formal systems and spacetime signatures

I wonder about spacetime signatures.


I don't think I've told anyone that I hypothesize that reality actually is a formal system, though what "formal system" would mean there might not be what it is elsewhere in literature so I would of course have to first define what I mean. 


Now, formal systems are very much like laboratories except these are mathematical laboratories.  In these laboratories, there must be four components:
1.  the symbols to be used (e.g., for communicating)


2.  a conglomerate of which utterances are considered in this laboratory to be grammatically correct


3. a conglomerate of which grammatically correct strings of symbols (i.e., statements) are axioms and assumed true in this laboratory


4.  a conglomerate of transformation rules that input so many statements (called premises) and output another set  of statements called conclusions (or the conclusion if there is only one statement in the output).  The transformation rules are called inference rules.


The only other related and relevant concept pertaining to formal systems is what could be called "the consequence closure of the formal system."  In order to discuss this, one needs to define what a proof is and what a theorem is.


A proof of a statement (i.e., grammatically correct utterances) is a finitely long sequence of statements such that, for all statements in the proof, that statement is either an axiom or the output of any of the transformation rules when applied to all previous statements in the proof.


A theorem is just any statement that is embeddable in a proof (which is a finite sequence of statements).  Intuitively, a theorem is true relative to the ambient formal system; within this formal system a theorem is merely a true statement (truth being relative here).


The consequence closure of a formal system is the ensemble of all theorems of that formal system.




Finally, back to spacetime signatures.  If reality is a formal system, then we must be among its theorems.  A parallel universe would just be any formal system embeddable within reality (viewed as a formal system).  Perhaps the consequence closure of that parallel universe is a good way to define its spacetime signature.