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You need a uniformity requirement in your limit characterization of $0'$, that is, the functions $f_k(n)$ need to be uniformly computable, in that there is a single computable function $f(k,n)$, such …

One consideration will be that the CH statement can become ambiguous in weaker foundations, since statements that are equivalent in ZFC are not always equivalent in weaker theories. For example, witho …

Here is a model of your theory. Start with a countably infinite set of objects $X=\bigsqcup_n X_n$, partitioned into infinitely many infinite sections. We inductively define $\in_n$. Consider the vari …

Unfortunately, the claim you have stated is not true. Regardless of the axiom of choice, every real is definable from a countable sequence of ordinal parameters, since the real is definable from the s …

Here is an example with Cohen forcing. Let $\mathbb{Q}$ be the forcing that adds two Cohen reals, viewed as binary sequences, along with their bitwise sum mod 2. So conditions are binary sequences $(p …

The answer is yes. This is a consequence of the non-amalgamation phenomenon. Let $M$ be a countable transitive model. Fix an $M$-generic real $z$ collapsing $\omega_1^M$, and let us define two reals $ …

Peter Koepke and Martin Koerwien developed the theory of sets of ordinals as a foundation of mathematics, showing senses in which it is equivalent to ZFC as a foundation. Peter Koeopke and Martin Koe …

There are several immediate things to say. Some instances are outright provable in ZFC: $\newcommand\FA{\text{FA}}\FA(\omega,\newcommand{\P}{\mathbb{P}}\P)$ is a theorem for every poset $\P$. $\FA(\k …

One might hope to handle proper classes as objects by working in one of the standard second-order set theories. For example, there is Gödel–Bernays set theory GBC, which has classes as objects, and in …

I have engaged with the Lewis-style set-theoretic mereology in a few papers, undertaken jointly with Makoto Kikuchi. My interest in this topic was inspired originally by a MathOverflow question, Why h …

The class of functions whose restriction to any interval is onto has only nonremovable discontinuities. An example of such a function is the Conway base 13 function. If a function remains onto when re …

This argument is essentially similar to the argument of Mel Fitting in his article, "Russell's paradox, Gödel's theorem" Chapter in book: Raymond Smullyan on self reference, 47–66, Outstanding Contri …

It is a decidable theory, because it is interpretable in the real-closed field $\langle\mathbb{R},+,\cdot,0,1\rangle$, which has a decidable theory. We can interpret complex numbers $a+bi$ as pairs of …

The elements of this partition are precisely the atoms of the complete Boolean algebra generated by the family.

Here is an answer concerning recursion, rather than induction, but they are of course related. Namely, the Ackermann function is defined by a double nested recursion $$A(m+1,n+1)=A(m,A(m+1,n))$$ w …