Timeline for What does it mean for a mathematical statement to be true?
Current License: CC BY-SA 2.5
14 events
| when toggle format | what | by | license | comment | |
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| Jun 1, 2021 at 13:49 | comment | added | Joel David Hamkins | Yes, as I said, one defines the standard model of arithmetic as a set-theoretic object. I generally have ZFC as my background theory. The standard model can be defined as the finite von Neumann ordinals. | |
| Jun 1, 2021 at 13:27 | comment | added | Joel Adler | @Joel: A dumb question, sorry. How is the intended model ⟨N,+,⋅,0,1,<⟩ defined? Don't you need some axioms of ZFC (among else the axiom of infinity) to define the set of natural numbers $\{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\},\ldots\}$? Thus defining "true" for a Peano statement cannot be done in Peano alone? | |
| Jul 2, 2011 at 1:36 | vote | accept | teil | ||
| Jan 20, 2011 at 13:21 | comment | added | Joel David Hamkins | Hans, yes, it is completely trivial, essentially by definition. I view model theory as taking place within the ZFC background theory. | |
| Jan 20, 2011 at 12:17 | comment | added | Hans-Peter Stricker | Another question: Have you any news on the proof of Macintyre concerning FLT in PA? | |
| Jan 20, 2011 at 12:16 | comment | added | Hans-Peter Stricker | But that's a model theoretic definition, not a ZFC theorem, is it? | |
| Jan 20, 2011 at 12:01 | comment | added | Joel David Hamkins | Hans, the recursive definition of the satisfaction relation holds that $\neg\varphi$ is true in a structure if and only if $\varphi$ is not. In other words, a statement is false iff it isn't true. | |
| Jan 20, 2011 at 10:14 | comment | added | Hans-Peter Stricker | @Joel: Dumb question: HOW does ZFC prove that every arithmetic statement is either true or false in the standard model of the natural numbers? | |
| May 15, 2010 at 10:43 | comment | added | Joel David Hamkins | The intended model is $\langle N,+,\cdot,0,1,\lt\rangle$, also known as the standard model of arithmetic. And you are right that one way to understand the Incompleteness Theorem is that it says we cannot write down a complete axiomatization of the theory of this structure. | |
| May 14, 2010 at 20:46 | comment | added | Kevin Casto | Can you clarify what the "intended model of arithmetic" is? It seems to me that the Incompleteness Theorem is really a statement about this model, that no theory can prove every statement that is true in it. | |
| May 12, 2010 at 18:35 | comment | added | Qfwfq | Interesting! Do you know any reference graspable by non-logicians? | |
| May 12, 2010 at 17:31 | comment | added | Joel David Hamkins | It is extremely difficult and is the subject of current philosophical work by Woodin, Koellner, Maddy, Hauser and others. But many large cardinal set theorists seem to espouse a view of this sort. | |
| May 12, 2010 at 17:27 | comment | added | Qfwfq | Can the phrase "these various stronger theories are approaching some kind of undescribable limit theory" be made formally precise in some sense? | |
| May 12, 2010 at 13:07 | history | answered | Joel David Hamkins | CC BY-SA 2.5 |