As Wojowu already pointed out $\mathsf{RCA}_0$ proves recursion theorem. You could find a proof in Simpson's book [1], Section II.3. In fact primitive recursion theorem is equivalent to $\Sigma^0_1\...

In the field of ordinal analysis people are typically interested in finding "natural" computable ordinal notation systems corresponding to various theories; those are called proof-theoretic ordinals ...

Yes, indeed this kind of choice in general doesn't imply $\mathsf{AC}$ over $\mathsf{ZF}-\mathsf{Reg}$. I will reason in $\mathsf{ZFC}$ and construct an interpretation of $\mathsf{ZF}-\mathsf{Reg}$, ...

Let me give an example of a theory that is computably axiomatizable but isn't axiomatizable by finitely many schemas. Fix any finite signature $\Omega$ with equality. Further by finite $\Omega$-models ...

I know two constructions of the chains of this sort that aren't based on explicit diagonalization. In a recent work by James Walsh and me https://arxiv.org/abs/1805.02095 we gave an example (Theorem ...

First let me note that one should be careful with formulation of $\mathsf{ZFCfin}$, for it to be bi-interpretable with $\mathsf{PA}$ (see the paper "On interpretations of arithmetic and set theory" by ...

The length of the least proof of contradiction in $\mathsf{Graham}+\forall n (n<g_{64})$ should be inbetween $(\log_2^*(g_{64}))^{1/N}$ and $(\ln^*(g_{64}))^{N}$, where $\ln^*(x)=\min\{n\mid \log_2^...

I'll provide a description of $(\alpha)^{\mathfrak{A}}$ in the case of countable $\mathfrak{A}$. I base my answer on observations by Emil Jeřábek (see discussion below the initial question), but of ...

This couldn't be achieved even by $\Sigma_3$ sentences. First note that $L_{\omega_1^{CK}}$ (as any other model of $\mathsf{KP}\omega+L=V$) satisfies the scheme of $\Sigma_3$-reflection: $$\varphi(\...

For any given finite signature $\Omega$ there is a second-order sentence $\varphi$ of the signature $\Omega$ such that $\mathsf{ZFC}+V=L$ proves that for any $\mathcal{L}_{\infty,\omega}$-formula $\...

It actually isn't true that the complexity of $\theta$ could be meaningfully bounded in the term of complexities of $\varphi$ and $\psi$. Let us fix an arbitrary recursive ordinal $\alpha$. Below I ...

$\mathfrak{Q}$ is the countable random distributive lattice. Emil Jeřábek has already pointed in his comments that there are only two possibilities for $\mathfrak{Q}$. Either there are no greatest ...

Let me show that for extensions $T\supseteq\mathsf{ACA}_0$ the usual proof-theoretic ordinal $|T|_{WO}$ coincide with $|T|_{WPO}$ that is the suprema of $\mathsf{o}(X)$, for recursive wpo $X$, for ...

In order to prove that any class $\mathrm{No}$ satisfying this definition is well-ordered by $\in$ it is crucial to use the axiom of foundation. Axiom of foundation tells us that $\in$ is a well-...

The ineventable consistency ordinals do not exist for all the computably axiomatizable extensions of $\mathsf{RCA}_0$. Indeed, for any given notation system $\alpha$ I'll construct a notation system $\...

Your system indeed couldn't prove even that $\mathcal{P}(\omega)$ is a set. Let $M$ be a countable transitive of $\mathsf{ZFC}+\mathsf{GCH}+\mbox{there exists an inaccessible}$. Let $\kappa\in M$ be ...

First, I'll discuss the case of empty $C$. Observe that the structures $(\varepsilon_{\alpha};+,\cdot,\mathsf{exp})$ are definitionally equivalent with the structures $HF(\alpha;<)$ (the structure ...

Let me denote as $\mathsf{K}(V)$ your system 1.+2.+3.+Super Transitivity. And as $\mathsf{K}^{+}(V)$ your system 1.+2.+3.+Limitation of size. Note that the well-founded part translation gives an ...

Let me sketch the proof that $\mathsf{ATR}_0+\Pi^1_1\textsf{-Ind}\vdash\mathsf{Con}(\mathsf{ATR}_0)$ (by Gödel's 2-nd incompleteness this implies that $\mathsf{ATR}_0$ doesn't prove $\Pi^1_1\textsf{-...

There is even an example of a cardinal $\kappa$ and an r.e. categorical second-order theory $T$ such that for no finitely axiomatized second-order theory $U$, the spectrum of $U$ has $\kappa$ as its ...

In short the answer is yes. Let us consider Schütte-style proof of completeness of $\omega$-logic. This proof works as follows. For any sequent $\Gamma$ we define it's canonical (cut-free) pre-proof ...

I don't have a general classification of this kind of models, but it is rather easy to construct quite a lot of models with this property. For example (for first-order variant of your system), ...

I don't have complete answer but I think that my remarks still may be useful. Let me consider theory $\mathsf{NT}$ which extends $\mathsf{PRA}$ by one unary predicate $X$ and axiom scheme of ...

First note that monadic second-order logic (i.e. the variant of second-order logic with second-order quantifiers only over unary predicates) isn't sufficient. This is implied by the fact that the ...