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ok, I define 1 as {∅} and 2 as {∅, {∅}}
proving the addition holds is slightly more complicated
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+ is a map from N×N to N where a + 0 = a and a + S(b) = S(a + b) (S is the successor function that gives the next number).
Then 1 + 1 = 1 + S(0) = S(1 + 0) = S(1) = 2.
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I really recommend the YouTube channel “Another Roof”. His first few videos were building up exactly this idea, as well as building up all the real numbers (possibly complex too if I’m remembering correctly). Sounds like a dry topic but he uses humour really well throughout. https://youtube.com/@anotherroof
Here is a playlist of the topic: https://youtube.com/playlist?list=PLsdeQ7TnWVm_EQG1rmb34ZBYe5ohrkL3t
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