On 23/06/06, Brian Hulley
This equation is just a shortcut, so I can't see how can it be ill-typed. It means something like: if you add one element to an infinite list, will it be longer?
What does your intuition say about this?
It won't be longer. How can it be? It's already infinite ;) It's like throwing things into bottomless hole and expecting it to get more "full."
But this explanation might just be vapid sophistry. Do you *really* want to trust it?
I perceive it as a way to explain to beginner students where bijection idea comes from. It's all it means to me. I suppose the whole idea is to start at something intuitive and then extend it to completely counter-intuitive notion of being infinite.
Just that there is a conflict with intuition no matter which option you choose: if I think that the list would be longer, I have to reject any proof to the contrary, but then my intuitions about valid proof are confounded, whereas if I accept the proof, my intuition about physical objects is confounded: if the list doesn't get longer, then where *is* the thing I added to it? Did it just disappear? So for these reasons I find that infinity is a troublesome concept.
I suppose infinity can't be totally intuitive in the end. We are not used to handle infinite objects and intuition as such was not developed to handle them. Regards, Piotr Kalinowski -- Intelligence is like a river: the deeper it is, the less noise it makes