Timeline for Does ZF+AD have any unusual arithmetic consequences?
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| Apr 6, 2020 at 18:37 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Apr 6, 2020 at 18:11 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Apr 6, 2020 at 15:59 | comment | added | Emil Jeřábek | No, being consistent in $\omega$-logic is a much stronger property than $\omega$-consistency. (The terminology is fairly confusing.) “Sound” is simply overloaded; in the context of theories of arithmetic (and perhaps other theories with a clearly distinguished “standard” model), it means “true in the standard model”, this is unrelated to the notion of a derivation system being sound with respect to a semantics. | |
| Apr 6, 2020 at 15:52 | comment | added | Tim Campion | @EmilJeřábek Thanks so much! I suppose the terminology "$\omega$-consistent" makes some sense -- it looks like a theory is $\omega$-consistent iff it is consistent in $\omega$-logic. But the terminology "sound" for "weakening of true arithmetic" strikes me as strange -- I'm used to "soundness" being a property of a logic rather than a property of a theory. | |
| Apr 6, 2020 at 15:30 | comment | added | Emil Jeřábek | See e.g. en.wikipedia.org/wiki/ω-consistent_theory for more background. | |
| Apr 6, 2020 at 15:23 | comment | added | Emil Jeřábek | The $\omega$-rule generates all of true arithmetic, yes, but you need repeated applications of the $\omega$-rule. Therefore your conclusion that any theory of arithmetic that is not sound (weakening of true arithmetic, as you call it) must prove $\exists x\,\neg\phi(x)$ and all instances $\phi(n)$ , $n\in\omega$, for some formula $\phi(x)$, is incorrect. Theories such that no such $\phi$ exists are known as $\omega$-consistent, and there exist unsound $\omega$-consistent theories. | |
| Apr 6, 2020 at 14:27 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| S Mar 31, 2020 at 18:51 | history | answered | Tim Campion | CC BY-SA 4.0 | |
| S Mar 31, 2020 at 18:51 | history | made wiki | Post Made Community Wiki by Tim Campion |