On 27 Sep 2011, at 20:02, meekerdb wrote:

On 9/27/2011 5:28 AM, Jason Resch wrote:



On Tue, Sep 27, 2011 at 6:49 AM, Stephen P. King <stephe...@charter.net > wrote:
On 9/26/2011 7:56 PM, Jason Resch wrote:






Okay, there may be other subjects, besides number theory and arithmetical truth where other forms of logic are more appropriate. For unambiguous propositions about numbers, do you agree with the law of the excluded middle?

Jason

I think this an assumption or another axiom. Consider the conjecture that every even number can be written as the sum of two primes. Suppose there is no proof of this from Peano's axioms, but we can't know that there is no proof; only that we can't find one. Intuitively we think the conjecture must be true or false, but this is based on the idea that if we tested all the evens we'd find it either true or false of each one. Yet infinite testing is impossible. So if the conjecture is true but unprovable, then it's undecidable.

Undecidable does not entails the negation of the law of the excluded middle.

Undecidable (by PA, say) = ~Bp & ~B~p. Law of the excluded middle = (p V ~p)

(~Bp & ~B~p) -> ~(p V ~p) is NOT a theorem of G*.

Bruno

http://iridia.ulb.ac.be/~marchal/



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