I've been doing some digging into Surreal Numbers, and I've seen this "inductive" proof of transitivity. Basically, we assume, that there exists the simplest triplet of surreal numbers, x, y, z, such that x<=y, y<=z, but x is not smaller or equal to z. Then, applying the axiomatic definition of what it means to be a number and the definition of <=, we conclude that there exists a simpler triplet satisfying the same property, and hence, contradiction, QED, GG EZ.
My issue is, that we seem to assume that an infinite chain of simpler surreal numbers is impossible, but I'm not convinced that it's true, since some surreal numbers are "created" on an infinite "day", having therefore an infinite sequence of surreal numbers that are simpler.
I am aware that just because a number has infinite "day" of "creation" it doesn't mean that it's possible to construct such a sequence of decreasing simplicity, but I'm not convinced otherwise either.
Here is the document in which I found the proof I'm talking about, maybe I'm missing something. The proof is outlined on page 3, in Theorem 1.1
Also, if anyone knows a better, more mathematically clean description of "day of creation" of a number, please let me know
TL; DR: I'm not convinced to the proof of transitivity of <= over surreal numbers
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from math https://ift.tt/3cRVld9
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