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.