## Tuesday, February 13, 2018

### Proper classes as merely possible sets

This probably won’t work out, but I’ve been thinking about the Cantor and Russell Paradoxes and proper classes and had this curious idea: Maybe proper classes are non-existent possible sets. Thus, there is actually no collection of all the sets in our world, but there is another possible world which contains a set S whose members are all the sets of our world. When we talk about proper classes, then, we are talking about merely possible sets.

Here is the story about the Russell Paradox. There can be a set R whose members are all the actual world’s non-self-membered sets. (In fact, since by the Axiom of Foundation, none of the actual world’s sets are self-membered, R is a set whose members are all the actual world’s sets.) But R is not itself one of the actual world’s sets, but a set in another possible world.

The story about Cantor’s Paradox that this yields is that there can be a cardinality greater than all the cardinalities in our world, but there actually isn’t. And in world w2 where such a cardinality exists, it isn’t the largest cardinality, because its powerset is even larger. But there is a third world which has a cardinality larger than any in w2.

It’s part of the story that there cannot be any collections with non-existent elements. Thus, one cannot form paradoxical cross-world collections, like the collection of all possible sets. The only collections there are on this story are sets. But we can talk of collections that would exist counterfactually.

The challenge to working out the details of this view is to explain why it is that some sets actually exist and others are merely possible. One thought is something like this: The sets that actually exist at w are those that form a minimal model of set theory that contains all the sets that can be specified using the concrete resources in the world. E.g., if the world contains an infinite sequence of coin tosses, it contains the set of the natural numbers corresponding to tosses with heads.

Angra Mainyu said...

I'd raise another issue: Let say that there is a possible world w1 at which there is some function F that does not exist in the actual world. Now on a common view of possible worlds (not mine, but on mine none of this applies so I'm leaving that aside), w1 exists at the actual world, and w1 is definite because it's a possible world. So, in which sense does F not exist at the actual world? Why can't we define a set using that function? The same, of course, goes for sets in general.

Granted, if, say, a three-headed dog exists at w1, that does not mean that it exists at the actual world. However, it seems we can properly define words, sets, etc., in terms of the dog in question, much in the way we can with mathematical assertions of existence.

Alexander R Pruss said...

Can you refer specifically to that function? If so, you might be able to use it to define a set in the actual world using the axiom of replacement (assuming that the axiom of replacement works for formulas in a language that goes beyond the language of set theory). But I suspect that in many cases, you won't be able to refer specifically to that function. In other words, you will know that at some world w1 there is a bunch of functions F satisfying some properties, but anchored as you are in the actual world you won't be able to single out a specific one. And to define a set with it via the axiom of replacement you'd need to single out one particular one.

Now, some sets that exist in other worlds, on this story, we can single out. For instance, the set of all of the actual world's sets. But the axiom of separation is limited to generating subsets of those sets that actually exist, so we can't generate the Russell set.

Brandon said...

Thinking about whether one could have an analogue to possible worlds in mathematics, it strikes me that Cantor's view of sets is arguably in the same very general family, although not exactly the same. If one takes consistent multiplicities to be related to possible worlds, and understands inconsistent multiplicities in Cantorian terms, I think it might be possible to argue that on the Cantorian view inconsistent multiplicities would be something like transworld 'regions' -- they cannot consistently be found in a possible world, but about which you can meaningfully speak (and, indeed, must be able to in a possible-worlds analysis). Thus on this view, proper classes would be the cross-world thing that can't be a set.

In pulling proper classes in as kinds of non-actual possibles, your suggestion seems harder to work out (although I don't know how well the previous works). Could one perhaps take a constructivist route for distinguishing sets that actually exist from the non-existent possible ones? I don't know enough about constructivism, unfortunately.

Angra Mainyu said...

Alex,

I can single it out in the way functions picked by the axiom of choice can be singled out. I gave an example in the other thread, but it applies here as well (at least, if we stipulate that the hypotheses in the other thread also hold; else, I would have to work more on it!). So, here goes (with more details this time):

O1: Let w9 be a possible world at which there is a function G from the set of natural numbers in w9 onto the set of real numbers in the interval (0,1) that exist at the actual world.
Let x be the following real number (which is logically possible and exists in some possible worlds).

x=a0*10^0+a1*10^1+....,

where a0=0, a1=2 if the second term (i.e., not counting the zero before the decimal point) in the decimal expansion of G(1) is 5, and a1=5 otherwise, a2=2 if the second term in the decimal expansion of G(2) is 5, and a2=5 otherwise, and so on (the set of natural numbers in w9 is still {1,2,3...}). It follows right away that from any n, G(n)=\=x. Thus, x is a real number between 0 and 1 which does not exist at the actual world!

Now, granted, I don't have an explicit formula for G at w9. However, that is generally the case both when we pick things (sets, functions, etc.) by the axiom of choice, and when we pick things in other possible worlds (where we suppose that something fills in the blanks so to speak). The main point is that while G does not exist at the actual world but it does at w9 (I concede I don't think I understand that, but it's a stipulation), it remains the case that G(n) has (in w9, but w9 is definite) a decimal expansion for every natural number n, so x is a run of the mill limit of sums of the form a_1/10^1+a_2/10^2...etc., where the a_j are in {0,...,9} for all j. Now, granted that the a_j for some reason exist in w9, but regardless, they are definite. That seems seriously weird.

Granted, also, I did not derive this by means of a first order proof. But then again, when talking about possible worlds, we're using more than just first order language.
Now I get that you say I can't single out the function, but why not? As far as I can tell, I'm singling it out in the way we single things out in other possible worlds. We suppose they're definite, but that's that. Also, I'm using the axiom of choice, but we're assuming AC is okay.

Still, if I can't single out the function for some reason, a guy Bob-9 who exists at w9 can do that, and then it still follows that x is a run-of-the-mill limit of sums, and yet it does not exist in the set of real numbers that exist at the actual world.

One alternative way to make this point is to pick an arbitrary possible world, say w10 (without stipulating that it is actual), and such that there is an injective function from the set of real numbers in w10 to the natural numbers in another possible world w11, and then a person at w11 makes the argument, and shows that some ordinary x is not in the set of real numbers at w10. Since w10 was arbitrary, that applies to every possible world (including ours), though the x's vary from world to world.

Of course, we can make the stipulation about proper classes without the other stipulation about a function into the set of natural numbers, but it seems to me we run into similar issues (if you think otherwise, I can further tailor the argument for that case).

Alexander R Pruss said...

Indeed, on stories like this, there will be sequences of 0s and 1st that don't actually exist but would exist if some other world w1 were actual. Now, if we could pick out a unique such function, then we could describe the sequence in *our* world, by saying that it is such-and-such a sequence that would exist if w1 were actual. But as long as we can't pick out a unique such function, we can't describe the sequence in our world.

Likely, too, on stories like this, there are worlds more impoverished than ours, so that we have sequences of 0s and 1s that they don't have. I don't see what's absurd about that.

Normally, the way we deal with things that we can't pick out uniquely is by quantification. We say: For some f, f is a choice function, and blah blah blah. But this only works if f exists in our world. For if f exists in another world w2, we can't quantify over f. What we can do is hypothetical quantification: If w2 were actual, then it would be the case that for some f, f is a choice function, and blah blah blah. But that doesn't generate a paradox.

Angra Mainyu said...

But why can't we quantify over f, if it exists in a determined world that exists in our world?

If we're using only the language of a first order predicate calculus (as in mathematical logic), we can't talk about possible worlds, either, so we're doing more than just that. I'm not sure why would quantification over that function be improper. Are you using some specific formal language?

Philip Rand said...
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Martin Cooke said...

Alex, you say "all the sets in our world," and therein lies the problem;
they already exist in this world, they all do: this world collects them.

Angra Mainyu said...

Philip,

That's not the kind of quantification we're talking about. We're talking about mathematics, and about "there exists an F", and things like that.

Angra Mainyu said...

Alex,

How about an argument from the assumptions to the nonexistence of some rational numbers (even with a finite, non-periodic decimal expansion) at every world?

Let w10 be a possible world, and F a funtion in another possible world w11 from the set of natural numbers in w11 onto the set of real numbers in the interval (0,1) that exist in w10. Let y be the following real number

y=a0*10^0+a1*10^1+a_2*10^2,....,

where a0=0, a1=2 if the second term in the decimal expansion of F(1) is 5, and a1=5 otherwise, a2=2 if the second term in the decimal expansion of F(2) is 5, and a2=5 otherwise, and so on. As before, it follows right away that from any n, F(n)=\=y. Thus, y is a real number between 0 and 1 which exists at w11, but not at w10 (if you like, we can posit that a person at w11 is making the argument, though I don't think that that is required). Now let M be a natural number greater than y, and let y(n)=a_0*10^0+a_1*10^1...+a_n*10^n for each natural number n (y(n) exists at w11).

If y(n) exists for every n at w10, so does the interval [y(n), M], and the intersection of those intervals for each n, which is the interval [y,M]. But that is a contradiction, since y does not exist at w10. Thus, there is a rational number that exists in w11 but not in w10. Since w10 was arbitrary, for every possible world w, there is a rational number that does not exist at w. I guess you might say that the numbers do exist, but not the intervals that have those rational numbers as extremes. But how could that be? And if the natural numbers do exist, how come some of the rationals don't? Why can't we construct them from the natural numbers?

Alexander R Pruss said...

Interesting argument, but that each individual interval [y(n),M] exists at w10 does not imply that the intersection of these intervals exists at w10. The only way I can think of proving that the intersection of these intervals exists at w10 using ZFC requires the { [y(n),M] : n is a natural number } to exist at w10. (Then the set of the complements of these exists by Replacement, and the union of the complements exists by Union, and hence the complement of the union exists by Separation, and that's the intersection.)

Angra Mainyu said...

It also suffices that the set A={y(n):n is a natural number} exists at w10 (unless of course the upper bound is greaten than y, but that would require excluding an entire interval!), though that might be equivalent ("might" because I'm still not sure I get how you're combining ZFC with possible worlds. Maybe that can be formalized, but I'd have to see it to be sure.). If I'm reading this right, would you say that even though each of the y(n) exists, they do not form a set?

If so, we can do that even for increasing unions of finite subsets of Q: Let A(n)= {y(1),...,y(n)}. They all exist at w10 (assuming all of the rationals do), and they are an increasing sequence of subsets of Q, but their union does not exist at w10. It seems to me that (at least) this shows that the ZFC formalism is not adequate to capture our intuitive notions, under this assumptions (i.e., surely, that union has to exist!). Then again, I have to admit that I don't see existence of sets as existence in possible worlds, so my view of possible worlds may be doing the intuitive work, rather than the mathematical stuff. Do you find this intuitively okay?

Angra Mainyu said...

On a different note, on this view of proper classes as possible sets, I would raise the question: why should one think that the AC holds? (i.e., what is the motivation?). I can see the intuitive motivation for the axiom of choice in the intuitive class of all hereditary sets. But in this context, I don't see why that would have to be so. It might well be that A is a set of pairwise disjoint sets in w, but there is no set in w that has one element of every one of the sets of A (there might be such a set in some other world w'). At least, I don't see why AC would hold.

Philip Rand said...
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Angra Mainyu said...

Philip,

No, I didn't mean to say anything about physics. I was talking about mathematics, and the philosophy of mathematics. If there is a set of disjoint sets, it's intuitive to me that in the class of all hereditary sets, there is a set B which has exactly one element from each of the elements of A. But if one does not have the class of all hereditary sets, but only the class of all hereditary sets in the actual world (all of which are elements of a set in another possible world), I have no intuition as to why there would be (in the actual world, not in some other world) a set like B.

I would be surprised if this were incompatible with any theory in physics.

Alexander R Pruss said...

Angra:

There is no special difficulty in combining ZFC with modality. (Possible worlds here are just a conveniently intuitive way of talking about modality.) Formally, one can do stuff like that. See, e.g., http://jdh.hamkins.org/the-set-theoretic-multiverse/ . (Hamkins is doing mathematics, not metaphysics, but the mathematical stuff he is doing shows that my metaphysical version is narrowly consistent.)

But your point about Choice is a really good one. There may be an intuitive tension between limiting what sets actually exist and allowing for Choice. It's not a formal logical

Philip Rand said...
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Alexander R Pruss said...

Probably, the view that best goes with the hypothesis I'm developing is that the Axiom of Choice is false, but for any set of non-empty sets, a choice function is *possible* (i.e., there is a possible world that has that choice function).

But I wish I could at least have the countable axiom of choice in this world, because it's hard to do much analysis without that. I don't know if I can argue for the countable axiom of choice without special pleading, though.

Angra Mainyu said...

Alex,

Regarding combining ZFC with possible worlds talk, the former is a way of formalizing mathematics, whereas the latter seems to me to be a rather informal way of talking about certain counterfactual scenarios. Granted, one can also partially formalize the latter with the boxes and diamonds and all, but as long as one is quantifying over something like propositions (or something equally "broad" in a sense), it doesn't seem to me very related to ZFC as far as I can tell, which is why I was asking for some formalization. Thanks for the link, I will take a look when I have more time, though I'm still not sure how you would connect the formalized math to the metaphysical part, because of propositions and stuff.

That said, after reading your posts on the matter, I think I'm getting an idea of how you see it, so I'm going to try to make arguments from within that context.

In re: the countable axiom of choice, it seems to me difficult to justify (on this view) as well. Also, I think there may be another issue (which is also more salient in the case of AC, but may not be limited to it):

You suggested something like "The sets that actually exist at w are those that form a minimal model of set theory that contains all the sets that can be specified using the concrete resources in the world".
However, with AC the set is not specified, at least in a significant (and I think very relevant) sense. Granted, AC guarantees existence of a set given a certain set, etc. But using some of the assumptions and also AC in other words, we can conclude that, say, some real numbers exist in other possible worlds but not in the actual world; the same goes for sequences of zeros and ones, and so on (e.g., my earlier arguments, though now I'm using them differently). But a question is: why would those require more resources?

I mean, the real numbers that only possibly exist are ordinary limits of series of the form a_1*10^{-1}+a_2*10^{-2}..., etc. (by the way, in the earlier posts, I miswrote that. I meant to use negative exponents), so why (or even how?) would the {a_i} that are picked in another world by AC but not in the actual world require more resources to specify than some other {b_i} picked by AC in our world, and whose limit does exist in our world?

But seems arbitrary to me: in both cases, the numbers are not specified beyond saying they exist or possibly exist. And they're ordinary limits in both cases, etc.
The same goes for other sorts of sets, of course.

Philip Rand said...
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Philip Rand said...
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Philip Rand said...
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Philip Rand said...
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Philip Rand said...
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Martin Cooke said...

Maybe proper classes are non-existent possible sets

The construction of the ordinals is quite definite, though; we then take all of those. What we then have looks a lot like an ordinal (whence the paradox). But if it was an ordinal in a non-actual but possible world, then what sort of construction were we doing in that possible world? If it was the same construction as ours then something that only looked like it should have arrived when we took all of the ordinals, so it must have been a different construction.

Although, that is intuitive; formally you can do anything!

Martin Cooke said...

I don't think that this, that you said, makes sense:
"there is actually no collection of all the sets in our world"
Consider our world, with all its sets, and ignore the non-sets:
is that not a collection of all of the sets in our world?
Surely our world collects all the sets in it,
but maybe that collection has no power-set,
in our world; intuitively, such a collection is possible,
so maybe it exists in a non-actual possible world,
is my opinion (I don't know why you don't draw the line there)