Timeline for How could a statement be true without proof?
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| Aug 1, 2016 at 7:41 | comment | added | user21820 | This notion is now implied by your updated answer. Note that it may well be the case that ZF is consistent and yet proves "not Con(ZF)", in which case ZF proves that ω does not satisfy "Con(ZF)", and yet if we move out one more level to a higher meta-system MS we can see that the natural numbers in MS do satisfy "Con(ZF)"! If it is difficult for you to grasp this, first consider the analogous situation of T = PA + not Con(PA), in which case you can check that T is consistent but proves "not Con(T)". | |
| Aug 1, 2016 at 7:35 | comment | added | user21820 | Well you said "we have a well-defined notion of truth in the natural numbers" and later "we 'know' what that means". Read ordinarily, this means that you are relying on the intuitive notion of natural numbers as a platonic collection, but the fact is that we do not actually have a way to pin down any such thing. Natural language is problematic (see math.stackexchange.com/a/1867336/21820), so we need some formalizable meta-system. Modern logicians use ZFC, in which ω is a model of PA given by the axiom of infinity, and then define arithmetical truth based on whether ω satisfies it. | |
| Jul 28, 2016 at 7:18 | comment | added | user21820 | For a more precise calibration of possible meta-systems, see my (very) brief outline at math.stackexchange.com/a/1808558/21820. | |
| Jul 28, 2016 at 7:15 | comment | added | user21820 | It's contentious from a philosophical point of view, that's true. But most modern logicians consider it to be so, and even go further than me by always asserting that the meta-system is ZFC. I don't even assume the meta-system to be ZFC, but I insist that one makes clear what is used, because otherwise everything one says is wishy-washy, and often people who refuse to specify their meta-system shift goal-posts. I don't agree with people who claim that mathematics can be done outside a formal system; their 'mathematics' is nothing but philosophical opinion (even if not necessarily false). | |
| Jul 28, 2016 at 7:06 | history | edited | Rob Arthan | CC BY-SA 3.0 | added 151 characters in body |
| Jul 28, 2016 at 4:36 | comment | added | user21820 | While your explanation is not wrong, it is not the one that logicians have in mind for (arithmetical) truth (which I've defined in my answer). One could hence argue that Godel's incompleteness theorems have much less philosophical impact than Tarski's undefinability theorem, though both are essentially the same from a purely mathematical viewpoint (because mathematics is always done in a formal system). Hence I suggest that you clearly specify that "true" and "false" is in mathematics ill-defined without reference to a structure, because it's wishy-washy to say "we know what that means". | |
| Jul 27, 2016 at 19:46 | history | answered | Rob Arthan | CC BY-SA 3.0 |