2
votes
3answers
292 views

Can a Decidable Theory Have Non-recursive Models?

Tennenbaums' theorem proves neither addition nor multiplication can be recursive in any countable non-standard model of arithmetic. Tennenbaum's proof applies to theories much weaker than PA. ...
2
votes
0answers
95 views

Peano (Dedekind) categoricity

What is the smallest fragment of second order logic such that $Th(\mathbb{N})$ in that logic is categorical (only one model, namely natural numbers, up to isomorphism). For example, can we do this in ...
0
votes
1answer
280 views

Recursive Non-standard Models of Modular Arithmetic? [closed]

Any algebraically closed field (ACF) is a model of Modular arithmetic (MA). (MA) has the same axioms as first order Peano arithmetic (PA) except $\forall x(Sx \neq 0)$ is replaced with $\exists ...
3
votes
3answers
164 views

Turing Functional and $\Sigma_1^0$-formulas in models of fragments of PA

In models of PA with restricted induction power (for example, only $I\Sigma_n$ is present), the failure of higher induction scheme is characterised by the existence of definable cuts (like $\Sigma_2$ ...
1
vote
1answer
773 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 ...
0
votes
0answers
269 views

Definitions for Oddness

In the thread Even Xor Odd Infinities I defined odd models of Modular Arithmetic (MA) as models satisfying the axioms of MA and two first order statements. Even XOR Odd Infinities? $O1) \forall x(x=0 ...
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)\ \ \&\ ...
12
votes
3answers
1k views

Is there a known natural model of Peano Arithmetic where Goodstein's theorem fails?

(I've previously asked this question on the sister site here, but got no responses). Goodstein's Theorem is the statement that every Goodstein sequence eventually hits 0. It turns out that this ...