作者lucses

證明１＋１＝２

We will proceed as follows: we define


0 = {}.


In order to define "1," we must fix a set with exactly one element;

thus


1 = {0}.


Continuing in fashion, we define


2 = {0,1},

3 = {0,1,2},

4 = {0,1,2,3}, etc.


The reader should note that 0 = {}, 1 = {{}}, 2 = {{},{{}}}, etc.

Our natural numbers are constructions beginning with the empty set.


The preceding definitions can be restarted, a little more precisely,

as follows. If A is a set, we define the successor of A to be the set

A^+, given by


A^+ = A ∪ {A}.


Thus, A^+ is obtained by adjoining to A exactly one new element,

namely the element A. Now we define


0 = {},

1 = 0^+,

2 = 1^+,

3 = 2^+...