Surreal number/Why isn't half-omega the (left) child of omega?

■ Surreal number/Why isn't half-omega the (left) child of omega? Click this link to see an answer given by Gemini, on 9 October 2024. This use of a chatbot raises two questions:

Is the answer correct? (Answer: I don't know)
Is it it appropriate to post something that might not be true on Wikiversity? (Answer: Yes! This is a lesson, not the Wikipedia article.)

This page should be viewed as the following student assignment:

Is Gemini's answer correct? If so, please elaborate. If not, either produce a better answer, or explain why Gemini is wrong.

Question:: On the left (negative) side of the tree starting at $\omega$ , why did Conway choose to assign $\omega -1$ to that branch instead of ${\tfrac {1}{2}}\omega$ ?

Gemini's answer:

That's a great question!

Conway's choice to assign ω-1 to the left branch of ω instead of ω/2 is based on the fundamental principles of surreal number construction.

Here's the reasoning:

Recursive Definition: Surreal numbers are constructed recursively, meaning they are defined in terms of previously constructed numbers.
Left and Right Sets: Each surreal number is defined by two sets: a left set (L) and a right set (R).
Ordering: The ordering of surreal numbers is determined by the relative positions of their left and right sets. If every element of L is less than every element of R, then the number is well-defined.

Now, let's consider ω. It is defined as follows:

Left set: {0, 1, 2, 3, ...}
Right set: {}

When constructing the left branch of ω, we need to find a surreal number that is less than every element in the left set. ω-1 satisfies this condition because it is less than every natural number.

Why not ω/2? While ω/2 is indeed less than every natural number, it's not a well-defined surreal number at this point. To define ω/2, we would need to construct its left and right sets, which would involve introducing new surreal numbers.

Therefore, Conway chose ω-1 for the left branch of ω because it is a well-defined surreal number that fits the recursive construction rules and maintains the correct ordering.