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
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