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Hopefully a simple question : As set theory includes only nested sequences of the empty set in the usual V, representation of an integer is easy. However what would the set that represents the number 1/2 look like precisely? I have seen some StackExchange responses describing how set theory handles Z, Q, R, Ordinal fractions, Surreal numbers etc. Presumably 1/2 is represented in different ways in these different number systems, but they went way over my head. I thought by asking what the set is that represents 1/2 would be straight forwards and kick off my trail to understand how set theory represents numbers less than a 'whole number'.

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    What about, the equivalence class of the pair of integers $(1,2)$? – FShrike Jan 13 '23 at 19:20
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    (I think you're overthinking this:) One of the usual implementations of arithmetic in set theory goes as follows: the natural numbers get implemented as the finite ordinals in the obvious way; an integer is then an equivalence class of ordered pairs of finite ordinals with $(\alpha,\beta)\sim(\gamma,\delta)$ iff $\alpha+\delta=\beta+\gamma$ (intuitively $(\alpha,\beta)$ represents $\alpha-\beta$); a rational is then an equivalence class of ordered pairs of integers; and so on. (And if you really want to have $\mathbb{N}\subseteq\mathbb{Z}\subseteq\mathbb{Q}$ etc., you do additional fudging.) – Noah Schweber Jan 13 '23 at 19:24
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    Crucially, this is not intended to be particularly natural; the way $1\over 2$ gets implemeneted set theoretically is not intended to "feel one-half-like," it's just supposed to get the job done so to speak. – Noah Schweber Jan 13 '23 at 19:26
  • @ Noah Schweber : I was led to believe that doing Z, Q, R etc. was very complicated - see https://math.stackexchange.com/questions/1711161/ordinal-fractions. An ordinal fraction is presumably a number less than a whole number ? –  Jan 13 '23 at 19:53

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