Dec 10, 2020 · ZFC doesn't have finitely many axioms because it has some axiom schemas. Let's look at Axiom Schema of Specification:

May 5, 2020 · On the other hand, some of us are interested in questions that demand significant strength beyond ZFC. For us, the appropriate theory to work on is an extension of ZFC (or NBG, or MK) with large cardinals, or a theory equiconsistent with such an extension, or a theory expected to be equiconsistent with such an extension.

What I mean by this is that saying ZFC + " κ κ is an inaccessible cardinal" ⊢ (Vκ ⊨ ┌ZFC┐) ⊢ (V κ ⊨ ⌜ ZFC ⌝), is ultimately about natural numbers, and we have shown Vκ ⊨ φ V κ ⊨ φ for φ φ, which are coded by standard natural numbers. So my question boils down to: Is ZFC ω ω …

Working in ZFC, does there exist a set Σ Σ of sentences which axiomatizes ZFC (i.e. every sentence in Σ Σ is provable from your favorite axiomatization of ZFC, and vice versa) and is minimal with respect to inclusion (i.e. no proper subset of Σ Σ axiomatizes ZFC)? André Nicolas's comments on What is the minimal axiomatization of a set of structures? seem to suggest the …

Mar 25, 2020 · I claim that ZFC + ‘‘ there exists a model of ZFCscheme2 " is equiconsistent with ZFC + an inaccessible (however, other than for second order ZFC, the existence of such a model does not outright imply the existence of an inaccessible). So assume that M is a transitive set-sized model of ZFCscheme2.

Oct 28, 2020 · It is an axiom of ZFC that ∪x exists for any set x, where ∪ denotes union. But how does one prove that for any non-empty set x, the intersection ∩x exists? We need x to be non-empty, because ∩∅ would be the set of everything, which does not exist in ZFC.

Nov 28, 2021 · ZFC with 1st order logic is known. My question may be formulated either way What is the definition of ZFC2 (ZFC+second order logic)? (I assume that formulas also have quantifications of two kind o...

Mar 31, 2018 · Here it is explained that the second order ZFC is the first order ZFC with second order variants of its axioms. I would like to see what second order variants look like and what intuitively they sa...

Apr 28, 2020 · I am teaching a course and want to provide students with a simple explanation of the ZFC axioms without technical jargon. I try to define most of the primitive words in the list with the following ...

Oct 16, 2018 · I have read that ZFC is recursively axiomatizable, and hence is incomplete by Gödel's theorem. Now why is this true? Consider in particular the axioms of replacement and separation. My guess is th...