R.E sets, Three valued logic and diagonal argument.
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From: Aatu Koskensilta on 9 Jul 2010 05:08 apoorv <sudhir_sh(a)hotmail.com> writes: > Similarly, given a number n, we can have a proof that it is not the > godel number (for PA) but we can never have a proof that it is the > godel number (for PA). What does "the godel number for PA" mean here? -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: apoorv on 10 Jul 2010 01:31
On Jul 9, 2:08 pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > apoorv <sudhir...(a)hotmail.com> writes: > > Similarly, given a number n, we can have a proof that it is not the > > godel number (for PA) but we can never have a proof that it is the > > godel number (for PA). > > What does "the godel number for PA" mean here? > Let phi(a) (where a is some constant)be a self referential sentence.such that phi(a) <->there is no PA proof of phi(a). By soundness, we have 'phi(a) and there is no PA proof of phi(a)' We want to test whether a given number, say 100 =a .So we have to prove that phi(100) holds, but if we have 100=a, then we also have that 'there is no proof of phi(100)'. In other words, if 100 is indeed the distinguished number we seek, then 'we have no proof that it is indeed the number that we seek' That number will forever elude us. -apoorv |