# Why can't mathematics be formalised in terms of classes rather than sets? [closed]

I've always been curious about the seeming compulsion to found mathematics upon sets, be it ZF(C) or some other system. Of course, there are other approaches these days like category theory and type theory (themselves inextricably linked by the Curry–Howard–Lambek correspondence), but these rather seek an entirely different approach. What I'm really getting at is: why can't we simply deal with (improper) classes? That is, throw away sets as reified mathematical objects and just deal with the classes implicitly defined by predicates in some logical system. Does this indeed make certain areas of mathematics inaccessible? Are there other problems I might not have considered? I would appreciate if someone could enlighten me here in a general way, albeit perhaps also with some specific cases within mathematical subfields.

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## closed as unclear what you're asking by Andreas Blass, Simon Thomas, Stefan Kohl, Monroe Eskew, Ricardo AndradeMay 25 at 23:06

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The question seems to admit many interpretations, but the most natural one seems to be that you want to deal only with classes and that these (being "improper") will not be elements of other classes. So the elements of classes will have to be some non-class entities that haven't been specified. If, on the other hand, you want to allow some (but not all?) classes to be members of others, then just define "set" to mean "class that can be a member of another class" and you're back to set-class theories like NBG. Maybe you intend something entirely different, but what? –  Andreas Blass May 25 at 14:16
@AndreasBlass: Well, I left this aspect open somewhat on purpose. But let's be more specific, for the sake of argument. First, we could envisage a universe of discourse being something like the naturals (recursively defined), with only first-order classes. Alternatively, we could work in some higher-order logic, thus allowing classes to be members of other classes, in a stratified manner. –  Noldorin May 25 at 14:21
I voted to close because there was no clear question here. Rhetorical questions in philosophical areas, with no clear content, may indeed be part of life, but they are not (nor should they be) part of MathOverflow. –  Andreas Blass May 25 at 19:28
Noldorin, you have an interesting question which could deserve a discussion from like-minded individuals. This forum is the wrong place for it. The closest place on the SE network for such a discussion is the chat rooms. This forum is more for question and answer format: read the FAQ for what formats and content are encouraged here. If you have a specific question that avails itself of a specific answer and is in line with the intent of this forum, you are encouraged to ask that kind of question. –  The Masked Avenger May 25 at 19:52
I'm going to remove comments that started us in the direction of incivility. Noldorin has asked why the votes to close, which is perfectly reasonable. So are responses from closers like Andreas Blass, who sets a nice tone for civility. Let's keep it that way, please. –  Todd Trimble May 25 at 20:01

just deal with the classes implicitly defined by predicates in some logical system.

and

we could envisage a universe of discourse being something like the naturals (recursively defined), with only first-order classes.

seems to me to yield an informal description of full second-order arithmetic, $Z_2$.

As to whether mathematics can be formalized in such a system, the answer is yes, virtually all mainstream mathematics can even be formalized in weak subsystems of $Z_2$. See the book Subsystems of Second Order Arithmetic by Stephen Simpson.

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Intriguing! Thank you for this answer. This is very much what I was hoping for. Do these formalisations avoid set theory altogether then, I take it? –  Noldorin May 25 at 14:41
I am reading you as wanting to "eliminate the ontology of sets" by letting every predicate function as a definable collection. $Z_2$ does this via the full comprehension scheme. –  Nik Weaver May 25 at 14:41
Yep, this is very much what I'm interested in. Thanks Nik. –  Noldorin May 25 at 14:46
I should add that full comprehension allows predicates which reference quantification over all predicates. If that kind of circularity bothers you, look at the subsystem $ACA$ (which still suffices for, arguably, all or nearly all mainstream mathematics). –  Nik Weaver May 25 at 14:48
@Noldorin: Because working with set theory is much easier than working with arithmetic; because ZFC allows you to "forget your foundation" when working in "virtually all mathematics"; because model theory is much easier to work with when you have access to actual semantics, even if you want to talk about uncountable models (e.g. $\Bbb R$) without reducing to their countable counterparts; because sets are not that hard to understand, and once you forget about them they're gone, but encoding things into integers is harder; and more. –  Asaf Karagila May 25 at 17:21