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byrnema comments on Supernatural Math - Less Wrong
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Until the time of writing this comment, I did believe that there would be a meta-logic that governs the math or logic that "can exist". However, I concede that such a statement requires proof and I do not have enough (read: any) background in logic to know if such a proof is forthcoming.
This is the question: if there were two physical universes realizing two distinct sets of mathematical ideas such that at least one mathematical idea is realized in one set and not the other, then would mathematics still provide a meta-framework for both sets of mathematical ideas?
More succinctly: Given any two models, can you always find a meta-model that includes them both? Or maybe it can be proven that this is unprovable within a given model?
A logician's input on this would be helpful.
I think this is a confused question. The answer is yes, but when I tell you what the meta-model is, I don't think your underlying curiosity will be satisfied.
Given 2 models A and B, we can construct a model C. The set of objects in C will be the union of the set of elements of the form (A, a) for all objects a in the model A, and the set of elements of the form (B, b) for all objects b in the model B. Then every proposition about A can can correspond to a proposition about C, particularly about its objects of the form (A, a), by simple substitution of (A, a) for a; and similarly propositions about B correspond to propositions about the objects of the form (B, b). This is, of course, a trivial combination in which the parts that correspond to the original models do not interact with each other at all, hence my belief that you will not be satisfied.
Perhaps a better question to ask would be: is there a meta framework that distinguishes some mathematical systems as "good" and others as "bad"? I think the answer to this question is the criteria that a mathematical system be self-consistent, that is, it does not produce contradictions. Of course, this criteria is not always possible to verify.