Why is the surreal number collection not a set

Question

I am currently writing an essay on the surreal numbers, to finalise said essay I want to talk about how the surreals are 'too big' (or so ive heard) to be a set. Is there a concrete way of showing this. I did find that if $N:=\{S|\}$ where S is the set of all surreal numbers, it can be shown that $N\ngeq{N}$. Is this valid/enough to conclude the Surreals cannot be a set?

Answer 1

So, your proof is correct. If there is a set $S$ containing all surreal numbers, then there is a number $N:=\{S \ | \ \varnothing\}$ which is strictly greater than all elements of $S$, i.e. all numbers. In particular $N>N$: a contradiction.

Answer 2

One way to show that No (the Surreal numbers) is not a set, is to break it into smaller steps, that are relatively easy to demonstrate:

1. Show that On (the Ordinal numbers) are contained in No.

2. Prove that On is not a set (that is, a proper class).

3. Observe that that any superclass of a proper class, must itself be proper.

I am pretty sure that each of the above three have easily searchable links on them, in the event you'd like to see more details. (As these are listed in decreasing order of difficulty, I suspect only #1 would be necessary to satisfy.)