Thread: ZFC and the foundation of mathematics

and what, pray tell, is the collection of objects of the category Set? i understand that Grothendieck introduced his notion of a universe to avoid using "proper classes", but in my point of view, this is just squeezing the "air bubble" from one room to the next. the existence of a Grothendieck universe is axiomatic, it's certainly not derivable from the ZF axioms.

and the point is, if you want to talk about category theory as an alternative to ZF(C) as a mathematical foundation, you want the weakest form (fewest assumptions) possible. sure, you can add "strongly inaccessible cardinals" to ZFC and within that framework, talk about categories, but adding such an axiom is not a conservative extension of ZFC, it is some larger theory (unlike say NBG).

in fact, the consistency of ZFC with Grothendieck universes, is not in and of itself a compelling reason for adopting the so-called ZFCU system. U has some of the same questionable flavor the axiom of choice does, or the continuum hypothesis.

anyway, by "already at hand" i meant using the language of (conventional) sets to describe category theory. i am aware that there are alternate approaches to founding category theory: this is discussed briefly in "Categories for the Working Mathematician" 1.6, and also in some book by Awodey, i can't recall which one.

in one sense you are correct, we don't "need" classes to be able to describe the category Set, but then what we have instead is a stronger theory (with Grothendieck universes, you can prove the consistency of ZFC, which by Godel's Theorem implies that it is indeed a stronger theory). whereas, if you use proper classes, you are using a theory that is "the same strength"
(true in ZFC iff true in GNB).

even GNB is insufficient to describe Cat (or what Maclane calls Cat'), however, because for each collection type of GNB, we can form a strictly larger category whose objects are that kind of type, and morphism functions between them.

Originally Posted by Deveno

this is just squeezing the "air bubble" from one room to the next. the existence of a Grothendieck universe is axiomatic, it's certainly not derivable from the ZF axioms.

It may be squeezing air bubbles or many things. The only point I've suggested is that it does not involve proper classes. Most of the axioms of ZF are assertions that there exist certain kinds of sets. The Grothendieck axiom is another such assertion. That it is not entailed by the axioms of ZF doesn't refute that it is a way to get the category of sets without resorting to proper classes. That's all I've claimed.

Originally Posted by Deveno

the consistency of ZFC with Grothendieck universes, is not in and of itself a compelling reason for adopting the so-called ZFCU system. U has some of the same questionable flavor the axiom of choice does, or the continuum hypothesis.

I don't ask anyone to adopt any axioms they don't care to adopt. My only point is that there is an axiomatization that provides for the category of sets without involving proper classes.

Originally Posted by Deveno

in one sense you are correct, we don't "need" classes to be able to describe the category Set,

And that's the only sense I suggested.