Sat, 27 May 2006 12:16:12 -0700 On 5/27/06, Taran Rampersad <cnd@knowprose.com> wrote: > Stephen J. Turnbull wrote: > >>>>>> "Taran" == Taran Rampersad <cnd@knowprose.com> writes: > >>>>>> > > > > Taran> I once had to do a mathematical proof that 1+1=2. It wasn't > > Taran> as simple as writing 'because it is', it was about 6 > > Taran> handwritten pages with all sorts of mathematical symbols. > > > > Wow, that's really efficient! It took Russell and Whitehead a couple > > hundred pages. ;-) > > > Peano Axioms. So I cheated. I didn't have to prove them. Um, you can't prove them. That's why they are axioms. BTW now I'm confused that it took so so long. In the Peano axioms, + can be recursively defined by the rules: a + 0 = a a + s(b) = s(a+b) In which case: 1 + 1 = s(0) + s(0) = s(s(0) + 0) = s(s(0)) = 2 Proving the basic algebraic properties of + is harder, but 1+1=2 is pretty easy. In sequence you prove that + is well-defined by single induction, commutative by a double induction then associative by triple induction. After + is properly established, you can define * and prove the same things about it in the same way. Then you can define <, and prove its basic properties, and basic algebraic relationships with + and *. All of that might take me 6 pages, but I suspect not. (Then again I have more experience than you did when you did that exercise.) Cheers, Ben