Let 0 = {}, 1 = {{}}, 2 = {{},{{}}}, etc.
In other words, 0 = {}, 1 = {0}, 2 = {0, 1}, ..., n = {0, ..., n-1}, ...
We say that a set S has cardinality n iff there exists a one-to-one mapping from the elements of S to the elements of n
(S has cardinality 0 iff S = 0)
((Can you define a one-to-one mapping before you define integers??))
(((Alright going back to bed)))

Natural numbers I mean
How would you define negative numbers 🤔
Adding an Element Of Negativity could work but I'd like to keep that -1 has cardinality 1
I also don't want to end up with a negative zero so 0 = {EoN} won't work

If you define a tuple as (x, y) = {{x}, {x, y}} then you could redefine 1 as (0, 1) and -1 as (1, 0) but then cardinality will have to be compared to the largest element of the tuple and you can't have the notion of a largest element before you have a cardinality......