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Your theory is true in the one-element universe $\{a\}$ in which $a<a$ is true and $a\in a$ is false. The order transitivity holds trivially in this case; the finiteness axiom holds vacuously; and the …

The theory PA fully proves the pigeon-hole principle, which how I like to refer to your principle, the claim that there is no injection from the predecessors of $n$ to those of $n-1$. (This implies of …

Yes, this function is obviously definable in PA and PA proves it is total. You are defining the tower of theories by recursion, which PA can do, and taking the Rosser sentence of each theory, which PA …

To my way of thinking, the arguments you mention seem not to distinguish sufficiently between the content of Goodstein's theorem as a universal claim $\forall p$ and Goodstein's theorem as a collectio …

Because your argument involves arithmetical classes at several points, as you noticed, it is not directly expressible in the first-order language of $\newcommand\PA{\text{PA}}\PA$, although as Noah me …

This theory is equivalent to the theory of a discrete linear order with a least element and no largest element, that is, the theory of $\langle\mathbb{N},<\rangle$. From max we can define the order an …

It is the same in Peano arithmetic, where the standard language is $\{+,\cdot,0,1,<\}$ for the standard model $\langle\mathbb{N},+,\cdot,0,1,<\rangle$, even though $0$, $1$, and $<$ are definable from …

Let me give half an answer. Specifically, we can interpret your theory in PA via the Ackermann encoding. Namely, in any model of PA, define $m\in x$ if and only if the $m$th bit of $x$ is $1$. This fu …

In the case of $\text{GBC}^-$, the answer is no, $\Diamond$ is not provable from the assumption that all reals are constructible. It is consistent with $\text{GBC}^-$ and even $\text{KM}^-$ that $\Dia …

This is a completely standard perspective in work on models-of-PA, a view that informs dozens of arguments. That is simply the nature of nonstandard models, that things they think are well founded are …

Let $T$ be the theory PA + ¬Con(PA), plus the axiom asserting that there is no proof of a contradiction in ZFC (or ZFC+LC etc.) of size below $k$, where $k$ is the smallest such that $\newcommand\PA{\ …

Without considering your system of arithmetic too closely, let me mention that ZFC is interpretable in Peano arithmetic, if one augments PA with the assertion that ZFC is consistent. That is, ZFC is i …

The answer is yes. The additive cluster structure knows the additive structure of the original model up to isomorphism. Theorem. The additive structure of any countable nonstandard model $M$ of $\newc …

If you intend literally to recover the addition operation of the given model $M$, then the answer is negative. For every nonstandard model of arithmetic $M$ there is another model $M'$ having exactly …

In fact one doesn't need the replacement axiom at all in order to implement this set-theoretic construction of the integers. The entire construction can be undertaken in Zermelo set theory, which lack …