Friday, February 02, 2018

The Essence of Cantor's Paradox


(1)     There are at least three things (e.g. these three words)

(2)     Given some things, there are possible selections from them (e.g. "these" and "three")

(3)     There are all the things given by reiterating (2), given (1)

Note that each possible selection is a thing, and that it was always possible.

(4)     Given some things, cardinally more selections from them are possible

That is shown by Cantor's diagonal argument.

(5)     There are cardinally more things of kind (3) than there are things of kind (3)

That follows from (4), given (3), but is contradictory, and hence false.

My resolution of that paradox (essentially Cantor's paradox) will begin by observing that apparently timeless possibilities could possibly become more numerous over time, because if possible selections become more numerous then that could change the meaning of (3) enough to avoid (5). Furthermore, were that the only logically possible resolution, we could conclude that possible selections do become more numerous; and if the only way they could do that was for a Constructive Creator to do some definitive selecting, then there would have to be such a God.

My example of apparently timeless possibilities becoming more numerous (from a few years ago) was as follows. You were always possible, but had you never existed that possibility would have been the possibility of someone just like you; it could not have been the possibility of you in particular were you not there to refer to. Looking back now, we can see that there was always the possibility of you in particular as well as the more general possibility, even before you came into being; but, had you never existed, there could have been no such distinction. Now, Presentism appears to be logically possible; and if Presentism is true then there will originally have been no such distinction, even though you were always possible; the distinction will have arisen when you came into being. So, it appears to be logically possible for apparently timeless possibilities – e.g. the possibility of you in particular – to emerge as distinct possibilities from more general possibilities.

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