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Below, I've focused on PA when lots of other theories would do. If replacing PA with a different theory leads to a more answerable question, feel free to do so. The standard system of a nonstandard ...

Recall that the Paris-Harrington Principle, $\mathsf{PH}$, is the statement that for each $e, r, k < \omega$ there is an $N < \omega$ so that given any coloring $c:[N]^e \to r$ there is an $H \...

Suppose $(M, \mathcal X) \models ACA_0$. Recall that a subset $A \subseteq M$ is $inductive$ over $M$ if $M$ satisfies all instances of induction in the expanded language with a predicate for $A$. ...

In the thread Even Xor Odd Infinities I defined odd models of Modular Arithmetic (MA) as models satisfying the axioms of MA and two first order statements. Even XOR Odd Infinities? $O1) \forall x(x=0 ...

It is known that to prove completeness of first-order logic for countable languages WKL0 is enough. But, is it the weakest subsystem where one can prove it? What about the incompleteness theorems? Is ...