Quote:
Originally Posted by nauthiz
I'd go so far as to say it's the very beginning of mathematics.
1. 0 is a natural number.
2. For every natural number x, S(x) is a natural number.
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You need to tell them that 1 is not S(x) for any naturan number x. And that if you have a set A with the property that 1 belongs to A and if whenever x is in A then so is S(x) then A is all of the natural numbers.
With that you, basic logic, the usual notions of a set you can build all of the usual number systems -- integers, rationals, reals and complex numbers. (See for instance
Foundations of Analysis by Landau).
Throw in the axiom of choice, and you can build virtually all of modern mathematics.
Yep, arithmetic is pretty fundamental.