I appreciate the conversation so I will try to build a common reference so
each additional step to my model can be built on that base and individually
commented on. As requested these are definitions and terms relevant to my
model not necessarily to established mathematics.
1a) FAS: A symbol string judge. It can judge all possible symbol strings
either "acceptable" or "not acceptable".
1b) My FAS contains a single symbol string that is given to be acceptable
called its axiom.
1c) The axiom contains all the allowed symbols - the alphabet.
1d) My FAS contains a set of rules for identifying additional acceptable
symbol strings with the axiom as the basis.
1e) My FAS contains the rule that any acceptable string contains the
encoded FAS as a prefix.
I believe my FAS meets the requirements to make it a FAS in the accepted sense.