```Date: Apr 6, 2013 1:38 PM
Author: mueckenh@rz.fh-augsburg.de
Subject: Re: Matheology § 224

On 6 Apr., 19:23, Nam Nguyen <namducngu...@shaw.ca> wrote:> On 06/04/2013 11:06 AM, WM wrote:>>>>>> > On 6 Apr., 18:25, Nam Nguyen <namducngu...@shaw.ca> wrote:>> >>>> Goldbach conjecture is false. <==> Counter example exists. <==>> >>>> Counter example can be found. <==> Goldbach conjecture is decidable.>> >>>> The second equivalence requires to neglect reality. But in mathematics> >>>> this is standard.>> >>> But, to start with, how would one _structure theoretically prove_ the> >>> 1st equivalence:>> >>> "Goldbach conjecture is false. <==> Counter example exists.">> >>> ?>> >>> Logically:>> >>> (A _specific_ counter example exists) => (Goldbach conjecture is false).>> > I disagree. There is an equivalence, not merely an implication. "GC is> > false" is the same statement as "There exist at least one counter> > example to GC".>> That's not precisely what you had claimed previously:>> "Counter example exists" and "There exist at least one counter example">> aren't necessarily the same,They are absolutely the same. since "Counter example exists" would also> mean "[The specific so and so] counter example exists".Every existing counter example is a specific one.>> 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. Differentnotation does not make different meaning.>> But, both ~P(SS.....S0) and Ex[~P(x)] can be interpreted as> "Counter example exists", right?>Both are the same. ~P(x) means necessarily that there is a number x orSS...S0 that fails to observe GC.Regards, WM
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