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". > > > 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.