An Extensive Categorisation of Infinite Paradoxes

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

What about if we use {left set|right set} intead of limits and infima? Well, there's now even less reason to believe that such a point has any relevance to this problem, but let's make a note of what we get. What is {|1, 1/2, 1/4, ...}? Well, it's 0, duh. OK, what if we exclude that by asking for {0|1, 1/2, 1/4, ...} instead? That's 1/ω. This isn't 1/2^ω; it's larger -- well, unless you meant "use ordinal exponentiation and then invert", in which case it is indeed equal and you need to be a hell of a lot clearer but it's all still irrelevant to anything. (Using ordinal exponentiation, 2^ω = ω; while using the surreal exponential, 2^ω = ω^(ω log 2) > ω.)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The headache: Skipping for now

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

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

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

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

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

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

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

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

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

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

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

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

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 $>\omega x$, 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 $>\omega x$ will have expected utility $>p\omega x$, 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 $2^{-\omega }$ 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 $[\frac{1}{n},n]$ 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 $\omega$, not a surreal number. You might object that there is also a surreal number called $\omega$, but this is different from the ordinal $\omega$. 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.

In "Trumped", it seems that if $n=\omega$, the first infinite ordinal, then on every subsequent day, the remaining number of days will be $\omega -k$ 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?

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

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

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

Great post!

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

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

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

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

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