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Homedefinition probably too strict

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# definition probably too strict

Hi,

There are examples of propositions which would usually be called equations, but do not fall within your definition of an equatoin as a proposition of the form E1=E2 with E1, E2 expressions with values in a groupoid. Using groupoids seems a bit arbitrary to me, since one could have for example an equation f(x) = y, where f is a continuous map between topological spaces X and Y with x in X, y in Y.

I guess a fairly general setting for equations is set theory, since it makes sense to say that two elements of a set are equal. For categories, proper classes, or other things that are not sets, matters become probably too complex (though I don't know much about these things). On the other hand, equations may involve expressions built from objects that are not in a set, e.g. the equation "dim V = 2", where V is a real vector space. (I also guess one _can_ speak of equality of morphisms of functors, as opposed to isomorphism of morphisms of functors, but that becomes quite complex again.)

Perhaps you could say something like the following: an equation is a proposition of the form E_1 = E_2, where E_1 and E_2 are expressions with values in some set. These expressions may contain indeterminates, and the equation can be said to hold for certain values of these indeterminates (which need not take values in a set). Examples of this definition are the equations f(x) = y and dim V = 2 mentioned above.

It's funny how hard it is to get a good idea what we actually mean by simple terms like "equation", "indeterminate", "theorem", or "proof" that we are using all the time... There are probably definitions of "equation" in logic, but I wonder what they look like.

Thanks,

Peter

**Status:**Accepted

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