changed, so the or­di­nal re­mains the same.

• Thanks for your feed­back. I’ll note that these are only in­for­mal hints/​thoughts on how sur­real num­bers could help us here and that I’ll be pro­vid­ing a more de­vel­oped ver­sion of some of these thoughts in a fu­ture post.

In­fini­tar­ian paral­y­sis: I con­sider Pas­cal’s Mug­ging to be its own seper­ate prob­lem. In­deed Pas­cal’s Mug­ging type is­sues are already pre­sent with the more stan­dard in­fini­ties. In any case, the mea­sure the­ory solu­tion is de­pen­dent on an in­di­vi­d­ual be­ing a finite frac­tion of the agents in the uni­verse. While this is an ex­tremely plau­si­ble as­sump­tion, there doesn’t seem to be any prin­ci­pled rea­son why our solu­tion to in­finite paral­y­sis should de­pend on this as­sump­tion.

Para­dox of the Gods, Banach-Tarski: Your com­plaint is that I’m dis­al­low­ing the a se­quence con­sist­ing of all finite in­verses/​all finite in­te­gers. I ac­tu­ally be­lieve that ac­tual and po­ten­tial in­finity mod­els of these prob­lems need to be treated sep­a­rately, though I’ve only out­lined how I plan to han­dle ac­tual in­fini­ties. Hope­fully, you find my next post on this topic more per­sua­sive.

Two-en­velopes para­dox: “The prob­a­bil­ity that there is a finite amount of money in the en­velope is in­finites­i­mal”—Hmm, you’re right. That is a rather sig­nifi­cant is­sue.

Sphere of suffer­ing: “Sur­real num­bers are not the right tool for mea­sur­ing the vol­ume of Eu­clidean space or the du­ra­tion of for­ever”—why?

St Peters­berg Para­dox: Ah, but you have an in­finites­i­mal chance of win­ning a higher in­finity. So it be­comes an even more ex­treme ver­sion of Pas­cal’s Mug­ging, but again that’s its own dis­cus­sion.

Magic Dart­board: Yes, I’m aware that re­ject­ing the ex­is­tence of magic dart­boards could have far-reach­ing con­se­quences. It’s some­thing I hope to look into more.

Par­ity: See my re­sponse to gjm. Or­di­nal num­bers lack re­s­olu­tion and so can’t prop­erly de­scribe the length of se­quence.

• In­deed Pas­cal’s Mug­ging type is­sues are already pre­sent with the more stan­dard in­fini­ties.

Right, in­finity of any kind (sur­real or oth­er­wise) doesn’t be­long in de­ci­sion the­ory.

“Sur­real num­bers are not the right tool for mea­sur­ing the vol­ume of Eu­clidean space or the du­ra­tion of for­ever”—why?

How would you? If you do some­thing like tak­ing an in­creas­ing se­quence of bounded sub­sets that fill up the space you’re try­ing to mea­sure, find a for­mula f(n) for the vol­ume of the nth sub­set, and plug in , the re­sult will be highly de­pen­dent on which in­creas­ing se­quence of bounded sub­sets you use. Did you have a differ­ent pro­posal? It’s sort of hard to ex­plain why no method for mea­sur­ing vol­umes us­ing sur­real num­bers can pos­si­bly work well, though I am con­fi­dent it is true. At the very least, vol­ume-pre­serv­ing trans­for­ma­tions like shift­ing ev­ery­thing 1 me­ter to the left or ro­tat­ing ev­ery­thing around some axis cease to be vol­ume-pre­serv­ing, though I don’t know if you’d find this con­vinc­ing.

• “In­deed Pas­cal’s Mug­ging type is­sues are already pre­sent with the more stan­dard in­fini­ties.”
Right, in­finity of any kind (sur­real or oth­er­wise) doesn’t be­long in de­ci­sion the­ory.

But Pas­cal’s Mug­ging type is­sues are pre­sent with large finite num­bers, as well. Do you bite the bul­let in the finite case, or do you think that un­bounded util­ity func­tions don’t be­long in de­ci­sion the­ory, ei­ther?

• The lat­ter. It doesn’t even make sense to speak of max­i­miz­ing the ex­pec­ta­tion of an un­bounded util­ity func­tion, be­cause un­bounded func­tions don’t even have ex­pec­ta­tions with re­spect to all prob­a­bil­ity dis­tri­bu­tions.

There is a way out of this that you could take, which is to only in­sist that the util­ity func­tion has to have an ex­pec­ta­tion with re­spect to prob­a­bil­ity dis­tri­bu­tions in some re­stricted class, if you know your op­tions are all go­ing to be from that re­stricted class. I don’t find this very satis­fy­ing, but it works. And it offers its own solu­tion to Pas­cal’s mug­ging, by in­sist­ing that any out­come whose util­ity is on the scale of 3^^^3 has prior prob­a­bil­ity on the scale of 1/​(3^^^3) or lower.

• There’s definitely a part of me won­der­ing if in­fini­ties ex­ist, but be­fore I even con­sider tack­ling that ques­tion, I need to figure out the most con­sis­tent in­ter­pre­ta­tion of in­fini­ties as­sum­ing they ex­ist.

“At the very least, vol­ume-pre­serv­ing trans­for­ma­tions like shift­ing ev­ery­thing 1 me­ter to the left or ro­tat­ing ev­ery­thing around some axis cease to be vol­ume-pre­serv­ing, though I don’t know if you’d find this con­vinc­ing”—Well there are non-mea­surable sets that do this with­out sur­re­als, but do sur­re­als add more ex­am­ples?

I’ll have to read more about how sur­re­als ap­ply to vol­umes. It may be hard get­ting con­ver­gence to the ex­act in­finites­i­mal, but I don’t know if the prob­lems will ex­tent be­yond that.

(Also, the abil­ity to in­te­grate is mostly be­sides the point. In­stead of the sphere of suffer­ing, we could have defined the ex­pand­ing cube of suffer­ing. This will then let us solve some spe­cial cases of the sphere of suffer­ing)

• There are mea­surable sets whose vol­umes will not be pre­served if you try to mea­sure them with sur­real num­bers. For ex­am­ple, con­sider . Say its mea­sure is some in­finite sur­real num­ber . The vol­ume-pre­serv­ing left-shift op­er­a­tion sends to , which has mea­sure , since has mea­sure . You can do es­sen­tially the same thing in higher di­men­sions, and the shift op­er­a­tion in two di­men­sions () can be ex­pressed as the com­po­si­tion of two ro­ta­tions, so ro­ta­tions can’t be vol­ume-pre­serv­ing ei­ther. And since differ­ent ro­ta­tions will have to fail to pre­serve vol­umes in differ­ent ways, this will break sym­me­tries of the plane.

I wouldn’t say that vol­ume-pre­serv­ing trans­for­ma­tions fail to pre­serve vol­ume on non-mea­surable sets, just that non-mea­surable sets don’t even have mea­sures that could be pre­served or not pre­served. Failing to pre­serve mea­sures of sets that you have as­signed mea­sures to is en­tirely differ­ent. Non-mea­surable sets also don’t arise in math­e­mat­i­cal prac­tice; half-spaces do. I’m also skep­ti­cal of the ex­is­tence of non-mea­surable sets, but the non-ex­is­tence of non-mea­surable sets is a far bolder claim than any­thing else I’ve said here.

• Well shift­ing left pro­duces a su­per­set of the origi­nal, so of course we shouldn’t ex­pect that to pre­serve mea­sure.

• What about ro­ta­tions, and the fact that we’re talk­ing about de­stroy­ing a bunch of sym­me­try of the plane?

• I’m happy to bite that bul­let and de­stroy the sym­me­try. If we pick a ran­dom point and line in the uni­verse, are there more unit points to the left or right? Well, that de­pends on where the point is.

• It’s a bad bul­let to bite. Its sym­me­tries are es­sen­tial to what makes Eu­clidean space in­ter­est­ing.

And here’s an­other one: are you not both­ered by the lack of countable ad­di­tivity? Sup­pose you say that the vol­ume of Eu­clidean space is some sur­real num­ber . Eu­clidean space is the union of an in­creas­ing se­quence of balls. The vol­umes of these balls are all finite, in par­tic­u­lar, less than , so how can you jus­tify say­ing that their union has vol­ume greater than ?

• “Its sym­me­tries are es­sen­tial to what makes Eu­clidean space in­ter­est­ing”—Isn’t the in­ter­est­ing as­pect of Eu­clidean space its abil­ity to model our world ex­clud­ing rel­a­tivity?

Well, I just don’t think it’s that un­usual for func­tions to have prop­er­ties that break at their limits. Is this any differ­ent from 1/​x be­ing defin­able ev­ery­where ex­cept 0? Is there any­thing that makes the change at the limit par­tic­u­larly con­cern­ing.

• I don’t fol­low the anal­ogy to 1/​x be­ing a par­tial func­tion that you’re get­ting at.

Maybe a bet­ter way to ex­plain what I’m get­ting at is that it’s re­ally the same is­sue that I pointed out for the two-en­velopes prob­lem, where you know the amount of money in each en­velope is finite, but the uniform dis­tri­bu­tion up to an in­finite sur­real would sug­gest that the prob­a­bil­ity that the amount of money is finite is in­finites­i­mal. Sup­pose you say that the size of the ray is an in­finite sur­real num­ber . The size of the por­tion of this ray that is dis­tance at least from is when is a pos­i­tive real, so pre­sum­ably you would also want this to be so for sur­real . But us­ing, say, , ev­ery point in is within dis­tance of , but this rule would say that the mea­sure of the por­tion of the ray that is farther than from is ; that is, al­most all of the mea­sure of is con­cen­trated on the empty set.

• As I un­der­stand it, there is not yet a good the­ory of in­te­gra­tion on the sur­re­als. Par­tial progress has been made, but there are also some nega­tive re­sults es­tab­lish­ing limi­ta­tions on the pos­si­bil­ities. Here is a re­cent pa­per.

• In “Trumped”, it seems that if , the first in­finite or­di­nal, then on ev­ery sub­se­quent day, the re­main­ing num­ber of days will be for some nat­u­ral . This is never equal to .

Put differ­ently, just be­cause we count up to doesn’t mean we pass through . Of course, the to­tal or­der on days has has for each finite , but this isn’t a well-or­der any­more so I’m not sure what you mean when you say there’s a se­quence of de­ci­sions. Do you know what you mean?

• “Put differ­ently, just be­cause we count up to n doesn’t mean we pass through n/​3”—The first pos­si­ble ob­jec­tion I’ll deal with is not what I think you are com­plain­ing about, but I think it’s worth han­dling any­way. n/​3 mightn’t be an Om­nific Num­ber, but in this case we just take the in­te­ger part of n/​3.

I think the is­sue you are high­light­ing is that all finite num­bers are less than n/​3. And if you define a se­quence as con­sist­ing of finite num­bers then it’ll never in­clude n/​3. How­ever, if you define a se­quence as all num­bers be­tween 1 and x where x is a sur­real num­ber then you don’t en­counter this is­sue. Is this valid? I would ar­gue that this ques­tion boils down to whether there are an ac­tu­ally in­finite num­ber of days on which Trump ex­pe­riences or only a po­ten­tial in­finity. If it’s an ac­tual in­finity, then the sur­real solu­tion seems fine and we can say that Trump should stop say­ing yes on day n/​3. If it’s only a po­ten­tial in­finity, then this solu­tion doesn’t work, but I don’t en­dorse sur­re­als in this case (still read­ing about this)

• If the num­ber of days is, speci­fi­cally, , then the days are num­bered 0, 1, 2, …, with pre­cisely the (or­di­nary, finite) non-nega­tive in­te­gers oc­cur­ring. They are all smaller than . The num­ber isn’t the limit or the least up­per bound of those finite in­te­gers, merely the sim­plest thing big­ger than them all.

If you are tempted to say “No! What I mean by call­ing the num­ber of days is pre­cisely that the days are num­bered by all the om­nific in­te­gers be­low .” then you lose the abil­ity to rep­re­sent a situ­a­tion in which Trump suffers this in­dig­nity on a se­quence of days with ex­actly the or­der-type of the first in­finite or­di­nal , and that seems like a pretty se­ri­ous bul­let to bite. In par­tic­u­lar, I think you can’t call this a solu­tion to the “Trumped” para­dox, be­cause my read­ing of it—even as you tell it! -- is that it is all about a se­quence of days whose or­der-type is .

Rather a lot of these para­doxes are about situ­a­tions that in­volve limit­ing pro­cesses of a sort that doesn’t seem like a good fit for sur­real num­bers (at least so far as I un­der­stand the cur­rent state of the art when it comes to limit­ing pro­cesses in the sur­real num­bers, which may not be very far).

• I already pointed above to the dis­tinc­tion be­tween ab­solute and po­ten­tial in­fini­ties. I ad­mit that the sur­real solu­tion as­sumes that we are deal­ing with an ab­solute in­finity in­stead of a po­ten­tial one, so let’s just con­sider this case. You want to con­ceive of this prob­lem as “a se­quence whose or­der-type is ω”, but from the sur­real per­spec­tive this lacks re­s­olu­tion. Is the num­ber of el­e­ments (sur­real) ω, ω+1 or ω+1000? All of these are pos­si­ble given that in the or­di­nals 1+ω=ω so we can add ar­bi­trar­ily many num­bers to the start of a se­quence with­out chang­ing its or­der type.

So I don’t think the or­di­nary no­tion of se­quence makes sense. In par­tic­u­lar, it doesn’t ac­count for the fact that two se­quences which ap­pear to be the same in ev­ery place can ac­tu­ally be differ­ent if they have differ­ent lengths. Any­way, I’ll try to un­tan­gle some of these is­sues in fu­ture posts, in par­tic­u­lar I’m lean­ing to­wards hy­per­re­als as a bet­ter fit for mod­el­ling po­ten­tial in­fini­ties, but I’m still un­cer­tain about how things will de­velop once I man­age to look into this more.

• So far as any­one knows, no ac­tual pro­cesses in the ac­tual world are ac­cu­rately de­scribed by sur­real num­bers. If not, then I sug­gest the same goes for the “near­est pos­si­ble wor­lds” in which, say, it is pos­si­ble for Mr Trump to be faced with the sort of situ­a­tion de­scribed un­der the head­ing “Trumped”. But you can have, in a uni­verse very much like ours, an end­less suc­ces­sion of events of or­der-type . If the sur­real num­bers are not well suited to de­scribing such situ­a­tions, so much the worse for the sur­real num­bers.

And when you say “I don’t think the or­di­nary no­tion of se­quence makes sense”, what it looks like to me is that you have looked at the or­di­nary no­tion of se­quence, made the en­tirely ar­bi­trary choice that you are only pre­pared to un­der­stand it in terms of sur­real num­bers, and in­deed not only that but made the fur­ther ar­bi­trary choice that you are only pre­pared to un­der­stand it if there turns out to be a uniquely defined sur­real num­ber that is the length of such a se­quence, ob­served that there is not such a sur­real num­ber, and then said not “Oh, whoops, looks like I took a wrong turn in try­ing to model this situ­a­tion” but “Bah, the thing I’m try­ing to model doesn’t fit my pre­con­cep­tions of what the model should look like, there­fore the thing is wrong”. You can’t do that! Models ex­ist to serve the things they model, not the other way around.

It’s as if I’d just learned about the or­di­nals, de­cided that all in­finite things needed to be de­scribed in terms of the or­di­nals, was asked some­thing about a countably in­finite set, ob­served that such a set is the same size as but also the same size as and , and said “I don’t think the no­tion of countably in­finite set makes sense”. It makes perfectly good sense, I just (hy­po­thet­i­cally) picked a bad way to think about it: or­di­nals are not the right tool for mea­sur­ing the size of a (not-nec­es­sar­ily-well-or­dered) set. And like­wise, sur­real num­bers are not the right tool for mea­sur­ing the length of a se­quence.

Don’t get me wrong; I love the sur­real num­bers, as an ob­ject of math­e­mat­i­cal study. The the­ory is gor­geous. But you can’t claim that the sur­real num­bers let you re­solve all these para­doxes, when what they ac­tu­ally al­low you to do is to re­place the para­dox­i­cal situ­a­tions with other en­tirely differ­ent situ­a­tions and then deal with those, while re­ject­ing the origi­nal situ­a­tions merely be­cause your way of try­ing to model them doesn’t work out neatly.

• Maybe I should re-em­pha­sise the caveat at the top of the post: “I will provide in­for­mal hints on how sur­real num­bers could help us solve some of these para­doxes, al­though the fo­cus on this post is pri­mar­ily cat­e­gori­sa­tion, so please don’t mis­take these for for­mal proofs. I’m also aware that sim­ply not­ing that a for­mal­i­sa­tion pro­vides a satis­fac­tory solu­tion doesn’t philo­soph­i­cally jus­tify its use, but this is also not the fo­cus of this post.”

You wrote that I “made the en­tirely ar­bi­trary choice that you are only pre­pared to un­der­stand it in terms of sur­real num­bers”. This choice isn’t ar­bi­trary. I’ve given some hints as to why I am tak­ing this ap­proach, but a full jus­tifi­ca­tion won’t oc­cur un­til fu­ture posts.

• OK! I’ll look for­ward to those fu­ture posts.

(I’m a big sur­real num­ber fan, de­spite the skep­ti­cal tone of my com­ments here, and I will be ex­tremely in­ter­ested to see what you’re propos­ing.)

• You want to con­ceive of this prob­lem as “a se­quence whose or­der-type is ω”, but from the sur­real per­spec­tive this lacks re­s­olu­tion. Is the num­ber of el­e­ments (sur­real) ω, ω+1 or ω+1000? All of these are pos­si­ble given that in the or­di­nals 1+ω=ω so we can add ar­bi­trar­ily many num­bers to the start of a se­quence with­out chang­ing its or­der type.

It seems to me that mea­sur­ing the lengths of se­quences with sur­re­als rather than or­di­nals is in­tro­duc­ing fake re­s­olu­tion that shouldn’t be there. If you start with an in­finite con­stant se­quence 1,1,1,1,1,1,..., and tell me the se­quence has size , and then you add an­other 1 to the be­gin­ning to get 1,1,1,1,1,1,1,..., and you tell me the new se­quence has size , I’ll be like “uh, but those are the same se­quence, though. How can they have differ­ent sizes?”

• Be­cause we should be work­ing with la­bel­led se­quences rather than just se­quences (that is se­quences with a length at­tached). That solves the most ob­vi­ous is­sues, though there are some sub­tleties there

• Why? Plain se­quences are a perfectly nat­u­ral ob­ject of study. I’ll echo gjm’s crit­i­cism that you seem to be try­ing to “re­solve” para­doxes by chang­ing the defi­ni­tions of the words peo­ple use so that they re­fer to un­nat­u­ral con­cepts that have been ger­ry­man­dered to fit your solu­tion, while re­fus­ing to talk about the nat­u­ral con­cepts that peo­ple ac­tu­ally care about.

I don’t think think your pro­posal is a good one for in­dexed se­quences ei­ther. It is pretty weird that shift­ing the in­dices of your se­quence over by 1 could change the size of the se­quence.

• the or­di­nary no­tion of sequence

I as­sume here you mean some­thing like “a se­quence of el­e­ments from a set is a func­tion where is an or­di­nal”. Do you know about nets? Nets are a no­tion of se­quence preferred by peo­ple study­ing point-set topol­ogy.

• Thanks for the sug­ges­tion. I took a look at nets, but their pur­pose seems mainly di­rected to­wards gen­er­al­is­ing limits to topolog­i­cal spaces, rather than adding ex­tra nu­ance to what it means for a se­quence to have in­finite length. But per­haps you could clar­ify why you think that they are rele­vant?

• A net is just a func­tion where is an or­dered in­dex set. For limits in gen­eral topolog­i­cal spaces, might be pretty nasty, but in your case, you would want to be some to­tally-or­dered sub­set of the sur­re­als. For ex­am­ple, in the trump para­dox, you prob­a­bly want to:

in­clude and for some in­finite
have a least el­e­ment (the first day)

It sounds like you also want some co­her­ent no­tion of “to­mor­row” at each day, so that you can get through all the days by pass­ing from to­day to to­mor­row in­finitely many times. But this is equiv­a­lent to hav­ing your set be well-or­dered, which is in­com­pat­i­ble with the prop­erty “closed un­der di­vi­sion and sub­trac­tion by finite in­te­gers”. So you should clar­ify which of these prop­er­ties you want.

• “But this is equiv­a­lent to hav­ing your set be well-or­dered, which is in­com­pat­i­ble with the prop­erty “closed un­der di­vi­sion and sub­trac­tion by finite in­te­gers”″ - Why is this in­com­pat­i­ble?

• An or­dered set is well-or­dered iff ev­ery sub­set has a unique least el­e­ment. If your set is closed un­der sub­trac­tion, you get in­finite de­scend­ing se­quences such as . If your se­quence is closed un­der di­vi­sion, you get in­finite de­scend­ing se­quences that are fur­ther­more bounded such as . It should be clear that the two lin­ear or­ders I de­scribed are not well-or­ders.

A small or­der the­ory fact that is not to­tally on-topic but may help you gather in­tu­ition:

Every countable or­di­nal em­beds into the re­als but no un­countable or­di­nal does.

• Okay, I now un­der­stand why clo­sure un­der those op­er­a­tions is in­com­pat­i­ble with be­ing well-or­dered. And I’m guess­ing you be­lieve that well-or­der­ing is nec­es­sary for a co­her­ent no­tion of pass­ing through to­mor­row in­finitely many times be­cause it’s a re­quire­ment for trans­finite in­duc­tion?

I’m not so sure that this is im­por­tant. After all, we can imag­ine get­ting from 1 to 2 via pass­ing through an in­finite num­ber of in­finites­i­mally small steps even though [1,2] isn’t well-or­dered on <. In­deed, this is the cen­tral point of Zeno’s para­dox.

• Yes, there are good ways to in­dex sets other than well or­ders. A net where the in­dex set is the real line and the func­tion is con­tin­u­ous is usu­ally called a path, and these are ubiquitous e.g. in the foun­da­tions of alge­braic topol­ogy.

I guess you could say that I think well-or­ders are im­por­tant to the pic­ture at hand “be­cause of trans­finite in­duc­tion” but a sim­pler way to state the same ob­jec­tion is that “to­mor­row” = “the unique least el­e­ment of the set of days not yet vis­ited”. If to­mor­row always ex­ists /​ is uniquely defined, then we’ve got a well-or­der. So some­thing about the story has to change if we’re not fit­ting into the or­di­nal box.

• Your sec­ond ex­am­ple, 1 > 12 > 14 > … > 0, is a well-or­der. To make it non-well-or­dered, leave out the 0.

• A well-or­der has a least el­e­ment in all non-empty sub­sets, and 1 > 12 > 14 > … > 0 has a non-empty sub­set with­out a least el­e­ment, so it’s not a well-or­der.

• Yes, you’re right.

In gen­eral, ev­ery sub­or­der of a well-or­der is well-or­dered. In a word, the prop­erty of “be­ing a well-or­der” is hered­i­tary. (com­pare: ev­ery sub­set of a finite set is finite)

• I think gjm’s re­sponse is ap­prox­i­mately the clar­ifi­ca­tion I would have made about my ques­tion if I had spent 30 min­utes think­ing about it.

• For Eve and her ap­ple pieces. She may eat one piece per sec­ond and stay in Par­adise for­ever be­cause at any given mo­ment only a finite num­ber of pieces has been eaten by her.

If her eat­ing pace dou­bles ev­ery minute, she is still okay for­ever.

Only if she, for ex­am­ple, dou­bles her eat­ing pace af­ter ev­ery say 100 pieces eaten, then she is in trou­ble. If she su­per­tasks.

• Great post!

Satan’s Ap­ple: Satan has cut a deli­cious ap­ple into in­finitely many pieces. Eve can take as many pieces as she likes, but if she takes in­finitely many pieces she will be kicked out of par­adise and this will out­weigh the ap­ple. For any finite num­ber i, it seems like she should take that ap­ple piece, but then she will end up tak­ing in­finitely many pieces.

Pro­posed solu­tion for finite Eves (also a solu­tion to Trumped, for finite Trumps who can’t count to sur­real num­bers):

After hav­ing eaten n pieces, Eve’s de­ci­sion isn’t be­tween eat­ing n pieces and eat­ing n+1 pieces, it’s be­tween eat­ing n pieces and what­ever will hap­pen if she eats the n+1st piece. If Eve knows that the fu­ture Eve will be fol­low­ing the strat­egy “always eat the next ap­ple piece”, then it’s a bad de­ci­sion to eat the n+1st piece (since it will lead to get­ting kicked out of par­adise).

So what strat­egy should Eve fol­low? Con­sider the prob­lem of pro­gram­ming a strat­egy that an Eve-bot will fol­low. In this case, the best strat­egy is the strat­egy that will lead to the largest amount of finite pieces be­ing eaten. What this strat­egy is de­pends on the hard­ware, but if the hard­ware is finite, then there ex­ists such a strat­egy (per­haps count the num­ber of pieces and stop when you reach N, for the largest N you can store and com­pare with). Gen­er­al­is­ing to (finite) hu­mans, the best strat­egy is the strat­egy that re­sults in the largest amount of finite pieces eaten, among all strate­gies that a hu­man can pre­com­mit to.

Of course, if we al­low in­finite hard­ware, then the prob­lem is back again. But that’s at least not a prob­lem that I’ll ever en­counter, since I’m run­ning on finite hard­ware.

• We can definitely solve this prob­lem for real agents, but the rea­son why I find this prob­lem so per­plex­ing is be­cause of the bound­ary is­sue that it high­lights. Imag­ine that we have an ac­tual in­finite num­ber of peo­ple. Color all the finite placed peo­ple red and the non-finite placed peo­ple blue. Every­one one t the right of a red per­son should be red and ev­ery­one one to the left of blue per­son should be blue. So what does the bound­ary look like? Sure we can’t finitely trans­verse from the start to the in­finite num­bers, but that doesn’t af­fect the in­tu­ition that the bound­ary should still be there some­where. And this makes me ques­tion whether the no­tion of an ac­tual in­finity is co­her­ent (I re­ally don’t know).

• My best guess about how to clear up con­fu­sion about “what the bound­ary looks like” is via math­e­mat­ics rather than philos­o­phy. For ex­am­ple, have you un­der­stood the prop­er­ties of the long line?

• Thanks for that sug­ges­tion. The long line looks very in­ter­est­ing. Are you sug­gest­ing that the bound­ary doesn’t ex­ist?

• Yeah, I’d agree with the “bound­ary doesn’t ex­ist” in­ter­pre­ta­tion.