All Questions

This 2007 paper presents a 5-quantifier $(\in, =)$-expression that is ZF-equivalent to the axiom of choice, but leaves open the 4-quantifier case: Thus the gap is reduced to the undecided case of a 4 ...

So, second order arithmetic, $Z_2$, is capable of proving quite a few things. One thing which would be of use is dependent choice for $\mathbb{R}$. Basically, dependent choice on $\mathbb{R}$ says ...

Say we have DC-λ where λ is some inaccessible cardinal. Is that enough to develop all of ordinary mathematics? If not, is there a strengthening that is but that nevertheless does not assume full ...

Let us assume $<$ is some class relation without minimal elements, meaning $\forall a, \exists b, b< a$. This means that for every $n\in\omega$, one can build a decreasing function $f$ with ...

[PLEASE SEE EDITS AT BOTTOM OF QUESTION] Consider the following set-theoretic axiom: For each set $X$ there exists a set-indexed collection $\{C_i \to X\}_{i\in I_X}$ of surjections such that for ...