# Tagged Questions

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### Non-standard proofs of standard theorems

In Richard Kaye's book Models of Peano arithmetic, one can read (page 13): We have proved that any nonstandard $M \models \mathrm{Th}(\mathbb{N})$ has a nonstandard $a \in M \models \theta(a)$ iff ...
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### Is this first order version of the Collatz conjecture decidable in peano arithmetic?

Let $\phi(x)$ be a first order formula in the language of arithmetic with one free variable $x$. Consider the sentence $\psi_\phi$, defined as: \phi(0)\wedge \phi(1) \wedge (\forall x \phi(x) \to ...
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### Primes in nonstandard models of PA

What is known about prime numbers in nonstandard models of PA? Restricted to true natural numbers the sets are identical, but does there always exist nonstandard primes? Can we explicitly define one ...
As in Godel's incompleteness theorem natural numbers encode proofs of theorems. Due to Godel's completeness theorem there is a natural number (in some nonstandard model) that proves $Con(PA)$. What ...
Why is (1) a copy of $\mathbb{N}$ "followed by" a copy of $\mathbb{Z}$ not a (non-standard) model of arithmetic, neither (2) a copy of $\mathbb{N}$ followed by an infinite sequence of copies of ...