EDIT:
The order of some collection of distinct objects has no influence on the number of the objects.
The power set of, for example, {0,1,2} is {{},{0},{1},{2},{0,1},{0,2},{1,2},{0,1,2}}.
Here is some example of the translation of a power set's members into <0,1> form, such that no "{X}" or "{}" forms are used anymore.
By this generalization the power set of some set is 2^(the number of the distinct objects of that set), for example:
{
000 ↔ {}
001 ↔ {0}
010 ↔ {1}
011 ↔ {2}
100 ↔ {0,1}
101 ↔ {0,2}
110 ↔ {1,2}
111 ↔ {0,1,2}
}
So by using <0,1>^X we can construct any power set, such that the members of the set and the members of power set are constructed by the same rule of <0,1> form, which enables us to use the diagonal method without any need of any extra set of indexes, because <0,1> form is its own index system.
Please pay attention that if we use the diagonal method on any arbitrary set of X members (and in this case X=3), which are based on <0,1> form, we get the members of the power set that are not in the range of some X arbitrary members of that set, for example:
If the arbitrary set of 3 distinct members is:
{
111,
100,
101
}
then the diagonal member of the power set that is not in the range of the arbitrary set of 3 distinct members, is 010.
We can change the arbitrary set of 3 distinct members, but always we get some diagonal member of the power set of the arbitrary set of 3 distinct members, which is not in the range of the arbitrary set of 3 distinct members.
By using the common constriction rule of <0,1> form, we are using the diagonal method also on the set of ∞ distinct members, as follows:
{
111… ,
100… ,
101… ,
…
}
The diagonal member of the power set that is not in the range of set of ∞ distinct members, starts (in this case) with 010… <0,1> form, even if X=∞ (also in this case the members of the set and the members of power set are constructed by the same rule of <0,1> form).
Because the set and the power set are based on the same constriction rule of <0,1> form and there are always members of the power set that are not in the range of the set (whether X is finite or not), then no set is complete exactly because every set has a power set and every power set has also power set etc... ad infinitum ...
Furthermore, we can be more precise by using the constriction rule of <0,1> form in order to use the diagonal method on the set of R members.
It is done by using a mirror image of <0,1> form in order to use R members, as follows:
For example, instead of using the right to left construction
{
…000,
…001,
…010,
…011,
…100,
…101,
…110,
…111,
…
}
we are using a left right mirror construction of the factional part of any given R member
{
.000…,
.100…,
.010…,
.110…,
.001…,
.101…,
.011…,
.111…,
…
}
The order of the distinct members has no influence on the result, which clearly demonstrates the incompleteness of any set w.r.t to its power set ad infinitum … , whether the set is finite or infinite.
Godel's first incompleteness theorem demonstrates the validity of a thing according to the rules of a given framework, which can't be proved within the given framework.
By using the diagonal method (without using any bijection) on the set of R members, we define an object that has the properties of that set (it obeys the construction rules of the given framework) but it is not in the range of R set (but can't be proved within this framework).
community simply 
its eyes and ignores Godel's first and second incompleteness theorems for the past 80 years.
↔ ϕn