What's stopping us from defining isomorphism in general?

abstract-algebra

This (appropriately further generalized to higher arities) is indeed the definition of isomorphism in the broad context of first-order structures, which vastly generalizes groups and rings; indeed, operations and relations of infinite arity are not problematic in this context either. However, the point is that some things are not (infinite-arity) first-order structures, or fruitfully interpreted as such: we don't want to a priori limit ourselves to any one particular notion of "mathematical object," and this prevents us from whipping up a precise definition of "same-ness" that we're totally certain will be applicable in all contexts.

This sort of concern crops up in a serious way in category theory, but that's a bit down the road; for now, I would simply take it as a given that there will always be more kinds of mathematical structure yet to see, and that a notion of isomorphism which is satisfactory in all contexts so far may not be applicable in all situations in the future.

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