Some theorems on sets
By Tom Brown
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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.
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Comments
Sorry, not my field at all.
Sorry, not my field at all. My husband is a mathematician, but remembers as a very young child having a dream where he was counting and suddenly realised that there wasn't an end to counting … ! Rhiannon
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