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