On 2006-06-22 at 15:45BST "Brian Hulley" wrote:
Jon Fairbairn wrote:
infinity+1 = infinity
Surely this is just a mathematical convention, not reality! :-)
I'm not sure how to answer that. The only equality worth talking about on numbers (and lists) is the mathematical one, and it's a mathematical truth, not a convention.
I don't see why induction can't just be applied infinitely to prove this.
because (ordinary) induction won't go that far.
I wonder why? For any finite list yq, |y| == |yq| + 1 So considering any member yq (and corresponding y) of the set of all finite lists, |y| == |yq| + 1
But the infinite lists /aren't/ members of that set. For infinite lists the arithmetic is different. |y| == |yq| +1 == |yq| If you don't use the appropriate arithmetic, your logic will eventually blow up.
Couldn't an infinite list just be regarded as the maximum element of the (infinite) set of all finite lists?
It can be, but that doesn't get it into the set. -- Jón Fairbairn Jon.Fairbairn at cl.cam.ac.uk