Hi,
Le Thursday 15 November 2007 14:45:24 Torgny Tholerus, vous avez écrit :
>> Bruno Marchal skrev:
> Le 14-nov.-07, à 17:23, Torgny Tholerus a écrit :
>>> What do you mean by "each x" here?
>
>
>
> >I mean "for each natural number".
>
>
> What do you mean by "each" in the sentence "for each natural number"? How
> do you define ALL natural numbers?
>
There is a natural number 0.
Every natural number a has a natural number successor, denoted by S(a).
There is no natural number whose successor is 0.
Distinct natural numbers have distinct successors: if a ≠ b, then S(a) ≠ S(b).
You need at least the successor axiom. N = {0 ,1 ,2 ,3 ,... ,N ,N+1, ..}
All natural numbers are defined by the above.
>
>
> How do you prove that each x in N has a corresponding number 2*x in E?
> If m is the biggest number in N,
By definition there exists no biggest number unless you add an axiom saying
there is one but the newly defined set is not N.
Quentin Anciaux
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