Quote:
Originally Posted by kovidgoyal
@beppe: I doubt you really need the Peano axioms, instead you can define natural numbers as properties of relations on sets of objects and derive the rules of arithmetic from the laws of logic.
I remember doing something along those lines at a maths summer school, way back when I had no facial hair.
IIRC Bertrand Russel did something along those lines as well.
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@ kovidgal: I like this way of addressing.
Absolutely. That's so nice about axioms. That you can substitute them with other starting points. The Peano's are fascinating because with so little you can build so much with just geometric considerations. In that way you can coast clear from the traps of logic.
I also like the expression facial hair. I was taught these things when you were just a twinkle in your mother's eyes.