May 9, 2009

自然數公設

# Peano Axiom #

The Natural Number satisfies the following properties:

1. For a set N, there exist n & n+ (after n) , n n+ (n+)+

2. The element "e" must be in N

3. There doesn't exist any one E s.t. E+ = e

4. For each element in N, a+ = b+ => a=b

5. Let S be a subset of N, "e" belongs to N; n is in N => n+ is in N

then S = N.

i.e. S is the so called "Natural Number set".

Also, e:=1 e+:= 2 (e+)+:=3 ......and so on


Define the additive operator <+> to be a function:

1. <+>(n,e) = n+ i.e. n <+> e = n+

2. <+>(n,m+) = (<+>(n,m))+ i.e. n <+> m+ = (n<+>m)+

<+>:= +

Existence: e, e+ ,(e+)+, ..... are in N

Uniqueness: For each n in N,

<+>(n,e) = n+
<+>(n,e+) = (<+>(n,e)) = (n+)+
<+>(n,(e+)+) = (<+>(n,(e+))+ = ( (<+>(n,e))+)+ = ((n+)+)+
...........


so <+>(e,e) = e+ ,say "1+1=2"

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  • 1樓

    1樓搶頭香

    回訪 :D

  • join702 at September 11, 2009 05:29 AM comment | prosecute
  • 2樓

    2樓頸推

    我都忘了我有去你那邊留言了
    !!!!!!

  • a31238 at September 17, 2009 09:10 PM comment | prosecute

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