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.

  • 1
    Formal systems are not constructed within a meta-theory, fixed or otherwise, natural language suffices to formulate the rules. Meta-theory is usually developed afterwards to study what goes on in the formal system, but one has no need for it to follow the rules themselves. So even if outputs of algorithms are "contained" in meta-theory, they do not "pre-exist" (or even "exist" at all, for nominalists), and strict finitists reject much of model theory anyway as platonistic. – Conifold Jan 27 at 5:53
  • To clarify, by meta-theory, I just mean any background theory with which one sets up and describes a formal system. So in this case, it would be natural language along with some basic assumptions, such as it's sensible to talk about collections of strings and they exist, etc. Maybe it's due to my training as a mathematician in which we always work within some domain of discourse, universe of sets, category, etc. but I'm confused by certain terminology in the setup of first order logic. For reference, c.f. for example the discussion in Tourlakis vol 1 Logic, pp 10-12. – user43961 Jan 28 at 1:18
  • My main confusions are about the precise meaning of "working within a theory/formal system" and having a "procedure to generate objects variables on demand." Namely, what am I missing when I assume that the object variables and everything we use to construct the formal system necessarily exist a priori within the meta-theory? I have no issue with assuming existence of certain primitives such as symbols, finite sets, etc., but I'm confused with how they do not have to exist a priori in the meta theory. – user43961 Jan 28 at 1:18
  • Aren't we implicitly assuming it's sensible to talk about such structures in the first place before we use it to construct the formal system? Ultimately, I'm assuming this is probably a philosophical question which is why I'm asking here. – user43961 Jan 28 at 1:18
  • The word "implicitly" is so often abused. We spend a long time analyzing a thing into conceptual pieces, and then declare that it was all there from the start, "implicitly". No, we do not "assume implicitly" that it is sensible to talk about X to use X. Nor do we have to, or we won't be able to start using anything due to definitional regress. We still do not know if arithmetic is consistent, but even if it is not, as Wittgenstein said, "it could render very good service; and it would be better for us to modify our concepts than to say that it really not yet have been a proper arithmetic". – Conifold Jan 28 at 1:37

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