Nam Nguyen wrote: > On 19/10/2013 11:32 AM, fom wrote:
>> And, the meaning of "impossible to know"? > > Right there: right in front of you. > > _A meta truth_ is said to be impossible to know if it's not in the > collection of meta truths, resulting from all available definitions, > permissible reasoning methods, within the underlying logic framework > [FOL(=) in this case].
We don't yet know if PA|-cGC or PA|-~cGC, so we don't know if "PA|-cGC" or "PA|-~cGC" is in the collection of meta truths. So we don't know if it's impossible to know cGC (or ~cGC). Why, then, do you claim that it's impossible to know cGC (or ~cGC)?
Do you know that both cGC and ~cGC are not in the collection of meta truths? If so you must know that neither PA|-cGC nor PA|-~cGC. You should publish your proof. And stop claiming that Gödel's incompleteness theorem is invalid, because if neither PA|-cGC nor PA|-~cGC, then that is an example of incompleteness.
Also if you know that neither PA|-cGC nor PA|-~cGC, then you've proved PA consistent. So you should stop claiming that its consistency is unprovable.
-- The world will little note, nor long remember what we say here Lincoln at Gettysburg