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, since "Counter example exists" would also mean "[The specific so and so] counter example exists".
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)].
But, both ~P(SS.....S0) and Ex[~P(x)] can be interpreted as "Counter example exists", right?
>> >>> How would one _prove_ ( i.e. _structure theoretically verify_ ) the >>> other-way-around? > > I do not claim that GC, i.e. the absence of a counter example could be > proved like FLT has been proved or like the sum of the first 10^20 > natural numbers can be calculated on a pocket calculator although most > of them cannot be written on that calculator. But it might be possible > that some bright head finds a way to prove GC, i.e., to decide GC > other than by its failure. > > Therefore I have to correct my above chain of equivalences: The last > one is only an implication. Counter example can be found. ==> Goldbach > conjecture is decidable. > > Regards, WM
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