My favorite number

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.

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4 thoughts on “My favorite number

  1. 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.

  2. “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.

  3. 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.

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