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I have the following, rather naive question:

To what extent can the a priori existence of mathematical objects be reasonably compared with the seemingly a posteriori existence of objects established for instance, by a computing language?

To elaborate:

When working in first order logic, one often imposes a collection of axioms, and then assumes in the meta-theory the a priori existence of some universe of discourse satisfying the axioms, namely a model, in order to attach semantics. In particular, the existence of such a platonic universe is necessarily established a priori.

On the other hand, when working with a computing language for example, one utilizes certain primitives with which one may generate "new" objects of interest, for instance new configurations of strings or new configurations of certain systems in an automata. The resulting configurations of such a system seem to exist a posteriori.

My question is then, to what extent can these two notions of existence be reasonably compared? In what sense can one system exhibit the notion of existence in the other? Is it the case that mathematics assumes the existence of abstract objects a priori, while the a posteriori existence of objects in a computing language as cited above is merely a physical instantiation of all possible configurations available to the computer a priori?

Or is it the case that in order to compare these notions of existence, that if we wish to model some universe describing the dynamics of a computing system, that we demand it be closed under all its in-built functions, and in that sense all possible configurations exist a priori? What is the corresponding notion of existence within a computing system that plays the role of the platonic universe? Does it exist a priori, or must it be explicitly given as the result of a computational process? (One can instead phrase the question in terms of the physical universe, but restricting to a computing system places clearer constraints.)

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In first order logic, we don't assume existence of any model, we are quantifying over all possibly existing models. If we show something follows from our axioms, it means that all models that satisfy our axioms also satisfy the conclusion. If, for some reason, there is no such model (because the axioms were inconsistent), then our proof is still valid, just not very useful maybe (it might still be useful if the proof can be applied to changed, consistent axioms).

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