Complement of a formal system

In set theory because it is easier for me.

I remember reading that Turing argued that a symbol set (alphabet) must be finite for the symbols to have meaning. Does this also apply to the complement of an alphabet? I may have a universal set E composed from the 3 members who I will name 1,2,3. I chose these from an infinite number of possible symbols (one for each counting number). So any universal set is going to be finite but with no upper bound to its possible size (maybe the number of atoms in the universe). The TransUniversal Set (T) will be the Union of all the possible universal sets. T will then also be finite! We can define the complement of a universal set E thus:

E U E` = T

Where E is formed by positive choice, E` is equally defined by negative choice. The same actions define E and E`.

Now I am deviating from standard set theory because I am dealing with meaningful symbols here. {x in N} is not allowed here. Need to work out details why.

So we can then construct our hierarchical system as the union of an infinite recursion with the Power set function. S = …P(P(E)) U P(E) U E. This set is now infinite.

Now if we use some system of Godel numbering then each set maps to a Natural number, and conversely each Natural number maps to a set.

e.g. I’ll use 2 bit bytes for simplicity. The = symbol means corresponds in the metasystem.
{ = 11
} = 00
, = 01

With an empty universal set we can then construct the natural numbers as cardinals of the sets as usual. Each set corresponds to the given code which can be interpreted as a binary number as given in decimal.
{}    = 1100 = 12
{ {} } = 11110000 = 240
{ {},{{}} } = 111100011111000000 = 247744
{ {},{{}},{{},{{}}} } = 11110001111100000111110001111100000000 = 259780452096

In reverse using the meta mapping, the number 15 = 1111 corresponds to the string {{ which isn’t a set in our system.

Worse the random number 267453 = 1000001010010111101 can’t even be mapped.

There are three levels then at which the system maps to Natural Numbers:
Level 1 – S1 – the set of numbers which correspond to valid set strings – this is S. {…,240,…}
Level 2 – S2 – the set of numbers which correspond to strings – which includes nonsense strings. {…,15,…,240,…}
Level 3 – S3 = N – which includes even numbers that cannot be mapped to strings at all. {N}

I imagine that S2 and S3 can be merged with a suitable “choice” of meta system. I’ll assume this and call it S*.

So the complement of S can be defined as:

S U S` = S*

So a system has a complement at two levels. The complement of the elements (character set) and the complement of the set of possible sets of these elements. And now I’ve run out of steam this morning… but can’t these two sets be put into correspondence?