On Saturday, 1 March 2014 17:46:16 UTC+2, Dan Christensen wrote: > On Saturday, March 1, 2014 1:48:26 AM UTC-5, John Gabriel wrote:
> A more modern approach based on the Peano axioms:
A monkey in a suit is a monkey no less. :-) Btw: Why a more "modern" approach? Would that be because Peano's axioms were crap to start with? :-) What Dannie boy means is that the first statement of the axioms is such rot, that it needs set theory to begin its journey to common sense. Only, set theory didn't cut it either. Tsk, tsk.
> Using this as the formal definition you could derive all the known theorems of number theory (Godel aside). Poor John Gabriel cannot even prove that n=/=n+1 in his system.
Gabriel does not have to prove it. It is assumed in the Peano axioms. Clue: successor function. I hardly think that it makes sense to assume P is true and then proceed to prove P is true. Ha, ha! :-)
> He will fume, you don't have to prove it -- it's "obvious!" By this he means, of course, that he can't prove it.
I am having a good laugh if you really must know! :-)
> > 1. Zero is a natural number. > 1. 0 in N (i.e. 0 is an element of N)
> > So, the first so-called "axiom" contains two undefined and unqualified terms. You shall see how rigorous and sound John Gabriel's axioms are at the end of this comment.
> > 2. Every natural number has a successor in the natural numbers. > 2. S: N --> N (S is function on N)
> > 3. Zero is not the successor of any natural number. > 3. For all x in N, S(x)=/=0
> > 4. If the successor of two natural numbers is the same, then the two original numbers are the same. > 4. S is injective (1-to-1)
> > 5. If a set contains zero and the successor of every number is in the set, then the set contains the natural numbers. > 5. For all subsets P of N, if 0 in P and for all x in P, we have S(x) in P, then every natural number is in P.
> Doesn't John Gabriel look silly?
:-) They are laughing at YOU silly! Ha, ha. It's you who looks silly because you are silly.
> Observe that John Gabriel cannot prove that n=/=n+1.
It needs no proof. :-) Except perhaps to a follower of Cantor. Shouldn't you be asking if I can prove oo =/= oo+1 or oo = oo + 1? Tsk, tsk. Ha, ha!
> Observe that John Gabriel cannot even prove the existence of a number other than 0. He mentions 0 in his "axioms" 2 and 6, but what about the others? Is this result, just "too obvious" to bother with as well, John Gabriel? Oh, well.
Nice try, but truly abysmal. Surely, you can do better? :-)