Timeline for Taking a theorem as a definition and proving the original definition as a theorem
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 6, 2021 at 22:29 | comment | added | shane.orourke | Also if $d$ satisfies S1, so does $-d$. | |
| Sep 29, 2021 at 8:28 | comment | added | user3840170 | S2 is not a good definition. It leaves gcd(0, 0) undefined where it should be 0. | |
| Sep 29, 2021 at 6:04 | comment | added | mrtaurho | @LSpice Ah, I see. Then I most likely misremembered something here. Thanks for clearing that up! | |
| Sep 28, 2021 at 23:14 | comment | added | LSpice | @mrtaurho, re, certainly I agree that induction for $\omega + \omega$ is stronger than well ordering for $\omega$ (though not than well ordering for $\omega + \omega$, I think?). But I think injectivity of the successor function (subsingleton preimages) is exactly about unique predecessors. Unique successors is just the statement that the successor function is a function. | |
| Sep 28, 2021 at 21:45 | comment | added | mrtaurho | @LSpice This version only guarentees the existence of a unique successor; not predecessor (or I'm confusing things right now). The example $\omega+\omega$ stands regardless, which satisfies well-ordering but not induction. | |
| Sep 28, 2021 at 21:15 | history | edited | LSpice | CC BY-SA 4.0 |
Mild proofreading and TeX
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| Sep 28, 2021 at 21:14 | comment | added | LSpice | @mrtaurho, re, I'm not sure what are the usual Peano axioms, but, in the formulation with which I'm familiar (and that Wikipedia uses), injectivity of the successor map is taken as one of the axioms. | |
| Sep 28, 2021 at 21:00 | comment | added | mrtaurho | As far as I remember, well-ordering is not able to prove the full strength of induction as witnessed by $\omega+\omega$. The crucial problem is that "every natural number has a unique predecessor" requires more than the usual Peano Axioms with Induction replaces by Well-Ordering. | |
| S Sep 28, 2021 at 16:53 | history | answered | Aeryk | CC BY-SA 4.0 | |
| S Sep 28, 2021 at 16:53 | history | made wiki | Post Made Community Wiki by Aeryk |