# 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.

__Resolution Paradoxes__

**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)] = ^{1}⁄_{4} 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.

__Last-Half Paradoxes__

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 ^{1}⁄_{10} 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 ^{1}⁄_{10}.

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 ^{50}⁄_{50} 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 ^{1}⁄_{10}.

**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.

__Parity Paradoxes__

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?

**Wi**th 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.

__Boundary Paradoxes__

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.

__Trivial Paradoxes__

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

^{1}⁄_{3}Picking a random radius then a random point on that radius gives probability of

^{1}⁄_{2}Picking a random point and using it as a midpoint gives

^{1}⁄_{4}

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.

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 x because it will always have higher expected utility to focus all of your attention on trying to find ways to increase utility by >ωx, since there’s some (however small) probability p>0 that such efforts would succeed, so the expected utility of focusing your efforts on looking for ways to increase utility by >ωx will have expected utility >pωx, which is higher than x. 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 2−ω miles if no gods erect any barriers until he advances 2−n miles for some finite n?

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 [1n,n] for an infinite surreal n, 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 ω, not a surreal number. You might object that there is also a surreal number called ω, but this is different from the ordinal ω. 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 t=−2, or add a −1 at the beginning of the sum, respectively), then you’ve added one thing, so the surreal number should increase by 1, 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.

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

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 f(ω), 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.

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 [0,∞)⊆R. Say its measure is some infinite surreal number n. The volume-preserving left-shift operation x↦x−1 sends [0,∞) to [−1,∞), which has measure 1+n, since [−1,0) has measure 1. You can do essentially the same thing in higher dimensions, and the shift operation in two dimensions ((x,y)↦(x−1,y)) 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 n. Euclidean space is the union of an increasing sequence of balls. The volumes of these balls are all finite, in particular, less than n2, so how can you justify saying that their union has volume greater than n2?

“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 [0,∞) is an infinite surreal number n. The size of the portion of this ray that is distance at least r from 0 is n−r when r is a positive real, so presumably you would also want this to be so for surreal r. But using, say, r:=√n, every point in [0,∞) is within distance √n of 0, but this rule would say that the measure of the portion of the ray that is farther than √n from 0 is n−√n; that is, almost all of the measure of [0,∞) 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 n=ω, the first infinite ordinal, then on every subsequent day, the remaining number of days will be ω−k for some natural k. This is never equal to n/3.

Put differently, just because we count up to n doesn’t mean we pass through n/3. Of course, the total order on days has has k<n/3<n for each finite k, 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, ω, then the days are numbered 0, 1, 2, …, with precisely the (ordinary, finite) non-negative integers occurring. They are all smaller than ω/3. The number ω isn’t the

limitor theleast upper boundof those finite integers, merely thesimplestthing bigger than them all.If you are tempted to say “No! What I mean by calling the number of days ω is precisely that the days are numbered by all the omnific integers below ω.” 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 ω, 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

youtell it! -- is that it isall abouta sequence of days whose order-type is ω.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

canhave, in a universe very much like ours, an endless succession of events of order-type ω. 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 butmade 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, thereforethe 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 ω but also the same size as 1+ω and ω2, 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

lovethe 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 withthose, 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.)

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 ω, 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 ω+1, 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.

I assume here you mean something like “a sequence of elements from a set X is a function f:α→X where α 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 f:I→X where I is an ordered index set. For limits in general topological spaces, I might be pretty nasty, but in your case, you would want I to be some totally-ordered subset of the surreals. For example, in the trump paradox, you probably want I to:

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

equivalentto having your set I 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

equivalentto 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 1>0>−1>−2>⋯ . If your sequence is closed under division, you get infinite descending sequences that are furthermore bounded such as 1>12>14>⋯>0. 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 f:I→X 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 >

^{1}⁄_{2}>^{1}⁄_{4}> … > 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 >

^{1}⁄_{2}>^{1}⁄_{4}> … > 0 has a non-empty subset without a least element, so it’s not a well-order.Yes, you’re right.

Adding to Vladimir_Nesov’s comment:

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!

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.