It is easy to overlook the fact that the existence of a given large cardinal provides us with a true arithmetical statement that would otherwise be false if the large cardinal notion were not consistent with ZFC (See On statements independent of ZFC + V=L ). The arithmetical statement that I'm r...

$\newcommand\Tr{\text{Tr}}$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 model of the first-order PA axioms, then a truth predicate on $\mathcal{N}$, also commonly called ...

In the paper ''A Nonstandard Model of Arithmetic Constructed by means of Forcing Method'', Zhang Jinwen states the following in his abstract: The first nonstandard model of arithmetic was given by Skolem. A. Robinson has introduced the concepts of standard, internal and external objects (sets...

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. Specifically, in any linear order $\langle L,\leq\rangle$, a cut is a partition $L=A\sqcup B$, such that ev...

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 $\newcommand\PA{\text{PA}}\PA_k$ refers to the theory with only $\Sigma_k$ induction. Question. If $M$ is...

Standard techniques (no pun intended) can be used to show that countable nonstandard models of Peano Arithmetic are order isomorphic to $\mathbb{N} + \mathbb{Z} \cdot \mathbb{Q}$. Once we have used the compactness theorem to verify that nonstandard models of PA exist, we can appeal to the Löwenh...

Let $\mathcal{L}_{\mathrm{exp}}$ be the language with signature $(0, ^\prime, <, \mathrm{exp})$ (with $0$ interpreted as zero, $^\prime$ as successor, and $\mathrm{exp}(x)$ as $2^x$) and let $\mathsf{TA}_{\mathrm{exp}}$ be true arithmetic in this signature; i.e. $\{ \phi \in \mathsf{Sent}(\mathca...

$\DeclareMathOperator{Cod}{Cod}$ $\DeclareMathOperator{Scl}{Scl}$ $\DeclareMathOperator{Def}{Def}$ $\DeclareMathOperator{Lt}{Lt}$ Background: All of the background to this question can be found in Kossak-Schmerl, The Structure of Models of Peano Arithmetic (2006) (referred hereafter as [KS]), s...

I know that every element of $\mathbb{N}$ is definable the standard model of Peano Arithmetic. Does there exist a countable non-standard model of PA where the same is true?

In Presburger Arithmetic there is no predicate that can express divisibility, else Presburger Arithmetic would be as expressive as Peano Arithmetic. Divisibility can be defined recursively, for example $D(a,c) \equiv \exists b \: M(a,b,c)$, $M(a,b,c) \equiv M(a-1,b,c-b)$, $M(1,b,c) \equiv (b=c)$...

Is Presburger arithmetic, augmented with two unary predicates P2, P3, for powers of 2 and powers of 3 respectively, decidable? I know that adding just one of P2, P3 to Presburger keeps it decidable, and I'm asking about both. If I understood correctly the table in the end of http://www.logique....

In a previous question, I asked whether there can be effectively axiomatizable set theories (at least as strong as, say, ZF) that are complete modulo first-order arithmetic, to which the answer is no; there will be sentences of second-order arithmetic that can't be proved even using the omega-rul...