Power set of set containing empty set, sets of empty set, and mixes of the former The title might be sort of confusing. The set is an infinite set like
$$
A = \{
\emptyset, \{ \emptyset \}, \{ \{ \emptyset \} \}, ...
\{ \emptyset, \{ \emptyset \} \}, \{ \{ \emptyset \}, \{ \{ \emptyset \} \} \}, ...
...
\}
$$
And it could be defined by the following rules


#1: ∅ ∈ A
#2: if a ∈ A, {a} ∈ A
#3: if a ∈ A and b ∈ A, a ⋃ b ∈ A


Consider a subset of 2 elements of A, like {x, y} ⊂ A, say, x ∈ A and y ∈ A, so {x} ∈ A and {y} ∈ A (#2), then {x} ⋃ {y} ∈ A (#3), as a result {x, y} is also an element of A.

Similar deductions could be applied to subsets of 3 or even more elements. But that should have no chance according to Cantor's theorem. What mistakes I've made and what is the power set of A?
I understand that the problem is that the "set" A appears to contain all subsets of itself and therefore P(A) ⊂ A  so that |P(A)| &lt;= |A| which would certainly upset Cantor.

Taking ZFC as the axiomatic framework to work in, then I don't think that your "set" is a valid contruction. The ZFC axioms were laid down to exclude pathological cases created by arbitrary definition, like Russell's "set" R of all sets which are not members of themseleves (is R ∈ R ?????): by the axioms of ZFC Russell's R is not a set, and nor do I think is yours.

You would have to demonstrate the construction of A quoting from the 10 ZFC axioms as you go along why each step is vaild.

1) ∅ the empty set exists by the axiom of the empty set.

2) {∅} esists by axiom of pairing

3) {∅, {∅}} exists by axiom of paring

4) there exists a set containing ∅ and the "successor" of each of it's elements by the axiom of infinity where the successor of a is {a}

Bit of a problem though to find axioms that will support your third rule. You could  construct a set which contains the union of two specific elements or even a finite number of such, but not I think an infinite collection that satisfied this condition.

On the other hand, if you start from the powerset (which exists by axiom of the powerset)  P(A) then your rule three is valid, but not rule 2: the powerset does not contain the sucessor of all of its elements.
I understand that the problem is that the "set" A appears to contain all subsets of itself and therefore P(A) ⊂ A  so that |P(A)| &lt;= |A| which would certainly upset Cantor.

Taking ZFC as the axiomatic framework to work in, then I don't think that your "set" is a valid contruction. The ZFC axioms were laid down to exclude pathological cases created by arbitrary definition, like Russell's "set" R of all sets which are not members of themseleves (is R ∈ R ?????): by the axioms of ZFC Russell's R is not a set, and nor do I think is yours.

You would have to demonstrate the construction of A quoting from the 10 ZFC axioms as you go along why each step is vaild.

1) ∅ the empty set exists by the axiom of the empty set.

2) {∅} esists by axiom of pairing

3) {∅, {∅}} exists by axiom of paring

4) there exists a set containing ∅ and the "successor" of each of it's elements by the axiom of infinity where the successor of a is {a}

Bit of a problem though to find axioms that will support your third rule. You could  construct a set which contains the union of two specific elements or even a finite number of such, but not I think an infinite collection that satisfied this condition.

On the other hand, if you start from the powerset (which exists by axiom of the powerset)  P(A) then your rule three is valid, but not rule 2: the powerset does not contain the sucessor of all of its elements.