Tl;dr version: there are two natural classes of cuts in the nonstandard model of arithmetic consisting of the Dedekind-finite sets (if, in fact, they constitute such a model); both these classes are ...
Modular arithmetic (MA) has the same axioms as first order Peano arithmetic (PA) except $\forall x(Sx \neq 0)$ is replaced with $\exists x(Sx=0)$. In Even XOR Odd Infinities? I asked if this statement ...
A well-behaved $A$ that is almost contained in every element of some filter for a countable arithmetically closed family $\mathfrak X$
The question has relevance for constructing Scott sets with certain extra desirable properties. Suppose that $\mathfrak X$ is a countable arithmetically closed family of subsets of $\mathbb N$: ...
Peano Arithmetic (PA) models have a prime number structure and commutativity of addition and multiplication. Presburger arithmetic (PrA) models of arithmetic have addition without multiplication and ...
Modular Arithmetic (MA) has the same axioms as first order Peano Arithmetic (PA) except $\forall x (Sx \ne 0)$ is replaced with $\exists x(Sx = 0)$. ...
Our question arises from wondering about the systems of natural numbers in models ZFC + Con(ZFC) and ZFC + $\neg$Con(ZFC). In thinking of the systems of natural numbers of these models, we came to ...
Consider Peano's axioms — in its first-order version and without addition and multiplication — with its single injective function $S$: $(\forall x) \neg Sx = 0$ $\Big(\phi(0)\ \ \&\ ...
Although, beyond any doubts, $ZFC$ is by and large the predominantly accepted theory of sets, there have been a few attempt to establish some serious competitors in town. I just quote two of them ...
This question is closely related to this one: Knuth's intuition that Goldbach might be unprovable. It stems from my ignorance about non-standard models of arithmetic. In a comment on the other ...
This is a question asked out of curiosity, and because I can't understand the wikipedia page. I have often been told that PA cannot prove the validity of induction up to $\epsilon\_0$, which has been ...