Some theorems on sets

This view of the infinite, which I consider to be the sole correct one, is held by only a few.

I entertain no doubts as to the truths of the transfinites, which I recognized with God’s help and which, in their diversity, I have studied for more than twenty years; every year, and almost every day brings me further in this science.

– Georg Cantor

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The rational numbers are countably infinite (can be written as a sequence).

| Q | = | N |

The idea is very simple, write the sequence rationals in [0, 1]

0 , 1 , 1/2 , 1/3 , 2/3 , 1/4 , 2/4 , 3/4 , 1/5 , 2/5 , 3/5 , 4/5 …

And similarly the union of countably many countable sets is countable.

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The cardinality of the real numbers is strictly greater than of the rationals (R cannot be written out as a sequence).

| R | > | Q |

Suppose R can be written out as a sequence, with Cantor's diagonal argument we construct a real number that not in that sequence.

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Russel's Paradox: There is no set of all sets, no ultimate universal set.

If we allow as sets things like A e A (A is an element of itself) it leads to logical contradiction.

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Cantor's Theorem: Given any set the power set is of strictly higher infinity. Thus there is no“highest” infinity or largest set.

| P(S) | > | S |

The idea of the proof is much the same as Russel's Paradox.

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Please note, text in italics is the formulation of a theorem, the rest is comments or ideas of the proofs.