Timeline for How can we know the well-foundedness of $\epsilon_0$?
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 24, 2020 at 18:16 | comment | added | Nik Weaver | @cody Oh, I see, you just meant that "PA is sound" is stronger than PA. That's right. | |
| Apr 24, 2020 at 17:16 | comment | added | cody | @NikWeaver I just mean that psychologically it's hard to imagine accepting PA without accepting PA + "PA is sound" which also implies $WF(\epsilon_0)$. | |
| Apr 24, 2020 at 8:16 | comment | added | Paul Sohn | Looks like I got the point. PA can prove $WF(\omega_n)$ for all $n$, and accepting the natural numbers gives us the $\omega$-conjunction rule(which lives outside PA), validating $\forall n. WF(\omega_n)$, that is $WF(\epsilon_0)$. I almost forgot about these $\omega$-stuffs. | |
| Apr 24, 2020 at 3:03 | comment | added | Nik Weaver | @cody I'm not sure what you mean. The consistency of PA + all true $\Pi_1$ sentences implies $WF(\epsilon_0)$. This is strictly weaker than "PA is sound". | |
| Apr 24, 2020 at 2:42 | comment | added | cody | @NikWeaver yes, somehow it's very hard to accept PA without implicitly accepting it's soundness, but this is strictly stronger! | |
| Apr 23, 2020 at 15:59 | comment | added | Nik Weaver | @JoshuaZ I don't have a source ... it's "clear" because you can give a uniform description of the $n$th proof $P_n$, and seeing that each of them is a proof in PA is easy. So I guess it would be pretty easy to write down a proof of this in PA, just very tedious. | |
| Apr 23, 2020 at 14:52 | comment | added | Paul Sohn | Might take a while to fully understand the scheme, but I think this is the answer I was looking for. Thanks! | |
| Apr 23, 2020 at 14:42 | vote | accept | Paul Sohn | ||
| Apr 23, 2020 at 14:34 | comment | added | JoshuaZ | @NikWeaver Can you give a source for the claim in your first sentence or a brief sketch of how one would show that? | |
| Apr 23, 2020 at 14:32 | comment | added | Nik Weaver | The funny thing is that you can even prove in PA that PA separately proves well-foundedness of $\omega_n$ for each n. You just can't go from this to "$\omega_n$ is well-founded for all $n$" in PA (by Lob). But if PA is sound then this conclusion does follow and therefore $\epsilon_0$ is well-founded. | |
| Apr 23, 2020 at 13:59 | history | answered | Andreas Blass | CC BY-SA 4.0 |