Search Results

Here is a somewhat different way to think about it, although the result is equivalent to the theories in the other answers. Begin with the observation that the structures $\langle\mathbb{N},+,\cdot,0, …

Let me assume that you also include the axioms $\vdash A$ whenever $A$ is a logical axiom, for some fixed standard proof system using modus ponens as the only inference rule. And let me also assume th …

No, both of these theories are equiconsistent with PA. The reason is that given any model of PA, we may expand it to a model in the language with $\vdash$ by interpreting $\vdash\psi$ as true in the m …

It doesn't contradict PA to have such a descending sequence. For example, you could instead simply keep substracting one from $g^{*1}$, and this would be provably descending, but there is no contradic …

Asymptotic density seems a very natural measure. The density of a set of sentences is the limit as $n\to\infty$ of the proportion of those sentences of length at most $n$ amongst all sentences of leng …

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 …

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 me give a simple example, which may help to clarify things. Consider the integer line $\langle\newcommand\Z{\mathbb{Z}}\Z,<\rangle$, and the natural number order $\langle\newcommand\N{\mathbb{N} …

This is a great question! Let me give a meager partial answer, for the case where we are talking about nonstandard models of true arithmetic. Theorem. No nonstandard model of true arithmetic arises …

For a certain project I am currently working on, I need to be able to find PA cuts in nonstandard models of PA, in desirable intervals. For example, I wonder if the following is true, where $\newcomma …

Victoria Gitman and I recently looked at the first-order version of this theory, which we called fPA, to contrast it with the theory FPA, which Woodin has looked into, which is a possibly stronger ver …

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 …

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 …

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 …

It seems that the Feferman-style description of PA will exhibit your requirements. Specifically, consider the theory $P$ defined as follows. Begin to enumerate the usual PA axioms, but include the n …