You are deliberately missing the point: you don't understand something, and comment about it. You say we can not comment about things we do not understand. Why can't we do something that you do all the time?
EDIT:
Because going deeper than some paradigm and change it by a paradigm-shift means that one understands the current paradigm as an inseparable factor of going beyond it (of fundamentally change it).
By your paradigm, given some definition it can't be changed, and this inability to be changed is a paradigm of your community.
On the contrary I show that by given deeper notions of some definition, it is changed by a paradigm-shift.
Here is again a concrete example of such a paradigm-shift about collections of distinct objects.
By using <0,1>^k(k=0 to ∞) as a common form for both S and P(S), we show that any non-empty collection of distinct objects is incomplete, no matter if the collocation is finite or not.
<0,1> is the common notation of the forms, ^ is the power operation on some given collection, and k is the cardinality of a given collection of distinct objects, whether it is finite or not.
By generalization, S and P(S) are based on <0,1>^k form, where k = 0 to ∞, as follows:
By <0,1>^0 (2^0)
P(S)=
(
{0}
or
{1}
)
and
S=
(
{}
)
By <0,1>^1 (2^1)
P(S)=
{0,1}
and
S=
(
{0}
or
{1}
)
By <0,1>^2 (2^2)
P(S)=
{00,01,10,11}
and
S=
(
{
10,
1
1
} →
00
or
{
10,
0
0
} →
01
or
{
00,
1
1
} →
10
or
{
01,
1
0
} →
11
)
etc… ad infinitum … and as can be seen, no subsets are used and no 1-to-1 mapping between S and P(S) is needed in order to conclude that by using a common form for both S and P(S), we are able to show an object (the diagonal object) that has the same form of S objects but it is not in the range of S collection of distinct objects. Since P(S) is S w.r.t P(P(S)) exactly as S is w.r.t P(S), then also in the case of P(S) we are able to show an object (the diagonal object) that has the same form of P(S) objects but it is not in the range of P(S) collection of distinct objects, etc… ad infinitum.
By using <0,1>^k as a common form of both S and P(S), it enables us to understand S P(S) beyond the difference of subset\set structures, and we are able to do it without using 1-to-1 mapping ( where 1-to-1 mapping is actually needed exactly because we do not use a common form for both S and P(S) ).
Let PR=“This sentence has no proof” within a given framework.
If PR is true, then there is PR that has no proof within a given framework.
If PR is false, then it is provable.
If PR is provable then there is a proof that contradicts PR, so also in this case it is true that there is PR that has no proof within a given framework.
In the case of S and P(S), by using the diagonal method on S members, we define an object that has the same form of S members but it is not a member of S and we need P(S) in order to define its membership.
But also in the case of P(S) and P(P(S)), by using the diagonal method on P(S) members, we define an object that has the same form of P(S) members but it is not a member of P(S) and we need P(P(S)) in order to define its membership.
Etc. … ad infinitum.
It is equivalent to the case where some framework has true PR but it is not provable within this framework and we need to extend the framework in order to prove it, but then there is a true PR of the extended framework that is not provable within the extended framework, which has to be extended in order to prove it, but then ... etc. … ad infinitum.
--------------------------------------
Since by your paradigm, P(S) is defined as the collection of subsets of S (and in this case you need 1-to-1 mapping between the different structures of S and P(S) objects), and also by your paradigm a given definition can't be changed, then you by your own mind determine the inability to understand OM's new paradigm on this fine and important subject, which rigorously shows that any given collection of distinct object is incomplete, whether it is finite or not.
