If $T$ is a theorem of ZF which says something only about reals, then one may want to prove $T$ using a theory like 2nd order PA or related theories like ZFC$^-$ or GBC$^-$ (minus accounts for the absence of the Power Set axiom). In many cases this goes through, sometimes rather straightforwardly...

Background essays (the material I've tried to understand in leading up to this question): Daghighi, et. al. [2014], "The foundation axiom and elementary self-embeddings of the universe." Dimonte [2017], "I0 and rank-into-rank axioms." To some extent, I've also tried to invoke Dougherty[92] in my...

I've only a shallow understanding of the relevant theory, but I don't understand how any internal proof of consistency is in any way satisfactory (even for systems that are so weak Gödel's incompleteness theorem doesn't apply). After all, suppose you've proven consistency from the axioms, both th...

It seems common amongst logicians to think of "truth" as being relative to a particular structure. Consider, for instance, the first-order theory of groups. The sentence $\forall x\forall y(x\cdot y=y\cdot x)$ is neither provable nor disprovable from the group axioms. However, rather than saying ...

While perusing p. 237 of the 3rd ed. of Marvin Greenberg's book on Euclidean and non-Euclidean geometries, I learned that it can actually be proven that "all possible models of hyperbolic geometry are isomorphic to one another, i.e., that the axioms for hyperbolic geometry are categorical". This ...

Let $\alpha$ be an algebraic integer of modulus 1, and $ R_\alpha z=\alpha z$. Is $$\lim_{n\to\infty}\frac{\log\sum_{k=1}^n \Re R_\alpha^k z}{\log n}=\frac12$$ for all $z\in S^1$?