There is a set A such that, given any set x, x is not a member of A.

A={1}

A={{},A}

p --> q

~p or q

{N}={{1,2,3,...}}

N={N}

Axiom of regularity (Axiom of foundation) implies that no set is an element of itself
Let A be a set such that A is an element of itself and define B = {A}, which is a set by the axiom of pairing. Applying the axiom of regularity to B, we see that the only element of B, namely, A, must be disjoint from B. But the intersection of A and B is just A. Thus B does not satisfy the axiom of regularity and we have a contradiction, proving that A cannot exist.

If n us in N then n+1 is in N

{ A, {A}, {{A}}, {{{A}}}, {{{{A}}}}... }

{ "A", "{A}", "{{A}}", "{{{A}}}", "{{{{A}}}}"... }

{ A, {A}, {{A}}, {{{A}}}, {{{{A}}}}... }

1 <--> A
2 <--> {A}
3 <--> {{A}}
...

A={A}

A={{A}}
A={{{A}}}
A={{{{A}}}}

{ A, {A}, {{A}}, {{{A}}}, {{{{A}}}}... }

a={a}

{a,b,c,…} = {{a},{b},{c},…} ?

x={x} .

N={N}

(http://en.wikipedia.org/wiki/Axiomatic_set_theory)

P(N)

P(P(N))

P(P(P(N)))

...

1 <--> {}
2 <--> {{}}
3 <--> {{{}}}
...

N, P(N), P(P(N)), P(P(P(N)))...

…P(P(P(N))) --> …{{{N}}}…

1 <--> {} <--> 0.9
2 <--> {{}} <--> 0.09
3 <--> {{{}}} <--> 0.009
...

0.9 = 0.9 a
0.99 = 0.9 + 0.09
0.999 = 0.9 + 0.09 + 0.009
0.9999 = 0.9 + 0.09 + 0.009 + 0.0009
... a

0.9999... = 0.9 + 0.09 + 0.009 + 0.0009... a

{} != {{}} a
{{}} != {{},{{}}}
{{{}}} != {{},{{}},{{{}}}}
{{{{}}}} != {{},{{}},{{{}}},{{{{}}}}}

...{{{}}}... = {{},{{}},{{{}}},{{{{}}}},{{{{{}}}}}...} a

0.9999... = 0.9 + 0.09 + 0.009 + 0.0009... a

...{{{}}}... = {{},{{}},{{{}}},{{{{}}}},{{{{{}}}}}...} a

{} = {{}} = 1 a
{{}} = {{},{{}}} = 2 a
{{{}}} = {{},{{}},{{{}}}} = 3 a
{{{{}}}} = {{},{{}},{{{}}},{{{{}}}}} = 4 a
...

|{}| = |{}| = 0 a
|{{}}| = |{ {} }| = 1 a
|{{{}}}| = |{ {},{{}} } = 2 a
|{{{{}}}}| = { {},{{}},{{{}}} } = 3 a
...

|{}| = |{}| = 0 a
|{{}}| = |{ {} }| = 1 a
|{{{}}}| = |{ {},{{}} } = 2 a
|{{{{}}}}| = { {},{{}},{{{}}} } = 3 a
...

...{{{}}}... = {{},{{}},{{{}}},{{{{}}}},{{{{{}}}}}...} a

...{{{}}}... = {{},{{}},{{{}}},{{{{}}}},{{{{{}}}}}...} a

N = {{},{{}},{{{}}},{{{{}}}},{{{{{}}}}}...} a

...{{{}}}... = N

"I know that my approach cannot be grasped easily by persons which are familiar with the standard approach about the successor concept, but at the moment you get it you can see that it is finer than the standard understanding of the successor concept."

"Don't worry about it. You will get it. It takes time to sink in"