Let's follow the traditional mathematical notion in order to show that given any non-empty set of distinct elements according to a given form, no possible set S of that form is complete, because:
a) There is no additional distinct element (there is nothing) between two distinct elements, which are constructed according to the given form.
b) Given the power set P(S) of distinct elements of a given form, there is a set S of distinct elements of that form, where S is a placeholder for any possible set of distinct elements of that form.
c) The largest power set P(S) of distinct elements of that form does not exist.
By using <0,1>^k(k=0 to ∞) as a common form for both S and P(S), we show that any possible non-empty S of distinct objects of a given form, is incomplete, no matter if S is finite or not.
<0,1> is the common notation of the forms, ^ is the power operation on some given S, and k is the cardinality of a given S 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>^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,
11
} → 00
or
{
10,
00
} → 01
or
{
00,
11
} → 10
or
{
01,
10
} → 11
)
...
Nothing is changed even if k=∞, because also in this case, given S with distinct elements of a given form, there is a distinct element of that form, which is an element of P(S) but it is not an element of S.
Since (every P(S) of distinct elements of a given form has a power set of distinct elements of that given form) AND (the largest power set of distinct elements of that form does not exist) then no given S with distinct elements of that given form, is complete (no matter if k is the cardinality of a finite or non-finite non-empty S).
