On 06/04/2013 12:27 PM, WM wrote: > On 6 Apr., 19:53, Nam Nguyen <namducngu...@shaw.ca> wrote: >> On 06/04/2013 11:38 AM, WM wrote: >> >> >> >> >> >>> On 6 Apr., 19:23, Nam Nguyen <namducngu...@shaw.ca> wrote: >> >>>> In details: >> >>>> We do have the logical equivalence: >> >>>> ~Ax[P(x)] <-> Ex[~P(x)] >> >>>> But we don't have this equivalence: >> >>>> ~P(SS.....S0) <-> Ex[~P(x)]. >> >>>> Right? >> >>> No. Unless SS...S0 is fixed it is the same as x for x in |N. Different >>> notation does not make different meaning. >> >> It was just unclear to you. In my presentation above SS.....S0 is >> a _fixed_ constant, _not_ a variable. > > Every counterexample of GC, if existing, is a fixed natural number. > But I do not pretend that I know such an SS...S0. And that statement > with, say 42, does not play a role in my arguing. >> > ~GC <==> Counter example with a fixed n in |N exists. <==> > Fixed n in |N can be found. ==> Goldbach conjecture is decidable.
You're incorrect in that:
~GC is syntactically of the form Ex[P(x)]. No "fixed n" is required. Period.
Nor is "fixed n" grammatically correct in FOL.
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