# [Haskell-cafe] Functional progr., images, laziness and all therest

Jon Fairbairn jon.fairbairn at cl.cam.ac.uk
Thu Jun 22 10:50:58 EDT 2006

```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

```