Maybe it’s well known to experts, but is there always an embedding $f$ of the standard model of Peano arithmetic into a nonstandard model? By Peano arithmetic I mean its first-order version, with the axiom schema of induction for each formula. I’m stuck at the step how to show $f$ is well defined. We can […]

It is easy to come up with objects that do not satisfy the Peano axioms. For example, let $\Bbb{S} = \Bbb N \cup \{Z\}$, and $SZ = S0$. Then this clearly violates the axiom that says that $Sa=Sb\to a = b$, unless we agree that $Z=0$, in which case what we have is exactly the […]

Gödel’s second incompleteness applies, for instance, to r.e. extensions of PA. I am wondering if it applies more generally to arithmetically definable extensions of PA. I see that there is a complete consistent $\Sigma^0_2$ extension of PA. Said theory therefore contains either the sentence “I am consistent” or “I am inconsistent”. However, just because it’s […]

This might be an easy question, but I still struggle to comprehend non-standard models for Peano axioms. I understand that Godel Theorem tells us that the theory defined by Peano axioms is not complete and therefore there exist propositions which are not provable with Peano axioms. So my question is how do we construct or […]

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