Re: Natural numbers

george wrote:
Jack Campin - bogus address wrote:

There are quite a few ways you could define "property" that lead to
a different answer - anything from aleph-0 to card(V).

Of course, but this one is the only reasonable way.
Everything else is just the sort of arrant pedantry up with
which we should not put.

Ah, arrant pedantry, my speciality! I have to side with Jack here, it is not at all obvious that the number of properties of naturals is 2^aleph. For example, to a predicativist a property of natural numbers is an intensional object, one which must be seen to be determinate on basis of the determinateness of the totality of natural numbers. Predicatively it doesn't even make sense to speak of the cardinality of the totality of properties, which isn't determined, although depending on the exact formulation it might or might not make sense to say that there are uncountably many properties.

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