We start off by defining the natural numbers (ie positive integers) in terms of sets.
For any set S, define a "successor" function f as f(S) = {S, {S}}
Then we define the natural integers as such: Define the number 0 to be the empty set, which I'll write here as O.
Then each successive integer (ie "n+1") is just the preceding integer put through the successor function, example:
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and so on