[zmgsabst]

> That is, an algebraic manipulation that we can reduce to an integer under certain circumstances, but it doesn't "exist" the same way that an integer or a rational exists.

This depends on your interpretation: some view the reals as completions of that process, in which those “verbs” are “nouns”.

But you can construct a coherent theory in which this is not the case — and nobody is much fussed, because mathematics is full of weird theories and interpretations.

And both integers and rationals are defined by their relations, eg, integers are equivalence classes of pairs of naturals and rationals as equivalence classes of pairs of integers — where the class obeys some algebraic manipulation properties. If you feel there’s some great difference in sequences (and where you find that difference, eg, allowing only constructibles) is a matter of perspective.