Confused: how does ZFC allow surreal numbers?

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arbiteroftruth
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Confused: how does ZFC allow surreal numbers?

Postby arbiteroftruth » Wed May 24, 2017 5:23 pm UTC

In Conway's construction of the surreal numbers, a surreal number is an equivalence class of number forms, where a number form is an ordered pair of sets of surreal numbers, with all members of the left set being strictly less than all members of the right set.

But as far as I can tell, this means that the equivalence class of number forms representing a particular surreal number form a proper class. The surreals form a proper class, and the number form {-x|x} for any positive surreal number x is a legitimate form of the number 0. So the equivalence class of forms of 0 is a proper class, and a similar argument could be made regarding any surreal number.

But if each equivalence class is a proper class, and each surreal number is such an equivalence class, then each surreal number would have to be a proper class itself. But a number form uses sets of surreal numbers, which would mean having proper classes as elements of those sets. Isn't that forbidden in most set theories in order to avoid paradoxes? What am I missing here?

Of course, there are other constructions of the surreal numbers that don't run into this perceived issue. I'm a fan of the sign expansion definition myself. But am I correct in thinking that Conway's construction in particular doesn't work in ZFC?

Nyktos
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Re: Confused: how does ZFC allow surreal numbers?

Postby Nyktos » Thu May 25, 2017 12:45 am UTC

It doesn't work in ZFC, in the same way that defining cardinals as isomorphism classes of sets doesn't work in ZFC. There exist set-theoretic tricks for making sense of this idea, which also work here if this MathOverflow answer is to be believed.

If I recall correctly, Conway himself considers the fact that the naive definition doesn't work formally in ZFC to be a failure on ZFC's part.

arbiteroftruth
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Re: Confused: how does ZFC allow surreal numbers?

Postby arbiteroftruth » Thu May 25, 2017 4:15 pm UTC

Interesting. Is there any standard choice of an alternative set theory that does allow the naive construction?

In order to talk about "the class of surreal numbers", when each surreal number is itself a proper class, it would be necessary to construct class-like collections of proper classes. But the naive way of doing so would fall prey to the same paradoxes as naive set theory. And if classes don't have something like an axiom of comprehension, then they're no different than sets. Perhaps the best option is to take a page from NFU, and say that proper classes have unrestricted comprehension over sets, but they only have 'stratified' comprehension over other proper classes?

radams
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Re: Confused: how does ZFC allow surreal numbers?

Postby radams » Fri May 26, 2017 4:56 am UTC

The usual move is to work in ZFC plus the axiom "There exists a Grothendieck universe".

https://en.wikipedia.org/wiki/Grothendieck_universe

Then let U be a Grothendieck universe, and replace 'set' with 'element of U', and 'class' with 'set'.

Incidentally, Wiles' proof of Fermat's Last Theorem assumed the existence of a Grothendieck universe. We still don't know if Fermat's Last Theorem is provable in ZFC (still less if it is provable in Peano Arithmetic).

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Eebster the Great
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Re: Confused: how does ZFC allow surreal numbers?

Postby Eebster the Great » Fri May 26, 2017 10:29 pm UTC

Mathematicians in the field seem pretty sure it would be straightforward to modify the proof to work in ZFC and probably in either Peano arithmetic or a slight extension.


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