I think it’s worth talking about how the language works here.
We say, “the natural numbers are a subset of the reals,” and this is a sensible thing to say.
We also might say, “you construct each successor number in the natural numbers as n + 1 = n ∪ {n}”. And then we say, “a real number is a Cauchy sequence of rational numbers, or a Dedekind cut of rational numbers.” From a set-theoretic perspective, “the natural numbers are a subset of the reals” is obviously untrue with these definitions, and it’s worth spending a moment to think what the statement actually means, or how you would have to interpret it in order to understand the truth of the sentence.
I might translate the sentence as “there is a left-cancellative morphism from natural numbers to real numbers,” but then I’d have to define what category I’m using, and what the morphisms are—which is usually implied. You end up having to stand on top of a surprisingly tall stack of proofs in order to say “the natural numbers are a subset of the reals” and actually explain what you mean by that, rigorously, from foundations.
Or, put another way, it’s sometimes useful to understand what you mean by “behave as the identity”.
Retrofitting the set theoretic definitions is trivial. Take the set of cauchy real numbers, remove all reals equal to a real-that-would-be-a-rational, i.e. constant sequences. Union together this set with the set of rationals defined as ratios of integers. Repeat as needed with smaller sets.
Obviously these reals are isomorphic to the cauchy reals. The reality, of course, is that no one actually works with foundations, they work with the intuitive understanding that 1:N is 1:Z is 1:Q is 1:R.
If your real numbers are Cauchy sequences of rational numbers, and your rational numbers are a subset of real numbers, then your rational numbers are a subset of Cauchy sequences of rational numbers (which violates the axiom of foundation).
This is not as trivial as it sounds, which is why we invented all these different tools for explaining what “is” or “equals” means in mathematics without resorting to set equality (equality, isomorphism, equivalence, etc).
We say, “the natural numbers are a subset of the reals,” and this is a sensible thing to say.
We also might say, “you construct each successor number in the natural numbers as n + 1 = n ∪ {n}”. And then we say, “a real number is a Cauchy sequence of rational numbers, or a Dedekind cut of rational numbers.” From a set-theoretic perspective, “the natural numbers are a subset of the reals” is obviously untrue with these definitions, and it’s worth spending a moment to think what the statement actually means, or how you would have to interpret it in order to understand the truth of the sentence.
I might translate the sentence as “there is a left-cancellative morphism from natural numbers to real numbers,” but then I’d have to define what category I’m using, and what the morphisms are—which is usually implied. You end up having to stand on top of a surprisingly tall stack of proofs in order to say “the natural numbers are a subset of the reals” and actually explain what you mean by that, rigorously, from foundations.
Or, put another way, it’s sometimes useful to understand what you mean by “behave as the identity”.