Given any two models, can you always find a meta-model that includes them both?
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.
Yes, the question was confused. I got distracted thinking about stuff I don’t know about (asking “how” instead of “whether”).
You don’t need “meta-logic”, whatever that might be, to know that 2+2=3 cannot be consistent with Peano arithmetic.
Here’s the “proof”. We know that 2+2=4 in “our” Peano Arithmetic. Suppose that 2+2=3 in “another” Peano Arithmetic in an alternate reality. Then the two Peano Arithemetics are actually different because they have a different set of “trues”. When we say that 2+2=4 in “our” Peano Arithmetic, we mean “our” PA and not the other. Whatever distinguishes the two arithemetics can be incorporated in our PA as an axiom – indeed it was included implicitly in what we meant by PA all along even if we lacked the imagination to explicitly identify it.
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.
Yes, the question was confused. I got distracted thinking about stuff I don’t know about (asking “how” instead of “whether”).
You don’t need “meta-logic”, whatever that might be, to know that 2+2=3 cannot be consistent with Peano arithmetic.
Here’s the “proof”. We know that 2+2=4 in “our” Peano Arithmetic. Suppose that 2+2=3 in “another” Peano Arithmetic in an alternate reality. Then the two Peano Arithemetics are actually different because they have a different set of “trues”. When we say that 2+2=4 in “our” Peano Arithmetic, we mean “our” PA and not the other. Whatever distinguishes the two arithemetics can be incorporated in our PA as an axiom – indeed it was included implicitly in what we meant by PA all along even if we lacked the imagination to explicitly identify it.