# On the importance of taking limits: Infinite Spheres of Utility

I had a dis­cus­sion re­cently with some Less Wrongers about a de­ci­sion prob­lem in­volv­ing in­fini­ties, which ap­pears to have a para­dox­i­cal solu­tion. We have been warned by Jaynes and oth­ers to be care­ful about tak­ing the proper limits when in­fini­ties are in­volved in a prob­lem, and I thought this would be a good ex­am­ple to show that we can get an­swers that make sense out of prob­lems that seem not to.

The prob­lem is the “In­finite Spheres of Utility.” To quote a de­scrip­tion from Philos­o­phy et cetera,

Imag­ine a uni­verse con­tain­ing in­finitely many im­mor­tal peo­ple, par­ti­tioned into two “spheres”. In one sphere [sphere A], all the in­hab­itants live a bliss­ful ex­is­tence, whereas the mem­bers of the other sphere [sphere B] suffer un­bear­able agony. Now com­pare the fol­low­ing two vari­a­tions:

1. Every­one starts off in the bliss­ful sphere. But each day, one more per­son gets per­ma­nently trans­ferred across to the agony sphere, where they reside for the rest of eter­nity.

2. Every­one starts off in the agony sphere. But each day, one more per­son gets per­ma­nently trans­ferred across to the bliss­ful sphere, where they reside for the rest of eter­nity.

At first con­sid­er­a­tion, the prob­lem ap­pears to cause a para­dox:

Which sce­nario is bet­ter? The an­swer, para­dox­i­cally, ap­pears to be “both”.

• At any mo­ment in time, there will be in­finitely many peo­ple in the origi­nal sphere, and only a finite num­ber who have been trans­ferred across. So op­tion 1 is bet­ter.

• How­ever, each par­tic­u­lar per­son will spend only a finite amount of time in the first sphere, whereas they will spend an eter­nity in their post-trans­fer home. So op­tion 2 is bet­ter.

Given these rea­son­able but hard-to-rec­on­cile view­points, how do we make a de­ci­sion?

## De­cid­ing how to decide

We first need to de­cide which kind of de­ci­sion anal­y­sis we want to use to choose a start­ing sphere. For ex­am­ple, we might sim­ply have an ar­bi­trary prefer­ence for putting ev­ery­one in the bad sphere. Para­dox over. Even if we want to use a form of util­i­tar­i­anism, we have more than one type to choose from. One of the sim­plest and most in­tu­itive is ad­di­tive util­i­tar­i­anism, in which we define a util­ity for each per­son, add them all to­gether, and make the choice with the larger to­tal util­ity. We can think of the para­dox above as re­sult­ing from our in­sis­tence that the solu­tion con­form to this kind of de­ci­sion anal­y­sis: it ap­pears that there should be more than one con­flict­ing solu­tion. In fact, we will see that there are three mu­tu­ally ex­clu­sive solu­tions con­form­ing to ad­di­tive util­i­tar­i­anism, and we can get to any one of them by solv­ing the finite prob­lem and choos­ing how to take the in­finite limit.

## The Finite Problem

We will be­gin by cre­at­ing an analo­gous finite prob­lem us­ing ad­di­tive util­i­tar­i­anism.

Define $u_A$ and $u_B$ as the util­ities for one per­son liv­ing for one day in spheres $A$ or $B$, re­spec­tively, which carry units of [util­ity/​(day*per­son)]. We might, for ex­am­ple, choose $u_A=1$ and $u_B=-1$, but we will cer­tainly choose $u_A > u_B$. If we make “choice 1,” we be­gin with all peo­ple in sphere A. If we make “choice 2,” we be­gin with all peo­ple in sphere $B$. The to­tal util­ity is the sum of $u_A$ and $u_B$ over all peo­ple and all days, or

$U =\sum_{t= 0}^{s}u_A n_{A,t} + u_B n_{B,t}$

where $t$ is the day, $s$ is the num­ber of days each per­son lives, $n_{A,t}$ and $n_{B,t}$ are the num­ber of peo­ple in spheres $A$ and $B$ on day $t$, re­spec­tively, and I’ve im­plic­itly summed over the num­ber of peo­ple for brevity. We have as­sumed a lin­ear util­ity func­tion that val­ues ev­ery per­son’s util­ity equally on ev­ery day, but we could eas­ily gen­er­al­ize to other func­tional forms. Define $r$ as the peo­ple-trans­fer rate, which the prob­lem dic­tates to be

$r=1[person/day]$.

For choice 1, we be­gin with $n$ peo­ple in sphere $A$ and lose $r$ of them per day so

$n_{A,t} = n - t r$

and

$n_{B,t} = t r$,

with these vari­ables only defined for $t < s$, be­yond which ev­ery­one would be dead. Note that even though $r$ is 1, we’ll end up with a unit er­ror if we don’t carry it around. If we go with choice 2 in­stead, these vari­ables are switched, so

$n_{A,t} = t r$

and

$n_{B,t} = n - t r$.

Ac­cord­ing to the prob­lem state­ment, the trans­fer rate is con­stant, but we could again eas­ily gen­er­al­ize to any trans­fer func­tion if we wanted to (even non-mono­tonic ones, or ones that de­pend on the num­ber of peo­ple in ei­ther sphere). Call the to­tal util­ities for each scheme $U_1$ and $U_2$. From here on out, I will re­place sums with in­te­grals, be­cause the graphs will look bet­ter and the math takes up less space. We take on a small er­ror in the pro­cess, but it won’t af­fect the con­clu­sion.

Then the to­tal util­ities are

$U_1 = \int_{0}^{s}dt\:\:u_A (n - t r) + u_B t r$

for choice 1, and

$U_2 = \int_{0}^{s}dt\:\:u_A t r + u_B (n - t r),$

for choice 2, so that the differ­en­tial util­ity is

$\Delta U = U_1-U_2 = \int_0^s dt\:\:u_A(n - 2t r) + u_B (2t r - n).$

In­te­grat­ing over $dt$, we have

$\Delta U = (ns - s^2 r)(u_A-u_B).$

When this func­tion is pos­i­tive, choice 1 is bet­ter. When it’s nega­tive, choice 2 is bet­ter. Keep in mind that the sec­ond fac­tor is always pos­i­tive, since we set $u_A>u_B$.

First, let’s con­firm that this re­sult makes sense for finite $s$ and $n$. Figure 1 shows what hap­pens as we in­crease the num­ber of peo­ple in the prob­lem, but keep their life spans fixed. The differ­en­tial util­ity forms a line, with the first part of the line be­low the hori­zon­tal axis. For a small num­ber of peo­ple, it’s bet­ter to start ev­ery­one in sphere $B$, since ev­ery­one will be quickly trans­ferred to $A$. But for a large num­ber of peo­ple, it’s bet­ter to let them live out most of their lives on av­er­age in sphere $A$ be­fore get­ting trans­ferred to sphere $B$. As $n$ ap­proaches in­finity, choice 1 is clearly bet­ter, since we have pos­i­tive $\Delta U$.

Figure 1: Fixed lifes­pan. As the pop­u­la­tion in­creases, choice 1 be­comes mono­ton­i­cally bet­ter.

In con­trast, Figure 2 shows what hap­pens as a fixed num­ber of peo­ple be­come long-lived. This time, we get a parabola. If ev­ery­one dies quickly, it’s bet­ter to start in sphere $A$, since many of them will die be­fore they get trans­ferred. If they live a long time, it’s bet­ter to start in sphere $B$, and then live out most of their lives af­ter be­ing trans­ferred to sphere $A$. As $s$ ap­proaches in­finity, choice 2 is clearly bet­ter, since we want peo­ple liv­ing out their im­mor­tal­ity in a good uni­verse, which cor­re­sponds to $\Delta U$ be­ing nega­tive.

Figure 2: Fixed pop­u­laiton. As life span in­creases, choice 2 be­comes bet­ter.

## The In­finite Limit

What hap­pens if both $s$ and $n$ go to in­finity, as stated in the prob­lem? If we just plug $\{s,n\}=\{\infty,\infty\}$ into our $\Delta U$ equa­tion, the an­swer is un­defined, and that is the sub­stance of the para­dox. But of course, math­e­mat­ics lets us care­fully take limits of func­tions of mul­ti­ple vari­ables. Figure 3 is a den­sity plot of $\Delta U$ as a func­tion of both $s$ and $n$. The value of $\Delta U$ is mapped on to color. When we re­fer to a sys­tem with “an in­finite num­ber of im­mor­tal peo­ple,” we’re talk­ing about a point far away from the ori­gin, where both $s$ and $n$ are pos­i­tive and in­finite. But from the den­sity plot, it’s clear that we could be talk­ing about many differ­ent points with differ­ent val­ues, and we have to spec­ify which di­rec­tion we go to get there. For ex­am­ple, if we fol­low the red dot­ted line out from the ori­gin, we will find that choice 1 gets bet­ter and bet­ter the more we in­crease $n$ and $s$. If we fol­low the blue dot­ted line, choice 2 is bet­ter for large $n$ and $s$. Fi­nally, if we fol­low the green dot­ted line out to in­finity, we find that $\Delta U = 0$ for all $n$ and $s$, mean­ing that our choices are equally good (or bad).

Figure 3: Utility vs. pop­u­la­tion and life span. The best choice de­pends on what path we fol­low to take the limit.

Prac­ti­cally, these lines rep­re­sent situ­a­tions in which we choose a finite num­ber of peo­ple and a finite life span, and then mon­i­tor what hap­pens to $\Delta U$ as we in­crease them both at a con­stant (but not nec­es­sar­ily equal) rate. We find that the an­swer de­pends on the differ­ence in rates. To for­mal­ize this re­sult, we can set $n = \alpha*s$ in the $\Delta U$ equa­tion, where $\alpha$ has units of [peo­ple/​day], and is al­lowed to be frac­tional. That way, ev­ery time we dou­ble $n$, we mul­ti­ply $s$ by $2\alpha$, and we move to­wards in­finity that way. Put an­other way, we con­strain our­selves to a lin­ear re­la­tion­ship be­tween $s$ and $n$ (we could take a non-lin­ear path out to in­finity if we cared to). We then have

$\Delta U_{linear} = s^2 (\alpha - r)(u_A - u_B).$

The limit of this func­tion as $s \rightarrow \infty$ is $+\infty$ if $\alpha , $-\infty$ if $\alpha>r$, and $0$ if $\alpha = r$, which cov­ers all of the pos­si­ble lin­ear paths to in­finite pop­u­la­tion and in­finite lifes­pan.

## Ex­pected Per­sonal Utility

Another in­ter­est­ing ques­tion to ask is “If I were one of the peo­ple in the prob­lem, what would I ex­pect my differ­en­tial util­ity to be?” We can an­swer this ques­tion us­ing the same method as above. I define my util­ity per day in each sphere as $u_{A}$ and $u_{B}$ in units of [util­ity/​day]. The ex­pected num­ber of days that I will be in the start­ing sphere for a finite pop­u­la­tion is $n/(2r)$, and the num­ber of days in the sec­ond sphere for a finite pop­u­la­tion and life span is $s - n/2r$. My ex­pected util­ity is there­fore given by

$U_1 = u_A\left(\frac{n}{2r}\right) + u_B\left(s-\frac{n}{2r}\right)$

for choice 1, and

$U_2 = u_A\left(s-\frac{n}{2r}\right) + u_B\left(\frac{n}{2r}\right)$

for choice 2. The differ­en­tial util­ity is the differ­ence be­tween these, or

$\Delta U = \left(u_A - u_B\right)\left(\frac{n}{r} - s\right).$

If we just plug in in­fini­ties, the an­swer is un­defined. But if we use the same method as above, and define $n = \alpha*s$, then we can write

$\Delta U = s\left(u_A - u_B\right)\left(\frac{\alpha}{r} - 1\right),$

and we have well defined limits for all slopes $\alpha\in\{0,\infty\}$. If I choose $\alpha, I find that my lifes­pan grows faster than the pop­u­la­tion grows, and I’m bet­ter off start­ing in the bad sphere, ex­pect­ing to be trans­fered be­fore half my life is over. If I choose $\alpha>r$, the pop­u­la­tion grows too quickly, and I would pre­fer to start in the good sphere, since on av­er­age I’ll die be­fore I live more than half my life in the bad sphere. And as be­fore, if $\alpha=r$, I’m in­differ­ent to the two plans, since on av­er­age I will live half my life in each sphere ei­ther way.

If we want to make a de­ci­sion based on ad­di­tive util­ity, the in­finite prob­lem is ill posed; it has no unique solu­tion un­less we take on ad­di­tional as­sump­tions. In par­tic­u­lar, we in­tro­duced an ad­di­tional pa­ram­e­ter $\alpha$, mak­ing clear three well defined solu­tions that span the space of pos­si­ble solu­tions. In do­ing so, we solved a similar well posed prob­lem, but not the origi­nal one.

So why bother, if we didn’t solve the prob­lem? It’s worth work­ing through the math be­cause tt gave us an in­tu­ition for how the sys­tem works, in­clud­ing a quan­ti­ta­tive un­der­stand­ing of cross­ing points, in­ter­cepts, and cur­va­tures of the util­ity curves. While the prob­lem state­ment cor­re­sponds to an im­pos­si­ble situ­a­tion, the finite prob­lem is quite pos­si­ble, and tak­ing lin­ear limits could very well cor­re­spond to a real phys­i­cal pro­cess. If we no­tice a para­dox and then stop think­ing, we lose an op­por­tu­nity to gain a bet­ter un­der­stand­ing of the de­ci­sion pro­cess.

We could also con­sider us­ing a differ­ent paradigm of de­ci­sion the­ory that might deal with in­finite quan­tities bet­ter. With that said, I think that ad­di­tive util­i­tar­i­anism rep­re­sents well the in­tu­itive para­dox that pre­sents each choice as bet­ter than the other. Our graphs with one vari­able fixed showed that these are both in­tu­itively valid view­points when taken alone, and their rec­on­cili­a­tion is the challeng­ing part.

If you’re in­ter­ested in similar prob­lems, the St. Peters­burg Para­dox also in­volves di­verg­ing util­lities, and it has has been “dis­pel­led” here.

Author’s notes: [last edit Oct.15, 2013] The sub­ject of un­bounded util­ities is not new here at Less Wrong. Stu­art_Arm­strong has a well writ­ten anal­y­sis of the Heaven and Hell prob­lem, among oth­ers. PhilGoetz has a use­ful note on in­fini­ties. And on a re­lated topic, in­finite set athe­ism abounds. I have been un­able to find on LW ex­am­ples of care­ful math­e­mat­i­cal treat­ments of ap­par­ent para­doxes that re­sult from mi­suse of in­finites in util­i­tar­ian calcu­la­tions, and I hope this ar­ti­cle serves as a con­crete ex­am­ple of how easy it is to defeat (some) such prob­lems. If there are other ex­am­ples at LW, please post them in the com­ments: I’d love to read them. It is in­ter­est­ing to note that Pas­cal’s Wager, a fa­mous re­lated prob­lem, is rather more difficult to solve. It in­volves not only in­fini­ties, but also a hy­poth­e­sis space whose car­di­nal­ity and par­tic­u­lar mem­bers are not ob­vi­ous.

Thanks to Me­stroyer for bring­ing this prob­lem to my at­ten­tion, Man­fred for helping me work through it, and Vin­cen­tYu for point­ing out the is­sues sur­round­ing ad­di­tive util­i­tar­i­anism as a de­ci­sion paradigm.

• The an­swer [ed­ited Oct 13, 2013]

As sev­eral com­menters have pointed out, the origi­nal prob­lem does not sup­ply a method for tak­ing limits. Our anal­y­sis shows that the prob­lem is ill posed: it has no unique solu­tion un­less we take on ad­di­tional as­sump­tions.

I dis­agree that the math­e­mat­ics of origi­nal prob­lem is ill-posed, and I think DanielLC made the same point. The point of con­tention seems to cen­ter on the use of in­fini­ties in the origi­nal prob­lem, which is in­deed an is­sue if they were ma­nipu­lated as real num­bers, but they were not. It is perfectly ac­cept­able and math­e­mat­i­cal to have a countably in­finite set of ob­jects, and to define a se­quence of sub­sets cor­re­spond­ing to the time evolu­tion of that set. In­finite sets are not defined as the limit of some se­quence of finite sets! There is no am­bi­guity in the math­e­mat­ics of origi­nal prob­lem.*

Be­cause the use of in­finity in the origi­nal prob­lem is not in the sense of a limit, there is no good rea­son to think that we should take limits, or that the limits of the solu­tions to the finite prob­lems should cor­re­spond in any way to the solu­tions of the origi­nal prob­lem.

Where there are am­bi­gui­ties are in the use of the word “util­ity” and similar con­cepts as though they were well-defined in this con­text. And in this sense, I agree that the origi­nal prob­lem is ill-posed.

* There are math­e­mat­i­cal am­bi­gui­ties in an un­fa­vor­able read­ing of the origi­nal prob­lem, but the fol­low­ing steel­man re­moves them: Bi­ject the peo­ple with the nat­u­ral num­bers, and then trans­fer the nth per­son on day n.

• I added a sec­tion called “De­cid­ing how to de­cide” that (hope­fully) deals with this is­sue ap­pro­pri­ately. I also amended the con­clu­sion, and added you as an ac­knowl­edge­ment.

• One of the sim­plest and most in­tu­itive is ag­grega­tive util­i­tar­i­anism, in which we define a util­ity for each per­son, add them all to­gether, and make the choice with the larger to­tal util­ity.

I sug­gest us­ing the phrase “ad­di­tive util­i­tar­i­anism” rather than “ag­grega­tive util­i­tar­i­anism”. It was en­tirely my fault for say­ing ag­grega­tive util­i­tar­i­anism in my com­ment, which was a mis­nomer; I got it mixed up with ag­grega­tive con­se­quen­tial­ism. (All fla­vors of util­i­tar­i­anism are by defi­ni­tion ag­grega­tive be­cause they take into ac­count the util­ities from some col­lec­tion of be­ings, but not all fla­vors are ad­di­tive.)

Note: Vin­cen­tYu has pointed out in the com­ments be­low that VNM util­ity may be able to deal with the in­finites in this prob­lem with­out tak­ing limits.

Un­for­tu­nately, I think that as­cribes too much power to VNM util­ity func­tions (that term it­self is a LessWrongism; el­se­where, they would be called car­di­nal util­ity func­tions or just util­ity func­tions). If we had our hands on a VNM util­ity func­tion, we would be okay (we sim­ply ask it which op­tion it prefers!), but the VNM the­o­rem sim­ply as­serts the ex­is­tence of a util­ity func­tion given cer­tain ba­sic ax­ioms, and it doesn’t give us the util­ity func­tion! So, un­for­tu­nately, VNM util­ity also falls flat on its face un­less we already know what we pre­fer. (An im­por­tant point is that VNM util­ity func­tions can­not work with the “util­ity” de­scribed in the prob­lem. It’s an un­for­tu­nate his­tor­i­cal ac­ci­dent that the word “util­ity” is over­loaded, be­cause VNM util­ity re­quires care­ful han­dling.)

If we fail to spec­ify the ex­act type of de­ci­sion the­ory we’re us­ing, it is en­tirely un­clear whether tak­ing in­finite limits would lead to a self-con­sis­tent solu­tion.

If we want to make a de­ci­sion based on [ad­di­tive] util­ity, the in­finite prob­lem is ill posed; it has no unique solu­tion un­less we take on ad­di­tional as­sump­tions.

Great, I think these are good clar­ifi­ca­tions!

• Un­for­tu­nately, I think that as­cribes too much power to VNM util­ity func­tions (that term it­self is a LessWrongism; el­se­where, they would be called car­di­nal util­ity func­tions or just util­ity func­tions).

I ac­tu­ally don’t re­call see­ing the us­age “VNM util­ity func­tions” on less wrong at all, prior to this thread. It may have oc­curred pre­vi­ously but cer­tainly not with suffi­cient fre­quency as to be a ‘less­wrongism’. As you say, the “VNM” is un­nec­es­sary in that con­text since is all the VNM part does is say “it must have a util­ity func­tion be­cause it ad­heres to these ax­ioms”.

It is some­times nec­es­sary to ex­plic­itly re­fer to things other than ‘util­ity func­tions’ with a ‘VNM’ qual­ifier. This is largely to pre-empt pedants who, when read­ing un­qual­ified us­age ‘con­se­quen­tial­ist’, are not will­ing to as­sume that it refers to the only kind of con­se­quen­tial­ist that is ever sig­nifi­cantly dis­cussed here (those that have util­ity func­tions).

VNM util­ity also falls flat on its face un­less we already know what we pre­fer.

Not quite, but the point stands. The ac­tual re­quire­ment is that there is any way to col­lect any ev­i­dence at all about our prefer­ences (or, to be even more gen­eral, any way to cause out­comes to be cor­re­lated to our prefer­ence).

• For the mo­ment, I’m go­ing to strike the com­ment from the post. I don’t want to as­cribe a view­point to Vin­cen­tYu that he doesn’t ac­tu­ally hold.

• I would like to in­clude this is­sue in the post, but I want to make sure I un­der­stand it first. Tell me if this is right:

It is pos­si­ble math­e­mat­i­cally to rep­re­sent a countably in­finite num­ber of im­mor­tal peo­ple, as well as the pro­cess of mov­ing them be­tween spheres. Fur­ther, we should not ex­pect a pri­ori that a prob­lem in­volv­ing such in­fini­ties would have a solu­tion equiv­a­lent to those solu­tions reached by tak­ing in­finite limits of an analo­gous finite prob­lem. Some con­fu­sion arises when we in­tro­duce the con­cept of “util­ity” to de­ter­mine which of the two choices is bet­ter, since util­ity only serves as a ba­sis on which to make de­ci­sion for finite prob­lems.

If that’s what you’re say­ing, I have a cou­ple of ques­tions.

1. Do you view the para­dox as there­fore un­re­solv­able as stated, or would you claim that a differ­ent re­s­olu­tion is cor­rect?

2. If I care­fully re­stricted my claim about ill-posed­ness to the ques­tion of which choice is bet­ter from a util­i­tar­ian sense, would you agree with it?

• Tell me if this is right:

It is pos­si­ble math­e­mat­i­cally to rep­re­sent a countably in­finite num­ber of im­mor­tal peo­ple, as well as the pro­cess of mov­ing them be­tween spheres. Fur­ther, we should not ex­pect a pri­ori that a prob­lem in­volv­ing such in­fini­ties would have a solu­tion equiv­a­lent to those solu­tions reached by tak­ing in­finite limits of an analo­gous finite prob­lem.

That’s an ac­cu­rate in­ter­pre­ta­tion of my com­ment.

Some con­fu­sion arises when we in­tro­duce the con­cept of “util­ity” to de­ter­mine which of the two choices is bet­ter, since util­ity only serves as a ba­sis on which to make de­ci­sion for finite prob­lems.

I do think that con­fu­sion arises in this con­text from the con­cept of “util­ity”, but not be­cause “util­ity only serves as a ba­sis on which to make de­ci­sion for finite prob­lems.” The “util­ity” in the prob­lem is clearly not that of VNM-util­ity (of which I pre­vi­ously gave a brief ex­pla­na­tion) be­cause we not as­sign­ing util­ity to ac­tions, de­ci­sions, or choices (a VNM-util­ity func­tion U would gen­er­ally have no prob­lem re­spond­ing to an in­finite set of choices, as it sim­ply says: do argmax_{choice}(U(choice))). This severely un­der­mines what we can do with the “util­ity” in the prob­lem be­cause we are left with the var­i­ous fla­vors of ag­grega­tive util­i­tar­i­anism, which suffer from in­tractable prob­lems even in finite situ­a­tions! At­tempt­ing to ex­tend them to the situ­a­tion at hand is prob­le­matic (and, as Kaj_So­tala re­marked, deal­ing with in­fini­ties in ag­grega­tive con­se­quen­tial­ism is the topic of one of Bostrom’s pa­pers).

1. Do you view the para­dox as there­fore un­re­solv­able as stated, or would you claim that a differ­ent re­s­olu­tion is cor­rect?

I think that the ap­pear­ance of the para­dox is a con­se­quence of un­fa­mil­iar­ity with in­finite sets, and that it is not too sur­pris­ing that our in­tu­ition ap­pears to con­tra­dict it­self in this con­text (by pre­sent­ing each op­tion as bet­ter than the other). The con­tra­dic­tory in­tu­itions don’t cor­re­spond to a log­i­cal con­tra­dic­tion, so the ap­par­ent para­dox needs no re­s­olu­tion. The ac­tual prob­lem (choos­ing be­tween the two op­tions) is a mat­ter of prefer­ence, just as the choice be­tween straw­berry and choco­late is a mat­ter of prefer­ence.

2. If I care­fully re­stricted my claim about ill-posed­ness to the ques­tion of which choice is bet­ter from a util­i­tar­ian sense, would you agree with it?

Ab­solutely. I think ag­grega­tive util­i­tar­i­anism (as a moral the­ory) is screwed even in finite sce­nar­ios, much less in­finite sce­nar­ios. (But I also think ag­grega­tive util­i­tar­i­anism is a good but ill-defined gen­eral stan­dard for com­par­ing con­se­quences in real life.)

• Ok, I think I’ve got it. I’m not fa­mil­iar with VNM util­ity, and I’ll make sure to ed­u­cate my­self.

I’m go­ing to edit the post to re­flect this is­sue, but it may take me some time. It is clear (now that you point it out) that we can think of the ill-posed­ness com­ing from our in­sis­tence that the solu­tion con­form to ag­grega­tive util­i­tar­i­anism, and it may be pos­si­ble to sidestep the para­dox if we choose an­other paradigm of de­ci­sion the­ory. Still, I think it’s worth work­ing as an ex­am­ple, be­cause, as you say, AU is a good gen­eral stan­dard, and many read­ers will be fa­mil­iar with it. At the min­i­mum, this would be an in­ter­est­ing finite AU de­ci­sion prob­lem.

Thanks for all the time you’ve put into this.

• How did you make those won­der­ful graphs?

• The plots were done in Math­e­mat­ica 9, and then I added the an­no­ta­tions in Pow­erPoint, in­clud­ing the dashed lines. I had to com­bine two color func­tions for the den­sity plot, since I wanted to high­light the fact that the line s=n rep­re­sented in­differ­ence. Here’s the code:

r = 1; ua = 1;ub = −1; f1[n, s] := (ns—s^2r ) (ua—ub); Show[Den­si­tyPlot[-f1[n, s], {n, 0, 20}, {s, 0, 20}, ColorFunc­tion → “Cher­ryTones”, Frame → False, PlotRange → {-1000, 0}], Den­si­tyPlot[f1[n, s], {n, 0, 20}, {s, 0, 20}, ColorFunc­tion → “BeachColors”, Frame → False, PlotRange → {-1000, 0}]]

• The Ross-Lit­tle­wood Para­dox is amus­ing.

You have an in­finite col­lec­tion of balls in a store­room, la­beled with the nat­u­ral num­bers (1, 2, 3, etc.) and a vase that can hold any num­ber, or all, of them.

At each in­te­ger time T, start­ing with T=1, you take the 10 low­est num­bered balls out of the store­room and put them in the vase, and then take the low­est num­bered ball out of the vase and de­stroy it. So at any finite time T, there are 9T balls in the vase and all the balls la­beled with a num­ber less than or equal to T have been de­stroyed.

Now, be­cause the num­ber of balls at any given time T is given by 9T, in the limit as T ap­proaches in­finity, there are in­finitely many balls in the vase. On the other hand, be­cause ev­ery ball has a time T at which it will be de­stroyed, the limit of the set of balls in the vase as T ap­proaches in­finity is the empty set. So at T = in­finity, you have an empty vase that con­tains in­finitely many balls.

The moral of the story is to be care­ful what limits you take, be­cause tak­ing two differ­ent limits can give two differ­ent an­swers even if they seem like they’re mea­sur­ing the same thing.

(Can this be used as an ar­gu­ment for the ex­is­tence of non­stan­dard num­bers?)

• This is ac­tu­ally a good ex­am­ple of the differ­ence be­tween poin­t­wise and uniform con­ver­gence. Con­sider the char­ac­ter­is­tic func­tion of the vase at time t, g_t : N → {0, 1}. Then g_t(n) = 1 if and only if ball n is in the vase at time t. It will ac­tu­ally help to con­vert g_t to a func­tion on the real num­bers, by mak­ing f_t(x) = g_t(floor(x)).

Now, for each ball n, there is a time when it will be de­stroyed, and there­fore will never be in the vase af­ter that time. So the char­ac­ter­is­tic func­tion f_t(x) con­verges poin­t­wise to f(x) = 0. This is pre­sum­ably what you mean by the limit of the vase be­ing the empty set.

But the crite­rion of uniform con­ver­gence is that for any ep­silon>0 there is a t such that f_t is within ep­silon of the limit ev­ery­where. Which is ob­vi­ously not true, be­cause at any time t there are some balls in the vase, and so the char­ac­ter­is­tic func­tion is 1 some­where. So f_t(x) does not uniformly con­verge to any­thing.

As it hap­pens, with­out uniform con­ver­gence, the limit of the in­te­grals of f_t(x) (which just so hap­pens to be the num­ber of balls in the vase, by my setup) is not gen­er­ally equal to the in­te­gral of the limit­ing func­tion f(x). So, in a way it is not re­ally true that you can say

in the limit as T ap­proaches in­finity, there are in­finitely many balls in the vase

as the in­te­gral does not trans­fer to the limit.

• Great prob­lem, thanks for men­tion­ing it!

I think the an­swer to “how many balls did you put in the vase as T->\in­fty” and “How many balls have been de­stroyed as T->\in­fty” both have well defined an­swers. It’s just a fal­lacy to as­sume that the “to­tal num­ber of balls in the vase as T->\in­fty” is equal to the differ­ence be­tween these quan­tities in their limits.

• On the difficul­ties that in­fini­ties pose to util­i­tar­i­anism, see also Nick Bostrom’s In­finite Ethics.

• You make a strong case for in­finite set athe­ism.

• Two of the best fun­da­men­tal prob­a­bil­ity guys, Jaynes and Wolpert, both ba­si­cally said that the ap­pli­ca­tions of their the­o­ries to in­finite sets were un­nec­es­sary and likely more trou­ble than they’re worth.

• Yeah I read PT:LOS and I’d like to be able to say that, but in­finite ethics doesn’t re­ally look to be so eas­ily swept un­der the rug.

Would you recom­mend The Math­e­mat­ics of Gen­er­al­iza­tion by Wolpert, and/​or some­thing else?

• Ama­zon doesn’t provide an in­dex, but the ti­tle was promis­ing enough that I bought one. The date looks good too, as it was af­ter I know much of the origi­nal pa­pers were com­pleted.

What you want are his gen­eral frame­work for an­a­lyz­ing gen­er­al­iza­tion prob­lems, and his ap­pli­ca­tion of that frame­work to Stacked Gen­er­al­iza­tion and No Free Lunch The­o­rems in ma­chine learn­ing and Search/​Op­ti­miza­tion.

Sorry I don’t have bet­ter de­tails, but the pa­pers are in stor­age, and I read them 15+ years ago.

• Thanks any­way, I’ll look up the pa­pers. :)

• Nice.

Bug re­ports:

Note that even though t is 1, we’ll end up with a unit er­ror if we don’t carry it around.

I think you meant r here.

And you have flipped the sign of \Delta U_{lin­ear}.

• Fixed. Thanks for read­ing so closely. It’s amaz­ing how many lit­tle mis­takes can sur­vive af­ter 10 read-throughs.

• Another small typo: Un­der “The Finite Prob­lem”, n_{A,s} and n_{B,s} should be n_{A,t} and n_{B,t} in­stead.

• Thanks! Do you guys want to copy edit my jour­nal pa­pers? ;)

• Does it mat­ter if the num­ber of peo­ple is countably in­finite, or un­countably in­finite?

If each per­son cor­re­sponds on a 1-1 ba­sis with the real num­bers, there are an in­finite num­ber peo­ple who will not be se­lected to change spheres on any of the in­te­ger-num­bered days. Those peo­ple will never change spheres.

• Just com­pare the car­di­nal­ity of the num­ber of days to the car­di­nal­ity of the num­ber of peo­ple. If |days| < |peo­ple| then start them in the heaven sphere. If |days| = |peo­ple| then it doesn’t mat­ter (by sym­me­try the first are last, and the last first, so to speak). If |days| > |peo­ple| then start them in the hell sphere.

My first im­pres­sion was the same as yours, but then I re­al­ized there was no guaran­tee about any of the car­di­nal­ity, even for the set of days. The post as­sumes the re­als, but com­par­ing the car­di­nal­ity should work for any sets (al­though if they’re big­ger than the re­als can we re­ally com­pare “util­ity” at all?)

• I think that’s a bet­ter state­ment of what I tried to say.

• In the above ex­am­ple, the num­ber of peo­ple and the num­ber of days they live were un­countable, if I’m not mis­taken. The take-home mes­sage is that you do not get an an­swer if you just eval­u­ate the prob­lem for sets like that, but you might if you take a limit.

Con­clu­sions that in­volve in­finity don’t map uniquely on to finite solu­tions be­cause they don’t sup­ply enough in­for­ma­tion. Above, “in­finite im­mor­tal peo­ple” refers to a con­cept that en­cap­su­lates three differ­ent an­swers. We had to in­vent a new pa­ram­e­ter, alpha, which was not sup­plied in the origi­nal prob­lem, to come up with a well defined re­sult. In essence, we didn’t ac­tu­ally an­swer the ques­tion. We made up our own prob­lem that was similar to the origi­nal one.

• Pro­vided you can as­sign a unique ra­tio­nal num­ber to each day each per­son lives, they are countable.

I will note that the ex­pected time for a given per­son to re­main in the sphere in which they started is in­finite, pro­vided they don’t know in what or­der they will be re­moved. The sum­ma­tion for each day be­comes (to­tal of an in­finite num­ber of peo­ple)+(to­tal of a finite num­ber of peo­ple); if we as­sume that a per­son-day in bliss is pos­i­tive and a per­son-day in agony is nega­tive, then the an­swer is triv­ial. An in­finite sum­ma­tion of terms of pos­i­tive in­finity is greater than an in­finite sum of terms of nega­tive in­finity- the car­di­nal­ities are ir­rele­vant.

• Thanks for clear­ing up the countabil­ity. It’s clear that there are some cases where tak­ing limits will fail (like when the util­ity is dis­con­tin­u­ous at in­finity), but I don’t have an in­tu­ition about how that is­sue is re­lated to countabil­ity.

• You said ‘dis­con­tin­u­ous at in­finity’. Did you mean ‘the in­finite limit di­verges or oth­er­wise does not ex­ist’?

• No, I mean a func­tion whose limit doesn’t equal its defined value at in­finity. As a triv­ial ex­am­ple, I could define a util­ity func­tion to be 1 for all real num­bers in [-inf,+inf) and 0 for +inf. The func­tion could never ac­tu­ally be eval­u­ated at in­finity, so I’m not sure what it would mean, but I couldn’t claim that the limit was giv­ing me the “cor­rect” an­swer.

• The func­tion could never ac­tu­ally be eval­u­ated at in­finity, so I’m not sure what it would mean, but I couldn’t claim that the limit was giv­ing me the “cor­rect” an­swer.

If you ac­cept the Ax­iom of In­finity, there’s no prob­lem at eval­u­at­ing a func­tion at in­finity. The prob­lem is rather that omega is a reg­u­lar limit car­di­nal, so there’s no way to define the value at in­finity from the value at the suc­ces­sor, un­less you in­clude in the defi­ni­tion an ex­plicit step for limit car­di­nals.
You can very well define a func­tion that has 1 as value on 0 and on ev­ery suc­ces­sor car­di­nal, but 0 on ev­ery limit car­di­nal. The func­tion will in­deed be dis­con­tin­u­ous, but its value at omega will be perfectly defined (I just did).

• The prob­lem with say­ing a func­tion is not con­tin­u­ous at in­finity is that the defi­ni­tions of ‘con­tin­u­ous’ re­quires the stan­dard defi­ni­tion of ‘limit’ (sigma-ep­silon), while the defi­ni­tion of limits at in­finity uses the same nomen­cla­ture and similar no­ta­tion, but ex­presses some­thing differ­ent.

Con­sider the case where F(X-ep­silon) is 1, F(X) is 0, and F(X+ep­silon) is ei­ther 0 or un­defined. The com­mon thought there is that the limit at X does not ex­ist; why is that any differ­ent just be­cause X is in­finite, with­out sac­ri­fic­ing the con­cept which al­lows us to talk about con­ti­nu­ity in terms of limits?

• The prob­lem with say­ing a func­tion is not con­tin­u­ous at in­finity is that the defi­ni­tions of ‘con­tin­u­ous’ re­quires the stan­dard defi­ni­tion of ‘limit’ (sigma-ep­silon), while the defi­ni­tion of limits at in­finity uses the same nomen­cla­ture and similar no­ta­tion, but ex­presses some­thing differ­ent.

Well… I guess you can see it that way if you want, but in set the­ory (again, ev­ery­thing I say is un­der the ax­iom of in­finity) both no­tions are unified un­der the no­tion of limit in the or­der topol­ogy.
In this way, you can define a con­tin­u­ous func­tion for ev­ery trans­finite or­di­nal.

Con­sider the case where F(X-ep­silon) is 1, F(X) is 0, and F(X+ep­silon) is ei­ther 0 or un­defined. The com­mon thought there is that the limit at X does not ex­ist; why is that any differ­ent just be­cause X is in­finite, with­out sac­ri­fic­ing the con­cept which al­lows us to talk about con­ti­nu­ity in terms of limits?

Yes, I un­der­stand that the con­cept of limit in calcu­lus and set the­ory means some­thing a lit­tle differ­ent. Pos­si­bly this is just ar­gu­ing over defi­ni­tions: in calcu­lus, it is said that a limit doesn’t ex­ist when the func­tion has a differ­ent value w.r.t. the value calcu­lated us­ing the topol­ogy of its do­main, but in set the­ory a limit is defined in a differ­ent way, us­ing only the or­der topol­ogy. In this sense, a func­tion can be defined at omega, at omega+1, omega+2, etc. After all, that’s the rai­son d’etre of the en­tire con­cept. Un­der this as­sump­tion, you just say that if the func­tion is defined at omega, and if it has a differ­ent value at omega than the one defined from its or­der topol­ogy, you just say that it’s dis­con­tin­u­ous.
Let me clar­ify with an ex­am­ple: the func­tion y = 2x, defined in set the­ory, would have at omega the limit omega (demon­stra­tion be­low). If you just define a similar func­tion but that has at omega the value 0, then you have a dis­con­tin­u­ous func­tion, be­cause the (topolog­i­cal) limit is differ­ent from the defined value.
Of course, if you want, you can just say that the (sim­ple) limit doesn’t ex­ists when this situ­a­tion arise, I was on the other side point­ing to the fact that a func­tion can be perfectly defined and con­tin­u­ous or dis­con­tin­u­ous at in­finity.

Th­e­sis
Omega is the limit of y=2x, defined on the nat­u­ral num­ber.
De­mon­stra­tion
The in­ter­sec­tion of the range of the func­tion with omega is still the range of the func­tion.
The range is un­bounded but ev­ery one of its mem­ber is finite: since omega is reg­u­lar limit, it is never reached by the se­quence. So, any or­di­nal greater then or equal to omega is an up­per bound. But omega is also an ini­tial or­di­nal, so is the least up­per bound.
That is, the least up­per bound of the in­ter­sec­tion of the range of the func­tion with omega is still omega, so by defi­ni­tion omega is the limit of the range of the func­tion. QED.

• TIL that math­e­mat­i­ci­ans com­monly strictly define terms in one con­text, then ex­tend them into other con­texts in ways that are not strictly com­pat­i­ble with the origi­nal con­text.

Now I un­der­stand that two peo­ple with sig­nifi­cantly differ­ent lev­els of math ed­u­ca­tion and un­der­stand­ing lack the com­mon vo­cab­u­lary re­quired to triv­ially com­mu­ni­cate ba­sic math-re­lated con­cepts.

It’s still go­ing to be hard to con­vince me that the sum of an in­finite num­ber of days, each of which has in­finite pos­i­tive util­ity and finite nega­tive util­ity, will ever be lower than zero.

• It’s still go­ing to be hard to con­vince me that the sum of an in­finite num­ber of days, each of which has in­finite pos­i­tive util­ity and finite nega­tive util­ity, will ever be lower than zero.

Re­mem­ber you’re not al­lowed to talk about in­finity ex­cept as a limit.

Con­sider the se­quence:

1. 1 per­son, 3 days—util­ity 1 − 1 − 1 = −1
2. 2 peo­ple, 6 days—util­ity 2 + 0 − 2 − 2 − 2 − 2 = −6
3. 3 peo­ple, 9 days—util­ity 3 + 1 − 1 − 3 − 3 − 3 − 3 − 3 − 3 = −15
4. 4 peo­ple, 12 days—util­ity 4 + 2 + 0 − 2 − 4 − 4 − 4 − 4 − 4 − 4 − 4 − 4 = −28


etc.

Clearly the limit of this se­quence is the sum you’re talk­ing about (that is, the util­ity of an in­finite num­ber of im­mor­tal peo­ple who start in sphere A, where we move one each day to sphere B). At the same time, clearly the limit of this se­quence is nega­tive.

(Of course the “real” an­swer is that it’s not well defined; there are many se­quences we can con­struct that come out as “in­finite im­mor­tal peo­ple” in the limit, and the util­ity is differ­ent de­pend­ing which we pick. But this is an ex­am­ple of why “lower than zero” is as le­gi­t­i­mate an an­swer as “higher than zero”).

• Ex­cept that in the origi­nal prob­lem, there can­not be more days than peo­ple;

...each day, one more per­son gets per­ma­nently trans­ferred across to...

Then again, rephras­ing the prob­lem in equiv­a­lent ways has in­ter­est­ing effects:

As­sume that the num­ber of peo­ple is countable; as­sign each of them a nat­u­ral num­ber, but don’t tell them which one it is.

Sup­pose that on the Nth day you move the per­son with the Nth prime num­ber to the op­po­site sphere; ev­ery in­di­vi­d­ual prefers case 1, where they have 100% chance of in­finite hap­piness.

Sup­pose that you tell ev­ery­one their num­ber and on the Nth day you move the per­son with the Nth num­ber to the op­po­site sphere; ev­ery in­di­vi­d­ual prefers case 2, where they have a finite pe­riod of agony fol­lowed by in­finite hap­piness.

What’s the Er­dos num­ber of an in­finite num­ber of mon­keys jug­gling an in­finite num­ber of ba­nanas?

• Um… -inf and +inf are not real num­bers. (Not­ing that your func­tion as de­scribed is un­defined at -inf.)

In ad­di­tion, the defi­ni­tion of con­tin­u­ous re­stricts it to points which ex­ist on an open in­ter­val; if the limit from be­low and limit from above are equal to the value at X, then the func­tion is con­tin­u­ous on an open in­ter­val con­tain­ing X. How do you de­ter­mine the limit as X ap­proaches +inf from above?

• MrMind ex­plains in bet­ter lan­guage be­low.

• That can also hap­pen if there are countably in­finite peo­ple. Sup­pose that for each n, on the n-th day the 2n-th per­son is moved from hell to heaven. All the odd-num­bered peo­ple will stay in hell for­ever.

• Con­clu­sions that in­volve in­finity can­not gen­er­ally be gen­er­al­ized to any finite solu­tion; this seems like a ‘each mon­key now has two ba­nanas’ mo­ment.

• The finite analogue come from the fact that in­finity minus in­finity is un­defined, and can be any­thing from “still in­finity” (like if you had in­finity peo­ple and then only sent away the even num­bered ones) to any num­ber you choose, to nega­tive in­finity. In a finite prob­lem, the an­swer be­comes well-defined, but there are mul­ti­ple pos­si­ble an­swers.

• Yeah, that’s the point.

• Con­sider the con­verse: Is there a way to ar­range the days and peo­ple such that it is bet­ter to start in hell? Triv­ially, it seems like the sim­ple solu­tion is that way, since each per­son leaves hell af­ter a finite num­ber of days and then spends in­finity days in heaven, but I lack the con­cept which al­lows me to find the amount of time the av­er­age per­son spends in hell.

• By the way, are you talk­ing about this meme, or is there an­other prob­lem with mon­keys and ba­nanas?

• I was talk­ing about the math that spawned that meme, yes.

• Thanks to whomever moved this to Dis­cus­sion. From the FAQ, I wasn’t sure where to put it. This is bet­ter, in ret­ro­spect.

• I’d pre­fer to see this in Main, it is in­ter­est­ing and im­por­tant.

• I’m not sure why it got moved: maybe not cen­tral to the the­sis of LW, or maybe not high enough qual­ity. I’m go­ing to add some dis­cus­sion of counter-ar­gu­ments to the limit method. Maybe that will make a differ­ence.

I no­ticed that the dis­cus­sion picked up when it got moved, and I learned some use­ful stuff from it, so I’m not com­plain­ing.

• Very nice work.

I feel it ac­com­plishes too much for most peo­ple, though, so that they might not get ev­ery­thing out of it.

Great as a demon­stra­tion of Jaynes’ point about para­doxes gen­er­ate by failure to iden­tify the limit­ing pro­cess for pro­posed in­fini­ties. But also great as a demon­stra­tion of how “do­ing the math” can re­solve a lot of these philo­soph­i­cal de­bates which just don’t spec­ify prob­lem con­di­tions suffi­ciently be­cause they don’t ex­plic­itly write them down in math­e­mat­i­cal form.

I think a lot of the self refer­en­tial prob­lems many like around here (Omega stuff, as an ex­am­ple) would be similarly dis­solved by the lat­ter.

• I agree that it’s a lot to cover, but I wanted to work a full ex­am­ple. We talk a lot on LW about de­ci­sion anal­y­sis and para­doxes in the ab­stract, but I’m com­ing from a math/​physics back­ground, and it’s much more helpful for me to see con­crete ex­am­ples. I as­sume some other peo­ple feel the same way.

Self-refer­en­tial prob­lems would be an in­ter­est­ing area to study, but I’m not fa­mil­iar with the tech­niques. I sus­pect you’re right, though.

• My prob­lem with this is how do you know the proper way to take the limit? The sce­nario isn’t a limit. It’s just some­thing with in­fini­ties in it. The limit is some­thing we force onto it.

Another, finite, prob­lem oc­curred to me:

Sup­pose there are two bliss spheres, that are equally good. You can ei­ther spend one year in sphere A or one year in sphere B.

If you spend 1-1/​n years in sphere A vs. one year in sphere B, sphere B is always bet­ter. The limit of these cases is the origi­nal prob­lem, so sphere B must be bet­ter. Similarly, sphere A must be bet­ter.

Per­haps if you can get two an­swers from two differ­ent limits, that just means that they’re the same. The prob­lem is that it’s re­ally easy to get this to hap­pen. Just use some­thing similar to Pas­cal’s wa­ger. This would mean that ev­ery­thing is the same, which is a com­pletely use­less util­ity func­tion.

• My prob­lem with this is how do you know the proper way to take the limit? The sce­nario isn’t a limit. It’s just some­thing with in­fini­ties in it. The limit is some­thing we force onto it.

And as such, it is am­bigu­ous. The limit is how we re­solve the am­bi­guity.

• If you take a par­tic­u­lar con­verg­ing limit, that gives you a re­s­olu­tion to the am­bi­guity, but the idea of tak­ing a limit does not. You’re just pick­ing your an­swer when you pick your limit.

De­cid­ing what to do in situ­a­tions in­volv­ing in­fini­ties by in­vok­ing limits is not a well-defined de­ci­sion the­ory.

• De­cid­ing what to do in situ­a­tions in­volv­ing in­fini­ties by in­vok­ing limits is not a well-defined de­ci­sion the­ory.

Well yes, be­cause those situ­a­tions are not well-defined with­out some ad­di­tional struc­ture cap­tur­ing some­thing which also de­scribes the limit­ing pro­cess.

• There is no ad­di­tional struc­ture. It’s not as if we can come upon two pairs of spheres, and no­tice while the end re­sult is the same, they’re the limits of two differ­ent pro­cesses, and there­fore differ­ent choices are bet­ter. There is only the in­finite case. If you want to con­sider se­quence that con­verge to it, there’s no clear way to de­cide which se­quence to look at.

Limits help you if you’re look­ing at an ex­treme value. If the limit as the pop­u­la­tion goes to in­finity is that a ran­dom sam­ple of X of them will give you Y con­fi­dence on a poll, then you can just use that if there’s a large pop­u­la­tion. If you’re deal­ing with the limit it­self, it doesn’t always help. You can start with a square, and then cut lit­tle squares off of the cor­ners, and then more squares off of those cor­ners etc. un­til you ap­proach a cir­cle. The per­ime­ter will always be four times the length, but this won’t be true of the cir­cle.

In this prob­lem, you can get liter­ally any an­swer if you take the limit ap­pro­pri­ately, so once you’ve de­cided on the right an­swer, there is some way to get to it with the limit, but de­cid­ing that the right an­swer is one that a limit con­verges to helps you not at all.

The prob­lem isn’t a se­quence of finite cases. It’s just the in­finite case all by it­self.

• You’re com­pletely right! As stated, the prob­lem is ill posed, i.e. it has no unique solu­tion, so we didn’t solve it.

In­stead, we solved a similar prob­lem by in­tro­duc­ing a new pa­ram­e­ter, \alpha. It was use­ful be­cause we gained a math­e­mat­i­cal de­scrip­tion that works for very large n and s, and which matches our in­tu­ition about the prob­lem.

It is im­por­tant to rec­og­nize, as you point out, that that tak­ing limits does not solve the prob­lem. It just elu­ci­dates why we can’t solve it as stated.

• The fi­nal sec­tion has been ed­ited to re­flect the con­cerns of some of the com­menters.