Comment by LudwigNagasena
1 year ago
Saying 1 is both an integer and a rational number is wildly different from saying it is both an integer and an ASCII character. Z is a subset of Q. ASCII characters don’t overlap with either.
When you construct numbers using sets under ZFC axioms or inside lambda calculus what you get is representation. But 1 is just 1.
Your keyboard has a button with ‘1’ printed on it. When you push that, you don’t always get an integer or a rational number. You can convert what you get to either. So there must be overlap with ASCII somehow?
There is some overlap in some sense somehow, but not in any literal sense unlike in the sense in which all elements of Z are elements of Q.
By any common set-theoretic construction of Q (e.g., equivalence classes of integer pairs under cross-multiplication), 1 as an element of Z is not literally an element of Q: 1 ∈ Q merely corresponds to 1 ∈ Z, and this correspondence is carefully designed to preserve the ring operations of Z within a certain subset of Q. In this case, the distinction is so negligible that eliding it is harmless, but the whole point of the article is that such elisions can become harmful in more complex constructions.
1 ∈ Z and ASCII '1' can similarly be seen as corresponding in terms of having the same glyph when displayed. But of course, there are far fewer meaningful operations common to the two.
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