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Since you allow arbitrary sets of ordinals in your second-order definitions, all sets will be in $\text{OD}^2$. The reason is that, for any set $x$, we can code $x$ into a set of ordinals as follows. …

Although you prefer models other than permutation models, let me point out an appearance of permutation models, particularly the basic Fraenkel model, at the border between computer science and logic. …

Do you have a reason to prefer NBG over MK (Morse-Kelley theory of sets and classes)? If not, you might look at the development of MK in the Appendix (if I remember correctly) of Kelley's book "Genera …

The answer to Question 1 is no. Your filter $W$ contains the set of inaccessible cardinals below $\kappa$, but the club filter $W'$ does not.

The dual of an abelian group $A$ is defined to be the group $\text{Hom}(A,\mathbb Z)$ of homomorphisms to the infinite cyclic group. As usual with such dualities, there's a canonical homomorphism from …

This is an answer not for the original question but for how to get the equivalence of (a) and (b) from the infinite Ramsey theorem (as requested in a comment). The implication from (a) to (b) is trivi …

By a theorem of Solovay, $|\mathbb R|$ can consistently be $\aleph_\alpha$ for any ordinal number $\alpha>0$ that does not have countable cofinality. Then the set $\{|A|:A\subseteq\mathbb R, |\mathbb …

This isn't an answer but it's too long for a comment. I suggest that $n-1$ is more important than $n$ in this context, for the following reason. Suppose $A$ is an $n$-group in a semigroup $S$. Then, f …

As indicated in Peter Vojtas's answer, this notion is a special case of what he called generalized Galois-Tukey connections (in the paper he linked to) and what I later called morphisms (in my chapter …

Yes. This is a weak form of Hindman's theorem. See, for example, https://en.wikipedia.org/wiki/IP_set .

Morse-Kelley set theory doesn't seem adequate for all the things one would like to do in category theory. It provides a nice treatment of proper classes, so it can deal with large categories like the …

As you've already noticed, this is essentially the conjunctive normal form, with the conjuncts separated as individual formulas of the sort usually called "clauses", i.e., disjunctions of atomic and n …

Well, let's compare the compactness example you cited with a non-standard models approach to the same result. Of course, since the result is about vertices of a graph, not just natural numbers, I'll …

At the risk of making things worse rather than better, let me point out that what appears to be one approach, say the Scott-Solovay version using complete Boolean algebras, is really a whole family of …

The following seems to be sort of a reversed version of Joel's example, categorical in uncountable cardinals rather than in $\aleph_0$. I'll use the theory of the set $\mathbb Z$ of integers with only …