Timeline for what ordinals were used in proving unique factorization over $\mathbb{Z}$
Current License: CC BY-SA 3.0
9 events
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| Mar 11, 2017 at 20:20 | comment | added | Noah Schweber | @johnmangual I've added a bit more about recursive proofs, I think it makes it clearer why the Shoenfield completeness theorem isn't obvious (and why the height of the proofs is $\omega^2$, not $\omega$) despite the ease of the $\Pi^0_1$ case. This obviously isn't directly relevant to your question, but I think it's useful for providing a full picture; if you disagree, let me know and I'll delete it. | |
| Mar 11, 2017 at 20:19 | history | edited | Noah Schweber | CC BY-SA 3.0 |
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| Mar 11, 2017 at 19:16 | vote | accept | john mangual | ||
| Mar 11, 2017 at 19:16 | comment | added | john mangual | well... it's a good ting I asked. | |
| Mar 11, 2017 at 19:15 | history | edited | Noah Schweber | CC BY-SA 3.0 |
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| Mar 11, 2017 at 19:12 | comment | added | Noah Schweber | @johnmangual I didn't say that everything is provable in PA. It turns out that PA with the recursive $\omega$-rule, though, does prove every true statement in the language of arithmetic, regardless of whether those statements are true in PA; I've edited to clarify that in the $\Pi^0_1$ case specifically. Also, the prime number theorem absolutely is provable in PA, and indeed much less. PA suffices for the vast, vast majority of number theory; indeed FLT is generally believed to be PA-provable. | |
| Mar 11, 2017 at 19:02 | comment | added | john mangual | +1 not everything is provable in Peano Arithmetic. the prime number theorem is not. Fermat's Little Theorem and Quadratic Reciprocity certainly are. So it sounds like I could be happy with structures in PA alone. | |
| Mar 11, 2017 at 17:48 | history | edited | Noah Schweber | CC BY-SA 3.0 |
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| Mar 11, 2017 at 17:41 | history | answered | Noah Schweber | CC BY-SA 3.0 |