gatekeeping

Nope, you fell for the classic sibling blunder:

What about INFINITY PLUS ONE!?

Last time I saw this kind of challenge it was on reddit and I just replied with ℝ, but people brought up that this leaves out complex numbers. I'll now contend, however, that any number not included in that isn't real.

I cast set theory?

Aren't there numbers past (plus/minus) infinity? Last I hear there's some omega stuff (for denoting numbers "past infinity") and it's not even the usual alpha-beta-omega flavour.

Come to think of it, is there even a notation for "the last possible number" in math? aka something that you just can't tack "+1" at the end of to make a new number?

Doesn't include "i"

Oh you like math? Name all the sets of sets that don’t include themselves.

0,1,2,3,4,5,6,7,8,9

I thought it was 'R', for the set of all real numbers.

U

You also have to remember to put the +C at the end

Pi

*

i

Fred.

(∀x:Number(x)=T)(Name(x)="Fred")

I name every number Fred.

Which negative infinity or positive infinity includes zero?

Where's my imaginary love?

Everyone is mentioning the imaginary (and, presumably complex) number domains, but not quaterions and other higher dimensional number sets.

I'm going with defining a describeable number as any number that, given any finite period of time and any finite amount of resources, could be uniquely described to another entity with the ability to read and understand the language it is being described in, then saying all numbers are either describeable numbers (Despite the fact that these are almost laughably uncommon in the scheme of all numbers, I have diligently prepared an example: "2"), or indescribeable numbers (so much more common, and yet I can't give even a single example).

Didn't it 0,1,2,3,4,5,6,7,8,9?

Depends on the bit limit

Where is i? Is it safe? Is it alright?