I finally have a favorite number. Well, actually, I have one favorite number for every well-ordering of the real numbers. If you don’t believe in the axiom of choice, I don’t have any favorite numbers.
Let (< ) be a well-ordering of the real numbers. There are aleph-0 many sequences of characters, and 2^aleph-0 (which is strictly larger than aleph-0) many real numbers. Therefore, there must be a real number which is not describable in words. My favorite number is the least such number according to the well-ordering.
Uh oh… I think I just described it in words.
Cool…! I’ve also found my favourite number (imitating you :-)_, that is the minimum number of elements one need
to remove from a set with aleph-1 cardinality to obtain aleph-0 cardinality set.
T spaces
Um, so you do believe in the axiom of choice.
“Do you believe in the axiom of choice?”, not knowing what “axiom of choice” means, sounds like a religious question to me. You know, like some kind of alien religion on star trek or something.
Hmm, there is one way that this may not be paridoxical: if there is no way to describe any well-ordering of the real numbers in words. However, even given that it would be easy to extend this construction to a paradox.
The great book “Meta Math” by Gregory Chaitin discusses your favourite number (and other excellent stuff); http://arxiv.org/find/grp_q-bio,grp_cs,grp_physics,grp_math,grp_nlin/1/all:+AND+Math+Meta/0/1/0/all/0/1