4
votes
1answer
74 views

How can I imagine a model of $\mathsf{(ZF-Inf)} + \neg \mathsf{Con(ZF-Inf)}$?

Gödel's second incompleteness theorem states that if $\mathsf{ZF-Inf}$ is consistent, then $\mathsf{ZF-Inf} \nvdash \mathsf{Con(ZF-Inf)}$. Moreover, if $\mathsf{(ZF-Inf)} + \neg \mathsf{Con(ZF-Inf)}$ ...
7
votes
1answer
157 views

Gödel's way of teaching non-standard models to Takeuti.

In Memoirs of a Proof Theorist, Gaisi Takeuti relates how Gödel taught him about nonstandard models in an "interesting" way: It went as follows. Let T be a theory with a nonstandard model. By ...
0
votes
1answer
81 views

Peano's theory of arithmetic and Gödel's 1st Incompleteness Theorem

Let $\mathcal{N}$ be Peano's 1st order theory of arithmetic and $\mathscr{A}$ it's standard model (which we assume exists). Infer from Gödel's 1st Incompleteness Theorem that there exists a closed ...
1
vote
1answer
137 views

Questions on a sentence in ZFC asserting that ZFC has a model

Question 1. Let $\operatorname{con}(\mathsf{ZFC})$ be a sentence in $\mathsf{ZFC}$ asserting that $\mathsf{ZFC}$ has a model. Let $S$ be the theory $\mathsf{ZFC}+\operatorname{con}(\mathsf{ZFC})$. ...
5
votes
4answers
147 views

Is every theorem of PA true in the standard model of number theory $N$?

My understanding is that every theorem $\phi$ of $PA$ is true in $N$ because $N$ is a model for $PA$, $N\models PA$. By completeness of first order logic, "$PA\vdash\phi$" implies that "if $N\models ...
5
votes
2answers
123 views

Running programs in nonstandard models of PA

I came across the following problem in several places, to paraphrase: Let $T$ be a recursively axiomatizable, consistent extension of PA. Then there exists some $e$ such that the $e'$th program ...
10
votes
2answers
394 views

Is Gödel's incompleteness theorem provable without any model-theoretic notion?

The entry on Gödel's incompletenss theorem in Wikipedia says: Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, ...
6
votes
2answers
185 views

How is the standard model of number theory specified, and why can't we use that specification to prove any number theoretical sentence of interest?

According to Gödel's incompleteness theorem, there exists a sentence $G$ in the vocabulary of number theory ($N$) which is not provable from any (recursively enumerable) consistent set of axioms $T$, ...
1
vote
2answers
204 views

Is this incompleteness result easier to get than incompleteness of PA?

Gödel's theorem for Peano Arithmetic shows that (under consistency hypothesis on PA) there is a statement which cannot be proved or disproved within PA that is true under the standard model ...