Friday, February 02, 2018

The Essence of Cantor's Paradox


(1)    There are at least three things

Clearly there are.

(2)    Given some things, there are possible selections from them

E.g. ‘clearly’ and ‘there’ are a pair of words.

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

Note that each possible selection is a thing.

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

That is shown by a Cantorian 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 begins by observing that apparently timeless possibilities could possibly become more numerous over time. It begins that way because if possible selections are becoming more numerous, then that could easily change the meaning of (3) enough to avoid (5). There is no other way of avoiding the contradiction that I can think of (and note that the main resolutions were constructivism, with its potential infinities, and going axiomatic, which means not addressing numbers of things directly); which is, after all, why this is a paradox.

Consequently this is essentially a proof by reductio ad absurdum that possible selections do become more numerous over time. And how could possible selections become more numerous, if not by a transcendent creator making definitive selections, constructing arithmetic as part of the creation of all things? There is no other way that I can think of; whereas this way is a serious possibility, because
A) mathematicians have taken constructivism surprisingly seriously, and constructivism would only be more Platonistic, and Millian, were the definitive constructions made by a transcendent creator, and
B) theologians have taken the idea of God being beyond our conception of number very seriously, e.g. the Trinity.

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