Timeline for Can transfinite induction be defined as axiom scheme in FOL on bin-tree structures?
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
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| Apr 8, 2011 at 10:51 | answer | added | Emil Jeřábek | timeline score: 2 | |
| Dec 17, 2010 at 5:05 | answer | added | none | timeline score: 0 | |
| Oct 28, 2010 at 19:22 | comment | added | Lucas K. | The deeper reason for my question is that I consider a pair operator (which makes bin-trees possible) more natural than PA with addition and multiplication. I do not object against ordinal numbers, but I consider them even less natural. So, I am curious if simple logic (FOL) can be build up in a different way, using pairs and if this is even stronger, because it can prove Goodstein. | |
| Oct 22, 2010 at 1:36 | comment | added | Henry Towsner | Transfinite induction up to a given ordinal can be added to FOL as an axiom scheme; certainly, enough induction to prove Goodstein's theorem can be added as an axiom. Therefore I expect that the answer to your question is certainly yes; if you're willing to do it really awkwardly, you could encode numbers in trees, then copy over all the axioms you'd need to prove Goodstein's theorem in an extension of PA. As for finding a more natural way, I don't see an easy way to do that using binary trees, but that doesn't mean there isn't one. | |
| Oct 8, 2010 at 0:48 | answer | added | Steven Stadnicki | timeline score: 0 | |
| Aug 12, 2010 at 22:54 | history | edited | Lucas K. | CC BY-SA 2.5 |
Clarifacation
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| Aug 12, 2010 at 3:38 | answer | added | Christoph-Simon Senjak | timeline score: 0 | |
| Aug 11, 2010 at 18:21 | history | asked | Lucas K. | CC BY-SA 2.5 |