if we construct our infinite squence of infinite sequences as simple counting with zero padding.

s(0) = 0...

s(1) = 10...

s(2) = 010...

...

and so on. Then all natural numbers will be enumerated. The diagonal would then be sd = 11...

A number in sd that at any sub index (i.e. from 1 to x) is going to be larger than any infinite series with the same index, i.e. s(x) = x but sd(x) = 2

^{x}- 1.

Now any number is going to be within the set s(x) as such 2

^{x}can be found at index s(2

^{x}) for any x.

So any subset of sd is going to be within s.

So is the impossibility of finding sd within the set of s(x) related to the faster growth of 2

^{x}?

and is the requirement of not just any faster growth (i.e. x

^{2}) related to the property of power sets?

i.e. that the cardinality of N is less than that of 2

^{N}but not so with N

^{2}?

If this is the case how about even faster growing functions (x

^{x}, busy beaver)?

Or are these functions less well defined in set theory and combinatorics perhaps?