Re: [asa] Goedel's theorem and religion

On Sat, 14 Oct 2006 philtill@aol.com wrote:

> There is a whole class of these self-referential arguments, and some are indeed trivial but others have deep consequences. As an example of how these "proofs" may be very consequential, consider this question. Can God's infinite mind think of a list containing all the real numbers that lie between 0 and 1? Well, if He could do so, then He could write them in base-2 (e.g., 0.001011010010...). He could write them in the form of a table, each row being one of the real numbers, with each row being infinitely long and there being an infinite number of such rows. Then He could take the numerals lying on the diagonal of that table and then take the binary complement of that diagonal (i.e., replace all its 1's with 0's and vice versa). This new number will also be a real number between 0 and 1. So had it already been recorded somewhere in the infinite list? Well Yes, by definition, if God recorded all of them. But No it cannot be in the list, because if it was in the list!
  th
> en it would cross itself on the diagonal somewhere, and at that crossing point it would need to be both 0 and 1 simultaneously. This proves that there exists a single number that cannot be on the list. Therefore, God cannot list all the real numbers.
>

Phil,

I don't think that you have made clear what you mean by "list". In what
you describe, your list consists of a first line, a second line, a third
line, etc., i.e. a countable set of lines. What you have given is the
classical proof that the set of real numbers is not countable. God is not
required to think only in terms of countable sets.

Gordon Brown
Department of Mathematics
University of Colorado
Boulder, CO 80309-0395

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Received on Sat Oct 14 14:49:27 2006