Friday, February 12, 2016

But you have disposed all things by ... number .... (Wisdom 11:20)

If God exists, the Axiom of Choice is true. For given any set of nonempty sets, surely God could choose a member of each of the nonempty sets. (This argument is by Meyer, Nous 21 (1987), 345-361.) But if the Axiom of Choice is true, then any set can be well-ordered. And any well-ordered set has the same cardinality as some ordinal. Ordinals are essentially numbers, albeit some of them are infinite. So, indeed, God has disposed all things by number, and necessarily so. (Wisdom 11:20 also says that God has disposed things by size and weight. Perhaps this indicates that the quantifiers in text are restricted to material things. Or perhaps immaterial things count as having zero size and zero weight.)

I actually find the necessity in the above argument quite surprising. Ordinals are pure sets: sets built up out of the empty set, not out of concrete ur-elements. I find it surprising that any infinite set of concrete objects that God could create can still be numbered with a pure set.

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