New answers tagged theories-of-arithmetic
2
votes
Is diamond consistent with 2nd order PA?
In the case of $\text{GBC}^-$, the answer is no, $\Diamond$ is not provable from the assumption that all reals are constructible. It is consistent with $\text{GBC}^-$ and even $\text{KM}^-$ that $\...
- 219k
15
votes
Con(PA) via non-well-foundedness?
This is a completely standard perspective in work on models-of-PA, a view that informs dozens of arguments. That is simply the nature of nonstandard models, that things they think are well founded are ...
- 219k
6
votes
In what sense does the sentence $\operatorname{con}(\mathsf{PA})$ "say" that $\mathsf{PA}$ is consistent?
"So, in light of these objections, what viewpoints do logicians have about the relationship between the assertion that $PA$
is consistent, and the arithmetic sentence $con(PA)
$?"
Logicians ...
- 2,317
8
votes
Accepted
In what sense does the sentence $\operatorname{con}(\mathsf{PA})$ "say" that $\mathsf{PA}$ is consistent?
The PA sentence “$\newcommand{\Con}{\text{Con}}\Con(\newcommand{\PA}{\text{PA}}\PA)$” says that PA is consistent in exactly the same ways that the PA sentences representing, say, the fundamental ...
- 18.8k
9
votes
In what sense does the sentence $\operatorname{con}(\mathsf{PA})$ "say" that $\mathsf{PA}$ is consistent?
There is really nothing peculiar about Con(PA) in this regard. Let's take a simpler statement, such as
$$(\exists x \exists y \exists z : xxx + yyy - zzz = 114) \vee (\exists x \exists y \exists z : ...
- 76.8k
3
votes
In what sense does the sentence $\operatorname{con}(\mathsf{PA})$ "say" that $\mathsf{PA}$ is consistent?
I am going to answer the question in the title - In what sense does the sentence con(PA) "say" that PA is consistent? - rather than the question in the body. This is not what the OP wants, ...
- 1,814
5
votes
In what sense does the sentence $\operatorname{con}(\mathsf{PA})$ "say" that $\mathsf{PA}$ is consistent?
Let's go back to the continuum hypothesis. Some, probably most, people think that it really is objectively true or false, despite having different truth values in different models, because they ...
- 41.5k
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