Why is the surreal number collection not a set 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?

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.