clarification of ideas

Let us take a set of symbols where S is {A, B, =}

Now take the set of all combinations of any number of these symbols, which is the inifinite set of all possible sentences containing these symbols where S* is

{A, B, =, AA, BB, ==, AB, A=,B=, AAA, BBB, ===, AAB, AA=, ABA, ABB, AB=, ..., A=B, ..., B=A, ... }

Now a logical system will adopt certain of these sentences and ignore the rest.

We might for example have a system with only two sentences

A=B
B=A

or we might have a Lindenmayer system

A=AB
B=A
A=AB=ABA=ABAAB=ABAABABA (http://en.wikipedia.org/wiki/L-system)

where = means "goes to" and the symbols A and B get replaced according to the two rules.

In the Lindenmayer system if A=AB and B=A is the general rule then an infinite subset of S* can be included in the system. So that the sentence:

A=ABAABABA could be considered a member of the L-system because it could be considered a short hand of the sentence above.

Another version of the L-system might be freedom to apply any of the rules, more like logic:

A=AB
B=A
A=AB=ABB=ABBB=ABBA

In such an odd system ABBA,ABAB,AABB could all be derived by applying rule 1 three times and then rule 2 on any of the B symbols.

The point here is to avoid "thinking" outside the system of symbols and simply to see them as sentences.

Now when we say that the sentences "A=AB=ABA=ABAAB=ABAABABA" and "A=ABAABABA" are in the L-system clearly our inclusion of them is quite arbitrary according to the set of sentences. The meaning and use we ascrive to these sentences is not present within the sentence, any more than the game of rugby can be deduced from the rugby ball.

Two sketched point to work on...

If a sentence A is included in a system and two sentence B and C which are not included in the system are derived from changing two different symbols, can we express non inclusion within the system itself?

A is a member of the system if conforms to some rules say (not not present in the system itself for how else would we decide if those rules were present in the system ad infinitum). B and C are not members if they do not conform to rules, if they cannot be derived from the axioms, i.e. proven. However if they cannot be proven then they are outside the system and yet we know that they were derived by changing two symbols.

Explore this firstly by creating a system in which arbitrary symbols can be changed at will.

Point 2... Godel theorem in short...

If a function was present in a system which could determine if a sentence was provable, i.e. could be derived from the axioms and rules, i.e. was in the system then would sentences containing that function themselves be part of the system. i.e. could they be derived from axioms? there is a problem here but I need a bit more brain power to get over the hurdle...