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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 …

Logicians are familiar with the variety of self-referential sentences expressible in the language of arithmetic: The Gödel sentence, "this sentence is not provable", which indeed is not provable in w …

There are many interpretations of arithmetic in set theory. The Zermelo interpretation, for example, begins with the empty set and applies the singleton operator as successor: $$0=\{\ \}$$ $$1=\{0\}$$ …

Presburger arithmetic admits elimination of quantifiers, if one expands the language to include truncated minus and the unary relations for divisibility-by-2, divisibility-by-3 and so on, which are de …

I am interested to know the full range of possibilities for the cofinality type of cuts in an ordered field and in other structures, such as nonstandard models of arithmetic. Definitions. Specifical …

Here is another way to do it. By the Gödel-Rosser theorem, there are continuum many distinct consistent completions of PA. One can see this by building a tree of finite extensions of PA, using the G …

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 …

$\newcommand\Tr{\text{Tr}}$My question is whether there can be a nonstandard model of PA having a unique inductive truth predicate. Background. If $\mathcal{N}=\langle N,+,\cdot,0,1,<\rangle$ is a m …

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, …

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 …

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{\ …

One of the usual ways of proving Tennenbaum's theorem also applies to many of the theories on your list, and so they can have no computable nonstandard models. The proof I have in mind is the followi …

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 …

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 …

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 …