Timeline for How is it possible for PA+¬Con(PA) to be consistent?
Current License: CC BY-SA 4.0
10 events
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| Oct 6, 2023 at 13:22 | history | edited | James E Hanson | CC BY-SA 4.0 |
Math error
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| Oct 5, 2023 at 13:41 | history | edited | James E Hanson | CC BY-SA 4.0 |
edited body
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| Oct 5, 2023 at 6:03 | comment | added | James E Hanson | A crucial part of the intuition one develops for non-standard models of arithmetic is the fact that internally everything seems finite. This makes it very subtle to apply ordinary logic about infinite objects to stuff inside a non-standard model of arithmetic. For example, normally if $\kappa$ is an infinite cardinal, then $\kappa = \kappa + 1$, but in a model of arithmetic $n$ is never equal to $n+1$. | |
| Oct 5, 2023 at 5:58 | history | edited | James E Hanson | CC BY-SA 4.0 |
Improved example
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| Oct 5, 2023 at 5:52 | comment | added | James E Hanson | In some sense it's a matter of 'internal' vs. 'external' induction. As far as anything definable using first-order logic is concerned, a non-standard model of PA satisfies induction, but it fails induction externally by virtue of not literally being isomorphic to $\mathbb{N}$. | |
| Oct 5, 2023 at 5:51 | comment | added | James E Hanson | I'm not sure you're using the phrase 'transfinite induction' in the sense that I'm used to. Since the linear order underlying a non-standard model (including this toy model) is never well-founded, transfinite induction (as in induction along ordinals) is never directly relevant. | |
| Oct 5, 2023 at 5:50 | history | edited | James E Hanson | CC BY-SA 4.0 |
edited body
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| Oct 5, 2023 at 5:50 | comment | added | E8 Heterotic | So (at least your toy-model) is akin to a sequence where finite induction works but transfinite induction fails? | |
| Oct 5, 2023 at 5:45 | history | edited | James E Hanson | CC BY-SA 4.0 |
edited body
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| Oct 5, 2023 at 5:39 | history | answered | James E Hanson | CC BY-SA 4.0 |