Timeline for A meta-mathematical question related to Hilbert tenth problem
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 26, 2011 at 18:34 | comment | added | Alexander Woo | I suppose I should insist that Y is relatively consistent with X. | |
| Sep 26, 2011 at 18:31 | comment | added | Alexander Woo | How about, "Given any axiomatic system X containing a model M of PA and any Turing machine T that takes first order sentences as input and produces TRUE or FALSE, there exists an axiomatic system Y extending X and a sentence A such that Y proves one answer for the interpretation of A in M and T gives the other answer on A." | |
| Sep 26, 2011 at 18:29 | comment | added | Joël | Or we interpret it as "provability" in some axiomatic of set theory, but then we get the problem that there will always be undecidable statement, that is with your definition, statements that are neither nor true nor false, and then what the algorithm we are talking about is suppose to do when such a statement is fed to it as input? So you say we can change our axiomatic of set theory to get rid of in decidable statement, but as you say it is a little tricky (plus it is not canonical) and I need to understand exactly what we're doing here. | |
| Sep 26, 2011 at 18:25 | comment | added | Joël | Thanks, I now understand that your answer is more or less equivalent to the other ones I received, and I need to read more in the link Emil gave me to be sure I understand them all, and then to see if I can accept them. I'll try to do that tonight. But meanwhile, let me just say that you put the finger exactly on where I have a problem with. Roughly speaking, you guys all define true as true in a model of arithmetic in set theory. But either you interpret the truth in this model as some absolute platonistic truth and we are not better than at the beginning... or you interpret it as what is | |
| Sep 26, 2011 at 18:06 | comment | added | Alexander Woo | No i don't need to believe in the absolute existence of sets - only the axiom system in which the model is constructed needs to believe in the absolute existence of sets. "True or false for that model" means "true or false for that model according to that axiomatic system". (Since I am not a Platonist, that's all it could mean!) (Yes the sentence could be undecidable, but one could just extend the axiomatic system - I guess one has to careful about the order of quantifiers to allow the axiomatic system to be extended after the statement is chosen.) | |
| Sep 26, 2011 at 13:24 | comment | added | Joël | Thanks for your answer, but I do not think that it solves the problem: You need to be a super-platonist to believe to the notion of truth in a model: that is, you need to believe in the absolute existence of sets, not just of integers. | |
| Sep 26, 2011 at 5:23 | history | answered | Alexander Woo | CC BY-SA 3.0 |