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 , why did Conway choose to assign to that branch instead of ?
Answer follows
[edit | edit source]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.