Almost nothing in this post is correct. This post displays not just a misuse of and failure to understand surreal numbers, but a failure to understand cardinals, ordinals, free groups, lots of other things, and just how to think about such matters generally, much as in our last exchange. The fact that (as I write this) this is sitting at +32 is an embarrassment to this website. You really, really, need to go back and relearn all of this from scratch, because going by this post you don't have the slightest clue what you're talking about. I would encourage everyone else to stop upvoting this crap.
This whole post is just throwing words around and making assertions that assume things generalize in a particular naïve way that you expect. Well, they don't, and certainly not obviously.
Really, the whole idea here is wrong. The fact that something does not extend to infinities or infinitesimals is not somehow a paradox. Many things don't extend. There's nothing wrong with that. Some things, of course, do extend if you do things properly. Some things extend in more than one way, with none of them being more natural than the others! But if something doesnt extend, it doesn't extend. That's not a paradox.
Similarly, the fact that something has unexpected results is not a paradox. The right solution for some of these is just to actually formalize them and accept the results. No further "resolution" is required.
In the hopes of making my point absolutely clear, I am going to take these one by one. ~(As per the bullshit asymmetry principle, I'm afraid my response will be much longer than the original post.)~ (OK, I guess that turned out not to be true.) Those that involve philosophical problems in addition to just mathematical problems I will skip on my first pass, if you don't mind (well, some of them, anyway; and I may have slightly misjudged some of the ones I skipped, because, well, I skipped them -- point is I'm skipping some, it hardly matters, the rest are enough to demonstrate the point, but maybe I will get back to the skipped ones later). Note that I'm going to focus on problems involving infinities somehow; if there are problems not involving infinities I'll likely miss them.
Infinitarian paralysis: Skipping for now due to philosophical problems in addition to mathematical ones.
Paradox of the gods: You haven't stated your setup here formally, but if I try to formalize it (using real numbers as is probably appropriate here) I come to the conclusion that yes, the man cannot leave the starting point. Is this a "paradox"? No, it's just what you get if you actually formalize this. The continuum is counterintuitive! It doesn't quite fit our usual notions of causality! Think about differential equations for a moment -- is it a "paradox" that some differential equations have nonunique solutions, even though it seems that a particle's position, velocity, and relation between the two ought to "cause" its future trajectory? No! This is the same sort of thing; continuous time and continuous space do not work like discrete time and discrete space.
But in addition to your "resolution" being unnecessary, it's also nonsensical. You're taking the number of gods as a surreal number. That's nonsense. Surreal numbers are not for counting how many of something there are. Are you trying to map cardinals to surreals? I mean, yeah, you could define such a map, it's easy to do with AC, but is it meaningful? Not really. You do not count numbers of things with surreals, as you seem to be suggesting.
Of course, there's more than one way to measure the size of an infinite set, not just cardinality. Since you translate the number into a surreal, perhaps you meant the set of gods to be reverse well-ordered, so that you can talk about its reverse order type, as an ordinal, and take that as a surreal? That would go a little way to making this less nonsensical, but, well, you never said any such thing.
Of course, your solution seems to involve implicitly changing the setting to have surreal-valued time and space, but that makes sense -- it does make sense to try to make such "paradoxes" make more sense by extending the domain you're talking about. You might want to make more of an explicit note of it, though. Anyway, let's get back to nonsense.
So let's say we accept this reverse-well-ordering hypothesis. Does your "resolution" follow? Does it even make sense? No to both! First, your "resolution" isn't so much a deduction as a new assumption -- that these reverse-well-ordered gods are placed at positions 1/2^α for ordinals α. I mean, I guess that's a sensible extension of the setup, but... let's note here that you actually are changing the setup significantly at this point; the original setup pretty clearly had ω gods, not more. But, OK, that's fine -- you're generalizing, from the case of ω to the case of more. You should be more explicit that you're doing that, but I guess that's not wrong.
But your conclusion still is wrong. Why? Several reasons. Let's focus on the case of ω-many gods, that the original setup describes. You say that the man is stopped at 1/2^ω. Question: Why? Is 1/2^ω the minimum of the set {1/2^n : n ∈ N } inside the surreals? Well, obviously not, because that set obviously has no smallest element.
But is it the infimum (or equivalently limit), then, inside the surreals, if not the minimum? Actually, let's put that question aside for now and note that the answer to this question is actually irrelevant! Because if you accept the logic that the infimum (or equivalently limit) controls, then, guess what, you already have your resolution to the paradox back in the real numbers, where there's it's an infimum and it's 0. So all the rest of this is irrelevant.
But let's go on -- is it the infimum (or equivalently limit)? No! It's not! Because there is no infimum! A subsets of the surreals with no minimum also has no infimum, always, unconditionally! The surreal numbers are not at all like the real numbers. You basically can't do limits there, as we've already discussed. So there's nothing particularly distinguishing about the point 1/2^ω, no particular reason why that's where the man would stop. (There's no god there! We're talking about the case of ω gods, not ω+1 gods.)
We haven't even asked the question of what you mean by 2^s for a surreal s. I'm going to assume, since you're talking about surreals and didn't specify otherwise, that you mean exp(s log 2), using the usual surreal exponential. But, since you're only concerned with the case where s is an ordinal, maybe you actually meant taking 2^s using ordinal exponentiation, and then taking the reciprocal as a surreal. These are different, I hope you realize that!
What about if we use {left set|right set} intead of limits and infima? Well, there's now even less reason to believe that such a point has any relevance to this problem, but let's make a note of what we get. What is {|1, 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. 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.
OK, time for the second half, where I get to the errors in the ones I initially skipped. And yes, I'm going to assert some philosophical positions which (for whatever reason) aren't well-accepted on this site, but there's still plenty of mathematical errors to go around even once you ignore any philosphical problems. And yeah, I'm still going to point out missing formalism, but I will try to focus on the more substantive errors, of which there are plenty.
So, let's get those philosophical problems out of the way first, and quickly review utility functions and utilitarianism, because this applies to a bunch of what you discuss here. Like, this whole post takes a very naive view of the idea of "utility", and this needs some breaking down. Apologies if you already know all of what I'm about to say, but I think given the context it bears repeating.
So: There are two different things meant by "utility function". The first is decision-theoretic; an agent's utility function is a function whose expected value it attempts to maximize. The second is the one used by utilitarianism, which involves (at present, poorly-defined) "E-utility" functions, which are not utility functions in the decision-theoretic sense, that are then somehow aggregated (maybe by addition? who knows?) into a decision-theoretic utility function. Yes, this terminology is terribly confusing. But these are two separate things and need to be kept separate.
Basically, any agent that satisfies appropriate rationality conditions has a utility function in the decision-theoretic sense (obviously such idealized agents don't actually exist, but it's still a useful abstraction). So you could say, roughly speaking, any rational consequentialist has a decision-theoretic utility function. Whereas E-utility is specifically a utilitarian notion, rather than a general consequentalist or purely descriptive notion like decision-theoretic utility (it's also not at all clear how to define it).
Anyway, if you want surreal E-utility functions... well, I think that's still probably pretty dumb for reasons I'll get to, but since E-utility is so poorly defined that's not obviously wrong. But let's talk about decision-theoretic utility functions. These need to be real-valued for very good reasons.
Because, well, why use utility functions at all? What makes us think that a rational agent's preferences can be described in terms of a utility function in the first place? Well, there's an answer to that: Savage's theorem. I've already described this above -- it gives rationality conditions, phrased directly in terms of an agent's preferences, that together suffice to guarantee that said preferences can be described by a utility function. And yes, it's real-valued.
(And, OK, it's real-valued because Savage includes an Archimedean assumption, but, well -- do you think that's a bad assumption? Let me repeat here a naive argument against infinite and infinitesimal utilities I've seen before on this site (I forget due to who; I think Eliezer maybe?). Suppose we go with a naive treatment of infinitesimal utilities, and A has infinitesimal utility compared to B. Then since any action you take at all has some positive (real, non-infinitesimal) probability of bringing about B, even sitting in your room waving your hand back and forth in the air, A simply has no effect on your decision making; all considerations of B, even stupid ones, completely wash it out. Which means that A's infinitesimal utility does not, in fact, have any place in a decision-theoretic utility function. Do you really want to throw out that Archimedean assumption? Also if you do throw it out, I don't think that actually gets you non-real-valued utilities, I think it just, y'know, doesn't get you utilities. The agent's preferences can't necessarily be described with a utility function of any sort. Admittedly I could be wrong about that last part; I haven't checked.)
In short, your philosophical mistake here is of a kind with your mathematical mistakes -- in both cases, you're starting from a system of numbers (surreals) and trying awkwardly to fit it to the problem, even when it blatantly does not fit, does not have the properties that are required; rather than seeing what requirements the problem actually calls for and finding something that meets those needs. As I've pointed out multiple times by now, you're trying to make use of properties that the surreal numbers just don't have. Work forward from the requirements, don't try to force into them things that don't meet them!
By the way, Savage's theorem also shows that utility functions must be bounded. That utility functions must be bounded does not, for whatever reason, seem to be a well-accepted position on this site, but, well, it's correct so I'm going to continue asserting it, including here. :P Now it's true that the VNM theorem doesn't prove this, but that's due to a deficiency in the VNM theorem's assumptions, and with that gap fixed it does. I don't want to belabor this point here, so I'll just refer you to this previous discussion.
(Also the VNM theorem is just a worse foundation generally because it assumes real-valued probabilities to begin with, but that's a separate matter. Though maybe here it's not -- since you can't claim to avoid the boundedness requirement by saying you're justifying the use of utilities with VNM rather than Savage, since you seem to want to allow surreal-valued probabilities!)
Anyway, so, yes, utilities should be real-valued (and bounded) or else you have no good reason to use them -- to use surreal-valued utilities is to start from the assumption that you should use utilities (a big assumption! why would one ever assume such a thing?) when it should be a conclusion (a conclusion of theorems that say it must be real-valued).
Ah, but could infinities or infinitesimals appear in an E-utility function, that the utilitarians use? I've been ignoring those, after all. But, since they're getting aggregated into a decision-theoretic utility function, which is real-valued (or maybe it's not quite a decision-theoretic utility function, but it should still be real-valued by the naive argument above), unless this aggregation function can magnify an infinitesimal into a non-infinitesimal, the same problem will arise, the infinitesimals will still have no relevance, and thus should never have been included.
(Yeah, I suppose in what you write you consider "summing over an infinite number of people". But: 1. such infinite sums with infinitesimals don't actually work mathematically, for reasons I've already covered, and 2. you can't actually have an infinite number of people, so it's all moot anyway.)
Yikes, all that and I haven't even gotten to examining in detail the particular mathematical problems in the remaining ones! You know what, I'll end this here and split that comment out into a third post. Point is, now in these remaining ones, when I want to point out philosophical problems, I can just point back to this comment rather than repeating all this again.
OK, time to actually now get into what's wrong with the ones I skipped initially. Already wrote the intro above so not repeating that. Time to just go.
Infinitarian paralysis: So, philosophical problems to start: As an actual decision theory problem this is all moot since you can't actually have an infinite number of people. I.e. it's not clear why this is a problem at all. Secondly, naive assumption of utilitarian aggregation as mentioned above, etc, not going over this again. Enough of this, let's move on.
So what are the mathematical problems here? Well, you haven't said a lot here, but here's what it's look like to me. I think you've written one thing here that is essentially correct, which is that, if you did have some system of surreal valued-utilities, it would indeed likely make the distinction you want.
But, once again, that's a big "if", and not just for philosophical reasons but for the mathematical reasons I've already brought up so many times right now -- you can't do infinite sums in the surreals like you want, for reasons I've already covered. So there's a reason I included the word "likely" above, because if you did find an appropriate way of doing such a sum, I can't even necessarily guarantee that it would behave like you want (yes, finite sums should, but infinite sums require definition, and who knows if they'll actually be compatible with finite sums like they should be?).
But the really jarring thing here, the thing that really exposes a serious error in your thought (well, OK, that does so to a greater extent), is not in your proposed solution -- it's in what you contrast it with. Cardinal valued-utilities? Nothing about that makes sense! That's not a remotely well-defined alternative you can contrast with! And the thing that bugs me about this error is that it's just so unforced -- I mean, man, you could have said "extended reals" rather than cardinals, and made essentially the same point while making at least some sense! This is just demonstrating once again that not only do you not understand surreals, you do not understand cardinals or ordinals either.
(Well, I suppose technically there's the possibility that you do but expect your audience doesn't and are talking down to them, but since you're writing here on Less Wrong, I'm going to assume that's not the case.)
Seriously, cardinals and utilities do not go together. I mean, cardinals and real numbers do not go together. Like surreals and utilities don't go together either, but at least the surreals include the reals! At least you can attempt to treat it naively in special cases, as you've done in a number of these examples, even if the result probably isn't meaningful! Cardinals you can't even do that.
And once again, there's no reason anyone who understood cardinals would even want cardinal-valued utilities. That's just not what cardinals are for! Cardinals are for counting how many there are of something. Utility calculations are not a "how many" problem.
Sphere of suffering: Once again we have infinitely many people (so this whole problem is again a non-problem) and once again we have some sort of naive utility aggregation over those infinitely many people with all the mathematical problems that brings (only now it's over time-slices as well?). Enough of this, moving on.
Honestly I don't have much new to say about the bad mathematics here, much of it is the same sort of mistakes as you made in the ones I covered in my initial comment. To cover those ones briefly:
But OK. The one new thing here, I guess, is that now you're talking about a "majority" of the time slices? Yeah, that is once again not well-defined at all. Cardinality won't help you here, obviously; are you putting a measure on this somehow? I think you're going to have some problems there.
Trumped: Same problems I've discussed before. Surreal numbers do not count time steps, you're changing the problem by introducing a variable, utility aggregation over an infinite set (this time of time-slices rather than people), you know the drill.
But actually here you're changing the problem in a different way, by supposing that Trump knows in advance the number of time steps? The original problem just had this as a repeated offer. Maybe that's a philosophical rather than mathematical problem. Whatever. It's changing the problem, is the point.
And then on top of that your solution doesn't even make any sense. Let's suppose you meant an ordinal number of days rather than a surreal number of days, since that is what you'd actually use in this context. OK. Suppose for example then that the number of days is ω (which is, after all, the original problem before you changed it). So your solution says that Trump should accept the deal so long as the day number is less than the surreal number ω/3. Except, oops! Every ordinal less than ω is also less than ω/3. Trump always accepts the deal, we're back at the original problem.
I.e., even granting that you can somehow make all the formalism work, this is still just wrong.
St. Petersburg paradox: OK, so, there's a lot wrong here. Let me get the philosophical problem out of the way first -- the real solution to the St. Petersburg paradox is that you must look not at expected money, but at expected utility, and utility functions must be bounded, so this problem can't arise. But let's get to the math, because, like I said, there's a lot wrong here.
Let's get the easy-to-describe problems out of the way first: You are once again using surreals where you should be using ordinals; you are once again assuming some sort of theory of infinite sums of surreals; getting infinitely many heads has zero probability, not infinitesimal (probabilities are real-valued, you could try to introduce a theory of surreal probabilities but that will have problems already discussed), what happens in that case is irrelevant; you are once again changing the problem by allowing things to go on beyond ω steps; and, minor point, but where on earth did the function n |-> n comes from? Don't you mean n |-> 2^n?
OK, that's largely stuff I've said before. But the thing that puzzled me the most in your claimed solution is the first sentence:
If we model this with surreals, then simply stating that there is potentially an infinite number of tosses is undefined.
What? I mean, yeah, sure, the surreals have multiple infinities while, say, the extended nonnegative reals have only one, no question there. But that sentence still makes no sense! It, like, seems to reveal a fundamental misunderstanding so great I'm having trouble comprehending it. But I will give it my best shot.
So the thing is, that -- ignoring the issue of unbounded utility and what's the correct decision -- the original setup has no ambiguities. You can't choose to make it different by changing what system of numbers you describe it with. Now, I don't know if you're making the mistake I think you're making, because who knows what mistake you might be making, but it looks to me that you are confusing numbers that are part of the actual problem specification, with auxiliary numbers just used to describe the problem.
Like, what's actually going on here is that there is a set of coin flips, right? The elements of that set will be indexed by the natural numbers, and will form a (possibly improper, though with probability 0) initial segment of it -- those numbers are part of the actual problem specification. The idea though that there might be infinitely many coin flips... that's just a description. When I say "With probability 0, the set of flips will be infinite", that's just another way of saying, "With probability 0, the set of flips will be N." It doesn't make sense to ask "Ah, but what system of numbers are you using to measure its infinitude?" It doesn't matter! The set I'm describing is N! (And in any case I just said it was an infinite set, although I suppose you could say I was implicitly using cardinals.)
This is, I suppose, an idea that's shown up over and over in your claimed solutions, but since I skipped over this particular one before, I guess I never got it so explicitly before. Again, I'm having to guess what you think, but it looks to me like you think that the numbers are what's primary, rather than the actual objects the problems are about, and so you can just change the numbers system and get a different version of the same problem. I mean, OK, often the numbers are primary and you can do that! But sometimes they're just descriptive.
Oy. I have no idea whether I've correctly described what your misunderstanding, but whatever it is, it's pretty big. Let's just move on.
Trouble in St. Petersburg: Can I first just complain that your numbers don't seem to match up with your text? 13 is not 9*2+3. I'm just going to assume you meant 21 rather than 13, because none of the other interpretations I can come up with make sense.
Also this problem once again relies on unbounded utilities, but I don't need to go on about that. (Although if you were to somehow reformulate it without those -- though that doesn't seem possible in this coin-flip formulation -- then the problem would be basically similar to Satan's Apple. I have my own thoughts on that problem, but, well, I'm not going to go into it here because that's not the point.)
Anyway, let's get to the surreal abuse! Well, OK, again I don't have much new to say here, it's the same sort of surreal abuse as you've made before. Namely: Using surreals where they don't make sense (time steps should be counted by ordinals); changing the problem by introducing a transfinite variable; thinking that all ordinals are successor ordinals (sorry, but with n=ω, i.e. the original problem, there's still no last step).
Ultimately you don't offer any solution? Whatever. The errors above still stand.
The headache: More naive aggregation and thinking you can do infinite sums and etc. Or at least so I'm gathering from your claimed solution. Anyway that's boring.
The surreal abuse here though is also boring, same types as we've seen before -- using surreals where they make no sense but where ordinals would; ignoring the existence of limit ordinals; and of course the aforementioned infinite sums and such.
OK. That's all of them. I'm stopping there. I think the first comment was really enough to demonstrate my point, but now I can honestly claim to have addressed every one of your examples. Time to go sleep now.
My primary response to this comment will take the form of a post, but I should add that I wrote: "I will provide informal hints on how surreal numbers could help us solve some of these paradoxes, although the focus on this post is primarily categorisation, so please don't mistake these for formal proofs".
Your comment seems to completely ignore this stipulation. Take for example this:
"Of course, your solution seems to involve implicitly changing the setting to have surreal-valued time and space... You might want to make more of an explicit note of it, though"
Yes, there's a lot of philosophical groundwork that would need to be done to justify the surreal approach. That's why I said that it was only an informal hint.
I'm going to assume, since you're talking about surreals and didn't specify otherwise, that you mean exp(s log 2), using the usual surreal exponential
Yes, I actually did look up that there was a way of defining 2^s where s is a surreal number.
Let's accept the premise that you're using a surreal-valued probability measure instead of a real one
I wrote a summary of a paper by Chen and Rubio that provides the start of a surreal decision theory. This isn't a complete probability theory as it only supports finite additivity instead of countable additivity, but it suggests that this approach might be viable.
I could keep going, but I think I've made my point that you're evaluating these informal comments as though I'd claimed they were a formal proof. This post was already long enough and took enough time to write as is.
I will admit that I could have been clearer that many of these remarks were speculative, in the sense of being arguments that I believed were worth working towards formalising, even if all of the mathematical machinery doesn't necessarily exist at this time. My point is that justifying the use of surreals numbers doesn't necessarily involve solving every paradox; it should also be persuasive to solve a good number of them and then to demonstrate that there is good reason to believe that we may be able to solve the rest in the future. In this sense, informal arguments aren't valueless.
My primary response to this comment will take the form of a post, but I should add that I wrote: "I will provide informal hints on how surreal numbers could help us solve some of these paradoxes, although the focus on this post is primarily categorisation, so please don't mistake these for formal proofs".
You're right; I did miss that, thanks. It was perhaps unfair of me then to pick on such gaps in formalism. Unfortunately, this is only enough to rescue a small portion in the post. Ignoring the ones I skipped -- maybe it would be worth my time to get back to those after all -- I think the only one potentially rescued that way is the envelope problem. (I'm still skeptical that it is -- I haven't looked at it in enough detail to say -- but I'll grant you that it could be.)
(Edit: After rechecking, I guess I'd count Grandi's series and Thomson's lamp here too, but only barely, in the sense that -- after giving you quite a bit of benefit of the doubt -- yeah I guess you could define things that way but I see absolutely no reason why one would want to and I seriously doubt you gain anything from doing so. (I was about to include god picking a random integer here, too, but on rechecking again, no, that one still has serious other problems even if I give you more leeway than I initially did. Like, if you try to identify ∞ with a specific surreal, say ω, there's no surreal you can identify it with that will make your conclusion correct.))
The rest of the ones I pointed out as wrong (involving surreals, anyway) all contain more substantial errors. In some cases this becomes evident after doing the work and attempting to formalize your hints; in other cases they're evident immediately, and clearly do not work even informally.
The magic dartboard is a good example of the latter -- you've simply given an incorrect proof of why the magic dartboard construction works. In it you talk about ω_1 having a first half and a second half. You don't need to do any deep thinking about surreals to see the problem here -- that's just not what ω_1 looks like, at all. If you do follow the hint, and compare the elements of ω_1 to (ω_1)/2 in the surreals, then, as already noted, you find everything falls in the first half, which is not very helpful. (Again: This is the sort of thing that causes me to say, I suspect you need to relearn ordinals and probably other things, not just surreals. If you actually understand ordinals, you should not have any trouble proving that the magic dartboard acts as claimed, without any need to go into the surreals and perform division.)
Meanwhile the paradox of the gods is, as I've already laid out in detail, an example of the former. It sounds like a nice informal answer that could possibly be formalized, sure; but if you try to actually follow the hint and do that -- switching to surreal time and space as needed, of course -- it still makes no sense for the reasons I've described above. Because, e.g., ω is a limit ordinal and not a successor ordinal (this is a repeated mistake throughout the post, ignoring the existence of limit ordinals), because in the surreals there are no infima of sets (that aren't minima), because the fact that a surreal exponential exists doesn't mean that it acts like you want it to (algebraically it does everything you might want, but this problem isn't about algebraic properties) or that there's anything special about the points it picks out.
In addition, some of the things one is expected to just go with would require not just more explanation to formalize (like surreal integration) but to even make even informal sense of (like what structure you are putting on a set, or what you are embedding it in, that would make a surreal an appropriate measure of its size).
In short, your hints are not hints towards an already-existing solution (or at least, not one that anyone other than you would accept); they're analogy-driven speculation as to what a solution could look like. Obviously there's nothing wrong with analogy-driven speculation! I could definitely go on about some analogy-driven speculation of mine involving surreals! But, firstly, that's not what you presented it as; secondly, in most of your cases it's actually fairly easy (with a bit of relevant knowledge) to follow the breadcrumb trail and see that in fact it goes nowhere, as I did in my reply; and, thirdly, you're purporting to "solve" things that aren't actually problems in the first place. The second being the most important here, to be clear.
(And I think the ones I skipped demonstrate even more mathematical problems that I didn't get to, but, well, I haven't gotten to those.)
FWIW, I'd say surreal decision theory is a bad idea, because, well, Savage's theorem -- that's a lot of my philosophical objections right there. But I should get to the actual mathematical problems sometime; the philosophical objections, while important, are, I expect, not as interesting to you.
Basically, the post treats the surreals as a sort of device for automatically making the infinite behave like the finite. They're not. Yes, their structure as an ordered field (ordered exponential field, even) means that their algebraic behavior resembles such familiar finite settings as the real numbers, in contrast to the quite different arithmetic of (say) the ordinal or cardinal numbers (one might even include here the extended real line, with its mostly-all-absorbing ∞). But the things you're trying to do here often involve more than arithmetic or algebra, and then the analogies quickly fall apart. (Again, I'd see our previous exchange here for examples.)
This is quite a long post, so it may take some time to write a proper reply, but I'll get back to you when I can. The focus of this post was on gathering together all the infinite paradoxes that I could manage. I also added some informal thoughts on how surreal numbers could help us conceptualise the solution to these problems, although this wasn't the main focus (it was just convenient to put them in the same space).
Unfortunately, I haven't continued the sequence since I've been caught up with other things (travel, AI, applying for jobs), but hopefully I'll write up some new posts soon. I've actually become much less optimistic about surreal numbers for philosophical reasons which I'll write up soon. So my intent is for my next post to examine the definition of infinity and why this makes me less optimistic about this approach. After that, I want to write up a bit more formally how the surreal approach would work, because even though I'm less optimistic about this approach, perhaps someone else will disagree with my pessimism. Further, I think it's useful to understand how the surreal approach would try to resolve these problems, even if only to provide a solid target for criticism.
Surreal numbers are useless for all of these paradoxes.
Infinitarian paralysis: Using surreal-valued utilities creates more infinitarian paralysis than it solves, I think. You'll never take an opportunity to increase utility by because it will always have higher expected utility to focus all of your attention on trying to find ways to increase utility by , since there's some (however small) probability that such efforts would succeed, so the expected utility of focusing your efforts on looking for ways to increase utility by will have expected utility , which is higher than . I think a better solution would be to note that for any person, a nonzero fraction of people are close enough to identical to that person that they will make the same decisions, so any decision that anyone makes affects a nonzero fraction of people. Measure theory is probably a better framework than surreal numbers for formalizing what is meant by "fraction" here.
Paradox of the gods: The introduction of surreal numbers solves nothing. Why wouldn't he be able to advance more than miles if no gods erect any barriers until he advances miles for some finite ?
Two-envelopes paradox: it doesn't make sense to model your uncertainty over how much money is in the first envelope with a uniform surreal-valued probability distribution on for an infinite surreal , because then the probability that there is a finite amount of money in the envelope is infinitesimal, but we're trying to model the situation in which we know there's a finite amount of money in the envelope and just have no idea which finite amount.
Sphere of suffering: Surreal numbers are not the right tool for measuring the volume of Euclidean space or the duration of forever.
Hilbert hotel: As you mentioned, using surreals in the way you propose changes the problem.
Trumped, Trouble in St. Petersburg, Soccer teams, Can God choose an integer at random?, The Headache: Using surreals in the way you propose in each of these changes the problems in exactly the same way it does for the Hilbert hotel.
St. Petersburg paradox: If you pay infinity dollars to play the game, then you lose infinity dollars with probability 1. Doesn't sound like a great deal.
Banach-Tarski Paradox: The free group only consists of sequences of finite length.
The Magic Dartboard: First, a nitpick: that proof relies on the continuum hypothesis, which is independent of ZFC. Aside from that, the proof is correct, which means any resolution along the lines you're imagining that imply that no magic dartboards exist is going to imply that the continuum hypothesis is false. Worse, the fact that for any countable ordinal, there are countably many smaller countable ordinals and uncountably many larger countable ordinals follows from very minimal mathematical assumptions, and is often used in descriptive set theory without bringing in the continuum hypothesis at all, so if you start trying to change math to make sense of "the second half of the countable ordinals", you're going to have a bad time.
Parity paradoxes: The lengths of the sequences involved here are the ordinal , not a surreal number. You might object that there is also a surreal number called , but this is different from the ordinal . Arithmetic operations act differently on ordinals than they do on the copies of those ordinals in the surreal numbers, so there's no reasonable sense in which the surreals contain the ordinals. Example: if you add another element to the beginning of either sequence (i.e. flip the switch at , or add a at the beginning of the sum, respectively), then you've added one thing, so the surreal number should increase by , but the order-type is unchanged, so the ordinal remains the same.
Thanks for your feedback. I'll note that these are only informal hints/thoughts on how surreal numbers could help us here and that I'll be providing a more developed version of some of these thoughts in a future post.
Infinitarian paralysis: I consider Pascal's Mugging to be its own seperate problem. Indeed Pascal's Mugging type issues are already present with the more standard infinities. In any case, the measure theory solution is dependent on an individual being a finite fraction of the agents in the universe. While this is an extremely plausible assumption, there doesn't seem to be any principled reason why our solution to infinite paralysis should depend on this assumption.
Paradox of the Gods, Banach-Tarski: Your complaint is that I'm disallowing the a sequence consisting of all finite inverses/all finite integers. I actually believe that actual and potential infinity models of these problems need to be treated separately, though I've only outlined how I plan to handle actual infinities. Hopefully, you find my next post on this topic more persuasive.
Two-envelopes paradox: "The probability that there is a finite amount of money in the envelope is infinitesimal" - Hmm, you're right. That is a rather significant issue.
Sphere of suffering: "Surreal numbers are not the right tool for measuring the volume of Euclidean space or the duration of forever" - why?
St Petersberg Paradox: Ah, but you have an infinitesimal chance of winning a higher infinity. So it becomes an even more extreme version of Pascal's Mugging, but again that's its own discussion.
Magic Dartboard: Yes, I'm aware that rejecting the existence of magic dartboards could have far-reaching consequences. It's something I hope to look into more.
Parity: See my response to gjm. Ordinal numbers lack resolution and so can't properly describe the length of sequence.
Indeed Pascal's Mugging type issues are already present with the more standard infinities.
Right, infinity of any kind (surreal or otherwise) doesn't belong in decision theory.
"Surreal numbers are not the right tool for measuring the volume of Euclidean space or the duration of forever" - why?
How would you? If you do something like taking an increasing sequence of bounded subsets that fill up the space you're trying to measure, find a formula f(n) for the volume of the nth subset, and plug in , the result will be highly dependent on which increasing sequence of bounded subsets you use. Did you have a different proposal? It's sort of hard to explain why no method for measuring volumes using surreal numbers can possibly work well, though I am confident it is true. At the very least, volume-preserving transformations like shifting everything 1 meter to the left or rotating everything around some axis cease to be volume-preserving, though I don't know if you'd find this convincing.
"Indeed Pascal's Mugging type issues are already present with the more standard infinities."
Right, infinity of any kind (surreal or otherwise) doesn't belong in decision theory.
But Pascal's Mugging type issues are present with large finite numbers, as well. Do you bite the bullet in the finite case, or do you think that unbounded utility functions don't belong in decision theory, either?
The latter. It doesn't even make sense to speak of maximizing the expectation of an unbounded utility function, because unbounded functions don't even have expectations with respect to all probability distributions.
There is a way out of this that you could take, which is to only insist that the utility function has to have an expectation with respect to probability distributions in some restricted class, if you know your options are all going to be from that restricted class. I don't find this very satisfying, but it works. And it offers its own solution to Pascal's mugging, by insisting that any outcome whose utility is on the scale of 3^^^3 has prior probability on the scale of 1/(3^^^3) or lower.
There's definitely a part of me wondering if infinities exist, but before I even consider tackling that question, I need to figure out the most consistent interpretation of infinities assuming they exist.
"At the very least, volume-preserving transformations like shifting everything 1 meter to the left or rotating everything around some axis cease to be volume-preserving, though I don't know if you'd find this convincing" - Well there are non-measurable sets that do this without surreals, but do surreals add more examples?
I'll have to read more about how surreals apply to volumes. It may be hard getting convergence to the exact infinitesimal, but I don't know if the problems will extent beyond that.
(Also, the ability to integrate is mostly besides the point. Instead of the sphere of suffering, we could have defined the expanding cube of suffering. This will then let us solve some special cases of the sphere of suffering)
There are measurable sets whose volumes will not be preserved if you try to measure them with surreal numbers. For example, consider . Say its measure is some infinite surreal number . The volume-preserving left-shift operation sends to , which has measure , since has measure . You can do essentially the same thing in higher dimensions, and the shift operation in two dimensions () can be expressed as the composition of two rotations, so rotations can't be volume-preserving either. And since different rotations will have to fail to preserve volumes in different ways, this will break symmetries of the plane.
I wouldn't say that volume-preserving transformations fail to preserve volume on non-measurable sets, just that non-measurable sets don't even have measures that could be preserved or not preserved. Failing to preserve measures of sets that you have assigned measures to is entirely different. Non-measurable sets also don't arise in mathematical practice; half-spaces do. I'm also skeptical of the existence of non-measurable sets, but the non-existence of non-measurable sets is a far bolder claim than anything else I've said here.
Well shifting left produces a superset of the original, so of course we shouldn't expect that to preserve measure.
What about rotations, and the fact that we're talking about destroying a bunch of symmetry of the plane?
I'm happy to bite that bullet and destroy the symmetry. If we pick a random point and line in the universe, are there more unit points to the left or right? Well, that depends on where the point is.
It's a bad bullet to bite. Its symmetries are essential to what makes Euclidean space interesting.
And here's another one: are you not bothered by the lack of countable additivity? Suppose you say that the volume of Euclidean space is some surreal number . Euclidean space is the union of an increasing sequence of balls. The volumes of these balls are all finite, in particular, less than , so how can you justify saying that their union has volume greater than ?
"Its symmetries are essential to what makes Euclidean space interesting" - Isn't the interesting aspect of Euclidean space its ability to model our world excluding relativity?
Well, I just don't think it's that unusual for functions to have properties that break at their limits. Is this any different from 1/x being definable everywhere except 0? Is there anything that makes the change at the limit particularly concerning.
I don't follow the analogy to 1/x being a partial function that you're getting at.
Maybe a better way to explain what I'm getting at is that it's really the same issue that I pointed out for the two-envelopes problem, where you know the amount of money in each envelope is finite, but the uniform distribution up to an infinite surreal would suggest that the probability that the amount of money is finite is infinitesimal. Suppose you say that the size of the ray is an infinite surreal number . The size of the portion of this ray that is distance at least from is when is a positive real, so presumably you would also want this to be so for surreal . But using, say, , every point in is within distance of , but this rule would say that the measure of the portion of the ray that is farther than from is ; that is, almost all of the measure of is concentrated on the empty set.
As I understand it, there is not yet a good theory of integration on the surreals. Partial progress has been made, but there are also some negative results establishing limitations on the possibilities. Here is a recent paper.
In "Trumped", it seems that if , the first infinite ordinal, then on every subsequent day, the remaining number of days will be for some natural . This is never equal to .
Put differently, just because we count up to doesn't mean we pass through . Of course, the total order on days has has for each finite , but this isn't a well-order anymore so I'm not sure what you mean when you say there's a sequence of decisions. Do you know what you mean?
"Put differently, just because we count up to n doesn't mean we pass through n/3" - The first possible objection I'll deal with is not what I think you are complaining about, but I think it's worth handling anyway. n/3 mightn't be an Omnific Number, but in this case we just take the integer part of n/3.
I think the issue you are highlighting is that all finite numbers are less than n/3. And if you define a sequence as consisting of finite numbers then it'll never include n/3. However, if you define a sequence as all numbers between 1 and x where x is a surreal number then you don't encounter this issue. Is this valid? I would argue that this question boils down to whether there are an actually infinite number of days on which Trump experiences or only a potential infinity. If it's an actual infinity, then the surreal solution seems fine and we can say that Trump should stop saying yes on day n/3. If it's only a potential infinity, then this solution doesn't work, but I don't endorse surreals in this case (still reading about this)
If the number of days is, specifically, , then the days are numbered 0, 1, 2, ..., with precisely the (ordinary, finite) non-negative integers occurring. They are all smaller than . The number isn't the limit or the least upper bound of those finite integers, merely the simplest thing bigger than them all.
If you are tempted to say "No! What I mean by calling the number of days is precisely that the days are numbered by all the omnific integers below ." then you lose the ability to represent a situation in which Trump suffers this indignity on a sequence of days with exactly the order-type of the first infinite ordinal , and that seems like a pretty serious bullet to bite. In particular, I think you can't call this a solution to the "Trumped" paradox, because my reading of it -- even as you tell it! -- is that it is all about a sequence of days whose order-type is .
Rather a lot of these paradoxes are about situations that involve limiting processes of a sort that doesn't seem like a good fit for surreal numbers (at least so far as I understand the current state of the art when it comes to limiting processes in the surreal numbers, which may not be very far).
I already pointed above to the distinction between absolute and potential infinities. I admit that the surreal solution assumes that we are dealing with an absolute infinity instead of a potential one, so let's just consider this case. You want to conceive of this problem as "a sequence whose order-type is ω", but from the surreal perspective this lacks resolution. Is the number of elements (surreal) ω, ω+1 or ω+1000? All of these are possible given that in the ordinals 1+ω=ω so we can add arbitrarily many numbers to the start of a sequence without changing its order type.
So I don't think the ordinary notion of sequence makes sense. In particular, it doesn't account for the fact that two sequences which appear to be the same in every place can actually be different if they have different lengths. Anyway, I'll try to untangle some of these issues in future posts, in particular I'm leaning towards hyperreals as a better fit for modelling potential infinities, but I'm still uncertain about how things will develop once I manage to look into this more.
So far as anyone knows, no actual processes in the actual world are accurately described by surreal numbers. If not, then I suggest the same goes for the "nearest possible worlds" in which, say, it is possible for Mr Trump to be faced with the sort of situation described under the heading "Trumped". But you can have, in a universe very much like ours, an endless succession of events of order-type . If the surreal numbers are not well suited to describing such situations, so much the worse for the surreal numbers.
And when you say "I don't think the ordinary notion of sequence makes sense", what it looks like to me is that you have looked at the ordinary notion of sequence, made the entirely arbitrary choice that you are only prepared to understand it in terms of surreal numbers, and indeed not only that but made the further arbitrary choice that you are only prepared to understand it if there turns out to be a uniquely defined surreal number that is the length of such a sequence, observed that there is not such a surreal number, and then said not "Oh, whoops, looks like I took a wrong turn in trying to model this situation" but "Bah, the thing I'm trying to model doesn't fit my preconceptions of what the model should look like, therefore the thing is wrong". You can't do that! Models exist to serve the things they model, not the other way around.
It's as if I'd just learned about the ordinals, decided that all infinite things needed to be described in terms of the ordinals, was asked something about a countably infinite set, observed that such a set is the same size as but also the same size as and , and said "I don't think the notion of countably infinite set makes sense". It makes perfectly good sense, I just (hypothetically) picked a bad way to think about it: ordinals are not the right tool for measuring the size of a (not-necessarily-well-ordered) set. And likewise, surreal numbers are not the right tool for measuring the length of a sequence.
Don't get me wrong; I love the surreal numbers, as an object of mathematical study. The theory is gorgeous. But you can't claim that the surreal numbers let you resolve all these paradoxes, when what they actually allow you to do is to replace the paradoxical situations with other entirely different situations and then deal with those, while rejecting the original situations merely because your way of trying to model them doesn't work out neatly.
Maybe I should re-emphasise the caveat at the top of the post: "I will provide informal hints on how surreal numbers could help us solve some of these paradoxes, although the focus on this post is primarily categorisation, so please don't mistake these for formal proofs. I'm also aware that simply noting that a formalisation provides a satisfactory solution doesn't philosophically justify its use, but this is also not the focus of this post."
You wrote that I "made the entirely arbitrary choice that you are only prepared to understand it in terms of surreal numbers". This choice isn't arbitrary. I've given some hints as to why I am taking this approach, but a full justification won't occur until future posts.
OK! I'll look forward to those future posts.
(I'm a big surreal number fan, despite the skeptical tone of my comments here, and I will be extremely interested to see what you're proposing.)
You want to conceive of this problem as "a sequence whose order-type is ω", but from the surreal perspective this lacks resolution. Is the number of elements (surreal) ω, ω+1 or ω+1000? All of these are possible given that in the ordinals 1+ω=ω so we can add arbitrarily many numbers to the start of a sequence without changing its order type.
It seems to me that measuring the lengths of sequences with surreals rather than ordinals is introducing fake resolution that shouldn't be there. If you start with an infinite constant sequence 1,1,1,1,1,1,..., and tell me the sequence has size , and then you add another 1 to the beginning to get 1,1,1,1,1,1,1,..., and you tell me the new sequence has size , I'll be like "uh, but those are the same sequence, though. How can they have different sizes?"
Because we should be working with labelled sequences rather than just sequences (that is sequences with a length attached). That solves the most obvious issues, though there are some subtleties there
Why? Plain sequences are a perfectly natural object of study. I'll echo gjm's criticism that you seem to be trying to "resolve" paradoxes by changing the definitions of the words people use so that they refer to unnatural concepts that have been gerrymandered to fit your solution, while refusing to talk about the natural concepts that people actually care about.
I don't think think your proposal is a good one for indexed sequences either. It is pretty weird that shifting the indices of your sequence over by 1 could change the size of the sequence.
Thanks for the suggestion. I took a look at nets, but their purpose seems mainly directed towards generalising limits to topological spaces, rather than adding extra nuance to what it means for a sequence to have infinite length. But perhaps you could clarify why you think that they are relevant?
A net is just a function where is an ordered index set. For limits in general topological spaces, might be pretty nasty, but in your case, you would want to be some totally-ordered subset of the surreals. For example, in the trump paradox, you probably want to:
include and for some infinite
have a least element (the first day)
It sounds like you also want some coherent notion of "tomorrow" at each day, so that you can get through all the days by passing from today to tomorrow infinitely many times. But this is equivalent to having your set be well-ordered, which is incompatible with the property "closed under division and subtraction by finite integers". So you should clarify which of these properties you want.
"But this is equivalent to having your set be well-ordered, which is incompatible with the property "closed under division and subtraction by finite integers"" - Why is this incompatible?
An ordered set is well-ordered iff every subset has a unique least element. If your set is closed under subtraction, you get infinite descending sequences such as . If your sequence is closed under division, you get infinite descending sequences that are furthermore bounded such as . It should be clear that the two linear orders I described are not well-orders.
A small order theory fact that is not totally on-topic but may help you gather intuition:
Every countable ordinal embeds into the reals but no uncountable ordinal does.
Okay, I now understand why closure under those operations is incompatible with being well-ordered. And I'm guessing you believe that well-ordering is necessary for a coherent notion of passing through tomorrow infinitely many times because it's a requirement for transfinite induction?
I'm not so sure that this is important. After all, we can imagine getting from 1 to 2 via passing through an infinite number of infinitesimally small steps even though [1,2] isn't well-ordered on <. Indeed, this is the central point of Zeno's paradox.
Yes, there are good ways to index sets other than well orders. A net where the index set is the real line and the function is continuous is usually called a path, and these are ubiquitous e.g. in the foundations of algebraic topology.
I guess you could say that I think well-orders are important to the picture at hand "because of transfinite induction" but a simpler way to state the same objection is that "tomorrow" = "the unique least element of the set of days not yet visited". If tomorrow always exists / is uniquely defined, then we've got a well-order. So something about the story has to change if we're not fitting into the ordinal box.
Your second example, 1 > 1/2 > 1/4 > ... > 0, is a well-order. To make it non-well-ordered, leave out the 0.
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!
Satan's Apple: Satan has cut a delicious apple into infinitely many pieces. Eve can take as many pieces as she likes, but if she takes infinitely many pieces she will be kicked out of paradise and this will outweigh the apple. For any finite number i, it seems like she should take that apple piece, but then she will end up taking infinitely many pieces.
Proposed solution for finite Eves (also a solution to Trumped, for finite Trumps who can't count to surreal numbers):
After having eaten n pieces, Eve's decision isn't between eating n pieces and eating n+1 pieces, it's between eating n pieces and whatever will happen if she eats the n+1st piece. If Eve knows that the future Eve will be following the strategy "always eat the next apple piece", then it's a bad decision to eat the n+1st piece (since it will lead to getting kicked out of paradise).
So what strategy should Eve follow? Consider the problem of programming a strategy that an Eve-bot will follow. In this case, the best strategy is the strategy that will lead to the largest amount of finite pieces being eaten. What this strategy is depends on the hardware, but if the hardware is finite, then there exists such a strategy (perhaps count the number of pieces and stop when you reach N, for the largest N you can store and compare with). Generalising to (finite) humans, the best strategy is the strategy that results in the largest amount of finite pieces eaten, among all strategies that a human can precommit to.
Of course, if we allow infinite hardware, then the problem is back again. But that's at least not a problem that I'll ever encounter, since I'm running on finite hardware.
We can definitely solve this problem for real agents, but the reason why I find this problem so perplexing is because of the boundary issue that it highlights. Imagine that we have an actual infinite number of people. Color all the finite placed people red and the non-finite placed people blue. Everyone one t the right of a red person should be red and everyone one to the left of blue person should be blue. So what does the boundary look like? Sure we can't finitely transverse from the start to the infinite numbers, but that doesn't affect the intuition that the boundary should still be there somewhere. And this makes me question whether the notion of an actual infinity is coherent (I really don't know).
My best guess about how to clear up confusion about "what the boundary looks like" is via mathematics rather than philosophy. For example, have you understood the properties of the long line?
Todo: Yablo's paradox
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:
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 a,b,a−1,b−1 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 S={1}∪S[a]∪S[b]∪S[a−1]∪S[b−1] .
Further S[a]={a}∪S[aa]∪S[ab]∪S[ab−1]
In other words, noting that there is a bijection between all sequences and all sequences starting with any symbol s, we can write S[a]≡S[a]∪S[b]∪S[b−1]
Breaking down S[b] similarly, we get:
S≡{1}∪(S[a]∪S[b]∪S[b−1])∪(S[a]∪S[b]∪S[a−1])∪S[a−1]∪S[b−1]
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?
With surreal numbers, this question will depend on whether the number of times that the switch is pressed is represented by an odd or even Omrific number, which will depend in a relatively complex manner on how we define the reals.
Grandi's Series: What is the sum of 1-1+1-1...?
Using surreal numbers, we can assign a length n to the series as merely saying that it is infinite lacks resolute. The sum then depends on whether n is even or odd.
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:
Now some of these method will count diameters multiple times, but even after these are excluded, we still obtain the same probabilities.
We need to bite the bullet here and say that all of these probabilities are valid. It's just that we can't just choose a "random chord" without specifying this more precisely. In other words, there isn't just a single set of chords, but multiple that can be defined in different ways.
Zeno's Paradoxes: There are technically multiple paradoxes, but let's go with the Dichtomy Paradox. Before you can walk a distance, you must go half way. But before you can get halfway, you must get a quarter-way and before that an eight of the way. So moving a finite distance requires an infinite number of tasks to be complete which is impossible.
It's generally consider uncontroversial and boring these days that infinite sequences can converge. But more interesting, this paradox seems to be a result of claiming an infinite amount of time intervals to diverge, whilst allowing an infinite number of space intervals to converge, which is a major inconsistency.
Skolem's Paradox: Any countable axiomisation of set theory has a countable model according to the Löwenheim–Skolem theorem, but Cantor's Theorem proves that there must be an uncountable set. It seems like this confusion arises from mixing up whether we want to know if there exists a set that contains uncountably many elements or if the set contains uncountably many elements in the model (the corresponding definition of membership in the model only refers to elements in the model). So at a high level, there doesn't seem to be very much interesting here, but I haven't dug enough into the philosophical discussion to verify that it isn't actually relevant.
This post was written with the support of the EA Hotel