On 10/19/2013 1:53 PM, Peter Percival wrote: > Nam Nguyen wrote: >> On 19/10/2013 12:10 PM, Peter Percival wrote: >>> 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. >> >> _Do you first understand the definition itself_ ? >> >> Would you please confirm you now do or still don't? Thanks. > > If I've understood it (the definition of "impossible to know") then my > argument above is valid. If it's valid then you're wrong about Gödel. > So you should be careful about what you ask to be confirmed. > > You have been caught out in a contradiction. Now, what's it to be: > i) you are too dim to recognize it, > ii) you are too dishonest to recognize it, > iii) you admit that your claims about cGC and Gödel are wrong? > Not iii) I bet. >
Goodness. Allow me to compliment you on that piece of reasoning. Given Nam's refusal to concede a non-classical basis for his logic, you are correct.
By the way, Shelah has that on his list of important things... a model theory for arithmetic comparable to the one for set theory.