3
votes
2answers
630 views

Neither Even Nor Odd Natural Numbers

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 ...
6
votes
1answer
151 views

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 $\mathbb N$: ...
1
vote
1answer
772 views

Are there non-commutative models of arithmetic which have a prime number structure?

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 ...
24
votes
3answers
1k views

Even XOR Odd Infinities?

Modular Arithmetic (MA) has the same axioms as first order Peano Arithmetic (PA) except $\forall x (Sx \ne 0)$ is replaced with $\exists x(Sx = 0)$. ...
0
votes
2answers
307 views

Models of the natural numbers in ultrapowers in the universe.

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 ...
1
vote
3answers
442 views

Generalizations of PA and its standard and non-standard models

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)\ \ \&\ ...
10
votes
5answers
1k views

Alternative Arithmetics

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 ...
35
votes
7answers
2k views

How would one even begin to try to prove that a simple number-theoretic statement is undecidable?

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 ...
14
votes
4answers
1k views

What is induction up to epsilon_0?

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 ...