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Every large cardinal property admits a formalization with the desired property. That is, every large cardinal property $\text{LC}$ admits a ZFC-provably equivalent formulation $A$ for which $\newcomma …

Yes, for the reasons you mention, it is important to define your Gödel coding in such a way that the syntactic operations you want to undertake with assertions in the language are indeed expressible i …

Set theory provides a good example. It is often convenient in set theory to work with the concept of "classes" and treat them as mathematical objects of their own kind. The standard axiomatization of …

This idea in play here is due to Rosser and is the main idea behind the Gödel-Rosser theorem. Specifically, Rosser proposes to consider the sentence $\rho$ asserting that for every proof of $\rho$ in …

The cautious enumeration idea in my paper has some affinity with your suggestion. Joel David Hamkins, Nonlinearity and illfoundedness in the hierarchy of large cardinal consistency strength, arxiv:22 …

These self-referential decision problems are already part of the subject of computational complexity. There are analogues of the halting problem, for example, for many of the various classes in the co …

The answer is no. Choose a fundamental sequence for $\epsilon_0$ itself in the usual way, which I think is $\epsilon_0[n]=\omega^{\omega^{{\vdots}^\omega}}$, and then modify the earlier fundamental se …

Logicians are familiar with the variety of self-referential sentences expressible in the language of arithmetic: The Gödel sentence, "this sentence is not provable", which indeed is not provable in w …

There are several things one can say. The theory of ZFC without powerset is often denoted by $\newcommand\ZFCm{\text{ZFC}^-}\ZFCm$. One has to be a little careful with what it means, since collection …

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 …

Although the other answers correctly explain the basic logical equivalence of the two proof methods, I believe an important point has been missed: With good reason, we mathematicians prefer a direct …

This does not answer your question, but I find it relevant. You ask for uniform incomparable statements $A_\tau$ and $B_\tau$, and then ask also for monotonicity. But I claim that if one asks for the …

You must read the charming essay lampooning this notation, while also giving a thorough logical analysis of it, by Adrian Mathias. Adrian Mathias, A Term of Length 4,523,659,424,929, Synthese 133 (20 …

There can be no such model. The first observation is that if there is a model as you describe, then I claim there will be an instance where $j$ is elementary. To see this, suppose that $M$ is as in y …

Here is perhaps a more relatable example, which doesn't use self-reference. (I once heard a similar such example from W. Hugh Woodin.)$\newcommand\Con{\text{Con}}\newcommand\ZFC{\text{ZFC}}$ Let $\ps …