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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3$\begingroup$ 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? $\endgroup$– Andreas BlassCommented May 25, 2014 at 14:16
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$\begingroup$ @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. $\endgroup$– NoldorinCommented May 25, 2014 at 14:21
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2$\begingroup$ 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. $\endgroup$– Andreas BlassCommented May 25, 2014 at 19:28
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2$\begingroup$ 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. $\endgroup$– The Masked AvengerCommented May 25, 2014 at 19:52
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1$\begingroup$ 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. $\endgroup$– Todd TrimbleCommented May 25, 2014 at 20:01
2 Answers
Putting together your two remarks
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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$\begingroup$ 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? $\endgroup$– NoldorinCommented May 25, 2014 at 14:41
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1$\begingroup$ 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. $\endgroup$ Commented May 25, 2014 at 14:41
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$\begingroup$ Yep, this is very much what I'm interested in. Thanks Nik. $\endgroup$– NoldorinCommented May 25, 2014 at 14:46
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$\begingroup$ 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). $\endgroup$ Commented May 25, 2014 at 14:48
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2$\begingroup$ @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. $\endgroup$– Asaf Karagila ♦Commented May 25, 2014 at 17:21
You ask
Why can't mathematics be formalised in terms of classes rather than sets?
The answer is that it can, and several standard accounts do precisely this. In particular, Gödel himself did this, for his version of what is now known as Gödel-Bernays set theory GBC (or von Neumann-Gödel-Bernays set theory NBGC) has only classes, not sets, as fundamental objects, and all the axioms refer only to classes. (One then introduces the concept of set as a defined term, a special kind of class, namely, a class that is a member of another class.) So this seems to be a central case that develops the theory as you like, and many contemporary accounts of GBC, such as Mendelson's, also use Gödel's version with only classes.
Meanwhile, it is also common, perhaps more common, to present the GBC theory as a two-sorted theory, as Bernays did, with both sets and classes. See page 14 of Kanamori's article Bernays and set theory for informed comparison and discussion. The two presentations of the theory are easily interpreted in one another, and so the difference is widely viewed as a mathematically unimportant cosmetic difference.
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$\begingroup$ Is this the same system referred to be en.wikipedia.org/wiki/…? If so, it seems to be discussing the two-sorted version. But apparently there's a logically equivalent one-sorted version that only deals with classes? $\endgroup$– NoldorinCommented May 25, 2014 at 14:43
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$\begingroup$ Yes, Wikipedia gives the common two-sorted version, but Goedel's version was one-sorted, using only classes. $\endgroup$ Commented May 25, 2014 at 14:44
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$\begingroup$ Aha. Now, it's equivalent to ZF in expressive power? $\endgroup$– NoldorinCommented May 25, 2014 at 14:47
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$\begingroup$ No, it is more expressive, because it is able to express statements about classes that may not be definable. For example, the Kunen inconsistency is a more powerful assertion in GBC than it is in ZFC, if one takes it in ZFC to refer only to definable classes. Meanwhile, it is conservative over ZFC for assertions about sets---that is, it doesn't prove any new facts about sets. $\endgroup$ Commented May 25, 2014 at 14:50
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1$\begingroup$ Yes, even though GBC refers to classes and not sets, it is really an account of set theory. It is finitely axiomatizable, however, but the subject of the theory is of course infinitary. $\endgroup$ Commented May 25, 2014 at 15:12