This question primarily concerns dynamical or time-dependent phenomena in philosophy and to what extent such heuristic discourse features in more precise mathematical settings.
In order to model dynamical phenomena, whether in set theoretic, categorical, etc. backgrounds, one often uses directed graph structures, e.g. transition system, which of course necessitates a priori knowledge of the possible outputs. Arguably, the a priori vs a posteriori existence of abstract structures is only a psychological distinction, and I'm wondering how one would model this psychological phenomena mathematically.
To be more precise, suppose we are constructing a formal system within a fixed background meta-theory. In other words, everything we use to construct the system must exist in the meta-theory a priori. Often, one observes dynamical processes built within a formal system which allows the user to generate new strings within the system on demand only beginning with primitives, without necessitating all pre-generated strings to be built into the formal system a priori.
1) Do all such possible output strings as a result of some built-in algorithmic process in the formal system, necessarily exist a priori in the meta-theory itself?
2) If so, is it accurate that a naive psychological analogy be that the human brain plays the role of the formal system which undergoes a dynamical procedure learning more strings and that the meta-theory contain the brain and some external stimuli which in some sense contains all the possible strings?
3) It seems that many finitist responses to constructing a formal system with arbitrarily many symbols are to have an in-built procedure within the formal system which generates new symbols on demand. (I am referring to the stronger finitist stance of objecting even to an algorithmic infinite sequence of symbols). But then, given the above considerations, wouldn't all such symbols necessarily be contained in the meta-theory itself, thus violating the objection to having infinitely many symbols in the meta-theory to begin with?
4) If the above is true, it seems that one necessitates an infinite meta-theory to model a finitist theory. This seems to correlate with the incompleteness theorems in that most of the time, weaker systems cannot prove consistency of stronger systems. Can the correlation between this consistency no-go theorem and the above considerations be made more precise?
Please let me know what I am missing if the above are incorrect due to a misunderstanding in the meta-theory/theory relationship and the definition of such algorithms/procedures.