LOL. Considering a finite case: if |S| = 3, then |P(S)| = 23. According to you, 3 = 23.
EDIT:
You are still missing my argument about S and P(S).
According to it the cardinality of natural numbers with any collection of different objects, exists between non-bijection to bijection.
When we are dealing with finite collections of different objects, if we understand each collection in terms of fixed cardinality, one of the results is that two finite collections have different amount of objects, exactly as can be seen in the particular case of {1,2,3} and {{},{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}.
By using the set of natural numbers together with Cantor's constriction method, we are no longer closed under the different amounts of {1,2,3} and {{},{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}, as follows:
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.
.
|
|
|-----------------------
|1 ↔ {1} |
|2 ↔ {2} | Provides
1 ↔ {}
|3 ↔ {3} |
|-----------------------
|
|
|-----------------------
|1 ↔ { } |
|2 ↔ {2} | Provides
2 ↔ {1}
|3 ↔ {3} |
|-----------------------
|
|
|-----------------------
|1 ↔ {1} |
|2 ↔ { } | Provides
3 ↔ {2}
|3 ↔ {3} |
|-----------------------
|
|
|-----------------------
|1 ↔ {1} |
|2 ↔ {2} | Provides
4 ↔ {3}
|3 ↔ { } |
|-----------------------
|
|
|-----------------------
|1 ↔ { } |
|2 ↔ {3} | Provides
5 ↔ {1,2}
|3 ↔ {1,3} |
|-----------------------
|
|
|-----------------------
|1 ↔ { } |
|2 ↔ {2} | Provides
6 ↔ {1,3}
|3 ↔ {1,2} |
|-----------------------
|
|
|-----------------------
|1 ↔ {1} |
|2 ↔ {3} | Provides
7 ↔ {2,3}
|3 ↔ {1,2} |
|-----------------------
|
|
|-----------------------
|1 ↔ { } |
|2 ↔ {1} | Provides
8 ↔ {1,2,3}
|3 ↔ {2} |
|-----------------------
|
|
.
.
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We actually deal with a DNA-like code between 3 and 8 objects, and by using it are able to define any mapping degree between |S| and |P(S)| for our purpose.
Such flexibility is impossible by the rigid |S| < |P(S)| particular case, which is wrongly taken as some universal principle of the possible relations between S and P(S).
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Furthermore, by using infinite sets we discover that the whole idea of mapping between sets is changed, because we are also able to define a bijection between S and its proper subset, as can be seen bet between the natural numbers and, for example, their proper subset of even (or odd) numbers.
We also are able to define a bijection between rational numbers and their proper subset of natural numbers.
This ability does not stop also in the case of real numbers, where natural numbers are their proper subset, and the possible bijection between real numbers (which are equivalent by their amount to the powerset of natural numbers) is shown by using Cantor's construction method, as follows:
Cantor's construction method constructs explicit |{{},...,{1,2,3,...}}l {{},...,{1,2,3,...}} members as follows:
1) Some defined explicit {{},...,{1,2,3,...}} member is the result of a 1-to-1 correspondence between {1,2,3,...} members and the same amount of members taken from {{},...,{1,2,3,...}}.
2) This explicit constructed {{},...,{1,2,3,...}} member ( notated as
D. Please read more about
D construction in
http://en.wikipedia.org/wiki/Cantor's_theorem ) includes a {1,2,3,...} member only if this {1,2,3,...} member does not exist as one of the members of the {{},...,{1,2,3,...}} member that is mapped with it.
By using this construction method with amount |{{},...,{1,2,3,...}}l , one enables to define a bijection between {1,2,3,...} members
and {{},...,{1,2,3,...}} members, where
D is actually a placeholder for |{{},...,{1,2,3,...}}l {{},...,{1,2,3,...}} members.
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In both cases, whether the considered set is finite or not, by using Cantor's construction method, one enables to define any wished degree of mapping between two sets, and use the resulted mapping for some useful purpose.
