Quentin Anciaux skrev:
Hi,
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.
Q.E.D.
--
Torgny Tholerus
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Received on Fri Nov 16 2007 - 04:14:32 PST