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I disagree but I'm being nitpicky I think the real funky business happens right between ℚ ⊂ ℝ but maybe all I'm really trying to say is "hey, look at me, I un

I disagree but I'm being nitpicky I think the real funky business happens right between ℚ ⊂ ℝ but maybe all I'm really trying to say is "hey, look at me, I understand how all of ℕ ⊂ ℤ ⊂ ℚ" I'm very very sure that it's possible to give a fully ZF set-theoretic construction of the rationals using sets. in fact IMO the annoying thing is that there are in fact two trivially equivalent constructions that I can think of the top of my head, and I find that even worse than there being none at all

klodolph 2y ago
What do you disagree with? Traditionally, you define ℚ as equivalence classes of pairs of integers. The integer 0 is an integer, it’s not an “equivalence class of pairs of integers” and therefore it’s not a rational, in the set-theoretic sense. Real numbers have lots more constructions. https://en.m.wikipedia.org/wiki/Construction_of_the_real_numbers https://en.m.wikipedia.org/wiki/Construction_of_the_real_num...
- ysofunny 2y ago
  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
- zeroonetwothree 2y ago
  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”
- anon291 2y ago
  This is a construction. The notion is the same. Not all math need be constructive. This is an implicit bias in most theorem provers.
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klodolph 2y ago

What do you disagree with? Traditionally, you define ℚ as equivalence classes of pairs of integers. The integer 0 is an integer, it’s not an “equivalence class of pairs of integers” and therefore it’s not a rational, in the set-theoretic sense. Real numbers have lots more constructions. https://en.m.wikipedia.org/wiki/Construction_of_the_real_numbers https://en.m.wikipedia.org/wiki/Construction_of_the_real_num...

ysofunny 2y ago
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
zeroonetwothree 2y ago
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”
anon291 2y ago
This is a construction. The notion is the same. Not all math need be constructive. This is an implicit bias in most theorem provers.

deleted 2y ago

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