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I am looking for examples of proofs of Godel's (First) Incompleteness Theorem which are essentially different from (Rosser's improvement of) Godel's original proof. This is partly inspired by question …

For most of my (mathematical) life, I believed that there was really no essential difference between set theory without urelements and set theory with urelements. However, while that may be true in ce …

Suppose $T$ is a complete first-order theory (in an finite, or at worst countable, language). Given any model $\mathcal{M}\models T$ of cardinality $\kappa$, we can ask whether $\mathcal{M}$ can be ma …

It depends what you mean by "second-order $\mathsf{PA}$." If you really mean full second-order $\mathsf{PA}$, then since that theory characterizes $\mathbb{N}$ up to isomorphism it's complete, and con …

For a structure $\mathcal{X}=(X;...)$, say that a cardinal $\kappa$ is $\mathcal{X}$-detectable iff there is some sentence $\varphi$ in the language of $\mathcal{X}$ together with a fresh unary predic …

Throughout, "computable structure" means "first-order structure in a computable language with domain $\omega$ whose atomic diagram is computable." Say that a computable structure $A$ is compact for co …

$\DeclareMathOperator\Eq{Eq}\DeclareMathOperator\Th{Th}$Originally asked at MSE without success: For a structure $\mathcal{A}$ whose signature only contains function and constant symbols, let $\Eq(\ma …

Given a binary operation $\star$ on $\mathbb{N}$, we can naturally extend $\star$ to a semicontinuous operation $\widehat{\star}$ on the set $\beta\mathbb{N}$ of ultrafilters on $\mathbb{N}$ as follow …

Expositionally, forcing is (usually) easier to understand with a c.t.m. This does indeed lead to somewhat different results, such as $(*)\quad$ If there is a countable transitive model of $\mathsf{ZF …

The answer is yes. I believe the following is best attributed to Feferman 1960; more generally, look up "arithmetized completeness theorem" (which is annoyingly different from the arithmetic completen …

This is a continuation of an MSE question which received a partial answer (see below). Given a topological space $\mathcal{X}$, let $C(\mathcal{X})$ be the ring of real-valued continuous functions on …

Here's a counterexample for $\kappa=\aleph_1$: let $B$ be the structure with underlying set $\mathbb{N}\sqcup\mathcal{P}(\mathbb{N})$, equipped with the usual ordering on $\mathbb{N}$ as well as the $ …

Originally asked and bountied at MSE: Question: Can $\mathsf{ZFC}$ prove that for every countably infinite structure $\mathcal{A}$ in a countable language there is an ultrafilter $\mathcal{U}$ on $\o …

EDIT: immediately after bountying the question (whoops ...) I found, while looking for something else entirely, that Sauro Tulipani gave an explicit algorithm for producing a Horn sentence $\varphi_p$ …

Vopěnka's principle (VP) is the statement that, given any proper class $\{\mathcal{A}_\eta: \eta\in ON\}$ of first-order structures in the same language, there are some $\alpha\not=\beta$ with $\mathc …