My personal favorite is Bertrand's paradox. Whenever somebody is talking about random, I ask him/her about this one. This leads always to some nice discussions, which I like.

Each number system has exactly 10 different digits.

Much of the math in Goedel, Escher & Bach went over my head or has since leaked out, but I think I remember that Goedel's contribution was to prove you can write "this is false" in any sufficiently expressive language, which shattered the hope of any complete and perfect number system. Does that sound right?

A good question is never answered. It is not a bolt to be tightened into place but a seed to be planted and to bear more seed toward the hope of greening the landscape of the idea. John Ciardi

hmmm... i seem to recall something about any language will allow a paradox. you need a 'superset' (my term) language, or a meta-language to resolve them.

but that higher level one will then have paradoxes, so you need a meta-meta-language, and so on.

Originally posted by Fred Rosenberger: hmmm... i seem to recall something about any language will allow a paradox. you need a 'superset' (my term) language, or a meta-language to resolve them.

I like it - it seems explaining the concept that new phrases are needed because the current ones are not sufficient required new phrases because the current ones are not sufficient. Recursion!

There will be glitches in my transition from being a saloon bar sage to a world statesman. - Tony Banks