Quentin Anciaux skrev:

Le Thursday 15 November 2007 14:45:24 Torgny Tholerus, vous avez écrit :
  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).

What do you mean by "Every" here?  Can you give a *non-circular* definition of this word?  Such that: "By every natural number I mean {1,2,3}" or "By every naturla number I mean every number between 1 and 1000000".  (This last definition is non-circular because here you can replace "every number" by explicit counting.)

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.

I can prove by induction that there exists a biggest number:

A) In the set {m} with one element, there exists a biggest number, this is the number m.
B) If you have a set M of numbers, and that set have a biggest number m, and you add a number m2 to this set, then this new set M2 will have a biggest number, either m if m is bigger than m2, or m2 if m2 is bigger than m.
C) The induction axiom then says that every set of numbers have a biggest number.


Torgny Tholerus

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