This is a construction. The notion is the same. Not all math need be constructive. This is an implicit bias in most theorem provers.

so yes, to answer one of my own questions. I was only really saying that I don't really know any of those constructions; but know all the "preceding" ones.

I gotta figure out dedekind cuts or at least learn tarski's fixed point theorem which I'm sure that would allow me to much better understand tarski's construction

all in the slow lifelong process of understanding and learning to draw post's lattice

finally, to answer your question, I disagree with saying that there's funky business between ℤ ⊂ ℚ because of my alleged claim that there is at least one construction which avoids the problem you describe, but as I was trying to say, these would have a non-unique way to construct number zero which nobody likes

You could construct the larger set and then define the smaller sets as specific subsets. For example Conway’s surreal numbers is a fun construction.

If you do it that way then there is only one version of “1”