In a Consistent System, Can a True Sentence Imply a False One?
in [Logic]
|
Prev: when math defines the boundary between finite versus infinite at 10^500 #696 Correcting Math
Next: FLT like 4Color Mapping, Poincare C. and Kepler Packing #697 Correcting Math
From: Daryl McCullough on 31 Jul 2010 14:42 Nam Nguyen says... >I've had a couple of related questions: > >Q1: Is it impossible (in principle, hence in practice) to know the truth > value of cGC (which is the FOL sentence "There are infinitely many > counter examples of GC")? cGC might be an example of a statement that nobody knows how to prove, or disprove. But there is also no good argument that it is impossible to prove or disprove. > >Q2: Should the answer of Q1 be a yes, is it reasonable to consider the truths > about the natural numbers relative, in general? I don't see why that would follow. -- Daryl McCullough Ithaca, NY
From: Nam Nguyen on 31 Jul 2010 14:56 Daryl McCullough wrote: > Nam Nguyen says... > >> I've had a couple of related questions: >> >> Q1: Is it impossible (in principle, hence in practice) to know the truth >> value of cGC (which is the FOL sentence "There are infinitely many >> counter examples of GC")? > > cGC might be an example of a statement that nobody knows how to prove, > or disprove. But there is also no good argument that it is impossible > to prove or disprove. Firstly, there seems to be a subtlety about mathematical knowledge here: there's something (truth value) possible to know even we don't know yet or practically couldn't know; but there might also be something that's impossible even in principle to know it. As long as the 2nd possibility is still a possibility, then there would be good arguments for the impossibility. > >> Q2: Should the answer of Q1 be a yes, is it reasonable to consider the truths >> about the natural numbers relative, in general? > > I don't see why that would follow. Suppose for the time being the answer for Q1 is a yes, then the truths of the formulas such as cGC or some of the theorems thereof would be relative: depending what we'd include or exclude such truths as arithmetics of the natural numbers. Note though some of the arithmetic truths are still absolute, common or invariant to the choice of the natural numbers: e.g. Ax[x=/=0 -> x > 0]. -- ----------------------------------------------------------- Normally, we do not so much look at things as overlook them. Zen Quotes by Alan Watt -----------------------------------------------------------
From: MoeBlee on 31 Jul 2010 15:05 On Jul 30, 8:42 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > they > (notably AK, CM, MoeBlee, etc...) would use the word "proof" of a 1st > order formula in 2 different senses: there's the syntactical sense via > the rules of inference for sure but they have been talking about proofs > through the intuitive knowledge of the natural numbers. I haven't advocated a sense of proof of a first order formula "through intuitive knowledge of the natural numbers". MoeBlee
From: MoeBlee on 31 Jul 2010 15:15 On Jul 31, 8:32 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Lighten up. "Lighten up," said by Aatu Koskensilta, belongs in a class with "Hey, I'm feeling you on that, bro," said by Kurt Godel, or something like that. MoeBlee
From: Aatu Koskensilta on 31 Jul 2010 15:31
MoeBlee <jazzmobe(a)hotmail.com> writes: > "Lighten up," said by Aatu Koskensilta, belongs in a class with "Hey, > I'm feeling you on that, bro," said by Kurt Godel, or something like > that. This is either slander or libel -- I never remember which is which -- and probably actionable. I expect a prompt and groveling apology. Should an apology not prove forthcoming, I will be forced to pout in impotent anger, and possibly stomp my feet. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |