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. Different
notation 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 or
SS...S0 that fails to observe GC.

Regards, WM