According to the traditional view the elements of a given powerset are subsets of a given set.
Cantor's theorem of powersets (which is based on comparison of mapping between the elements of a given set (notated as S) and the elements of its powerset (notated as P(S)) is based of two steps ( here
http://www.mathacademy.com/pr/prime/articles/cantor_theorem/index.asp you can find a simple explanation of it):
1) It shows that the number of elements of P(S) are at least equal to the number of elements of S.
2) It enables to construct an explicit element of P(S), which any attempt to define mapping between this element and some S element, necessarily leads to logical contradiction, which enables to conclude that no S element is mapped with this explicit element of P(S), or in other words, there are more P(S) elements than S elements even if we deal with infinitely many elements.
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We are not dealing here with Cantor's theorem of powersets.
Instead we research its construction method, which enables us to construct explicit P(S) elements which are always beyond the range of the mapping of S elements.
In this case S and P(S) elements can be mapped multiple times, according to the following rules (let's demonstrate it on S = {a,b,c}):
1) Under multiple given mappings each element of S (under a given mapping) is mapped with some element of P(S), such that under a given mapping (which is some case of multiple mappings), no element of S is mapped more than once with some element of P(S).
2) Under a given mapping if the element of S is not included in the element of P(S), then we define an element of P(S), which includes this element.
Each given mapping is resulted by some P(S) element that is beyond the range of the mapping of S elements.
Here is some example of such multiple given mappings:
a ↔ {a,b,c}
b ↔ {b,c}
c ↔ {a,c}
is resulted by {}, which is a P(S) element that is beyond the range of the mapping of S elements.
a ↔ {b,c}
b ↔ {a,b}
c ↔ {a,c}
is resulted by {a}, which is a P(S) element that is beyond the range of the mapping of S elements.
a ↔ {a,c}
b ↔ {c}
c ↔ {b,c}
is resulted by {b}, which is a P(S) element that is beyond the range of the mapping of S elements.
a ↔ {a}
b ↔ {b}
c ↔ {}
is resulted by {c}, which is a P(S) element that is beyond the range of the mapping of S elements.
a ↔ {b}
b ↔ {a}
c ↔ {c}
is resulted by {a,b}, which is a P(S) element that is beyond the range of the mapping of S elements.
a ↔ {}
b ↔ {b}
c ↔ {a,b}
is resulted by {a,c}, which is a P(S) element that is beyond the range of the mapping of S elements.
a ↔ {a}
b ↔ {c}
c ↔ {a,b}
is resulted by {b,c}, which is a P(S) element that is beyond the range of the mapping of S elements.
a ↔ {b}
b ↔ {a}
c ↔ {}
is resulted by {a,b,c}, which is a P(S) element that is beyond the range of the mapping of S elements.
Please be aware of the fact that this construction method is equivalent to the results of the S and P(S) elements that are based on the same type of forms, for example:
1) The forms are constructed by 0;1 symbols.
2) Instead of using one S and one P(S) with explicit elements (as done in the mapping case), we are using one P(S) with explicit elements, and multiple sets which are not necessarily have the same elements, but their elements have the same form of the P(S) elements.
3) Instead of using mapping between S and P(S) elements (and research mapping ranges) we define P(S) elements that are not one of the considered S elements, by using Diagonalization, for example:
100
010
001
is resulted by 000, which is a P(S) element that is not one of the given S elements.
000
010
001
is resulted by 100, which is a P(S) element that is not one of the given S elements.
000
010
001
is resulted by 100, which is a P(S) element that is not one of the given S elements.
000
001
101
is resulted by 110, which is a P(S) element that is not one of the given S elements.
100
011
110
is resulted by 001, which is a P(S) element that is not one of the given S elements.
010
011
110
is resulted by 101, which is a P(S) element that is not one of the given S elements.
110
001
010
is resulted by 011, which is a P(S) element that is not one of the given S elements.
010
001
110
is resulted by 111, which is a P(S) element that is not one of the given S elements.
The privilege of the second method is the ability to conclude that no given set is complete, exactly because P(S) and S elements are actually the same type of form (and also no P(S) is incomplete, since the largest P(S) does not exist).
This conclusion can't be achieved by the first method, because S and P(S) elements do not share the same type of form.
