in reply to Re^4: What do you know, and how do you know that you know it?
in thread What do you know, and how do you know that you know it?
Incidentally your playing around with the Peano axioms mislead you. The Peano axioms allow you to reason about the positive integers. (Proving that every positive integer is either even or odd is simple induction.) To get from there to the integers, you model each integer as an equivalence class of pairs of integers (p, n) with (p, n) in the same equivalence class as (p', n') if p'+n = p+n'. (Think of (p, n) as p-n.) Now you have to prove that the equivalence relation is well-defined, and that definition of addition is likewise. Define the usual multiplication, and prove that that works. And that you can map the positive integers into the integers with p->(p+1, 1). Then you can start proving other things, such as that every integer is either positive or negative.
Once you have all of that, then you're in a position to prove things about the integers (like every integer is even or odd).
You have to go through the same process again to define rationals as equivalence classes of pairs of integers. You can then define real numbers as equivalence classes of Cauchy sequences of rationals. Once you've done that you then have to reprove everything. But, amazingly, you can get all of the way through Calculus with just the Peano axioms. (It takes a lot of work though.)
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Re^6: What do you know, and how do you know that you know it?
by hv (Parson) on Aug 03, 2004 at 23:20 UTC