
CONTINUUM
HYPOTHESIS

An example of confusion is in George Gamow's One Two Three...Infinity. On page 34 he says:The sequence of numbers (including the infinite ones!) now runs:First note that Gamow's sequence of finite and transfinite numbers is not correct. It should be:1 2 3 4 5 ... aleph_{1} aleph_{2} aleph_{3} ...
and we say "there are aleph_{1} points on a line" or "there are aleph_{2} different curves" ...
1 2 3 4 5 ... aleph_{0} aleph_{1} aleph_{2} aleph_{3} ...
Also note that he is assuming CH when he says "there are aleph_{1} points on a line" and GCH when he says "there are aleph_{2} different curves."
Another example of confusion is on page 46 of Michael Guillen's Bridges to Infinity:an aleph_{0} set ... has precisely 2^{aleph0} conceivable subsets ... It is the first stepping stone beyond infinity, the first transfinite number, which Cantor named aleph_{1} ... A set with aleph_{1} elements in turn, has precisely 2^{aleph1} conceivable subsets. This is the second stepping stone, the second transfinite number aleph_{2}, and so forth.Note that Cantor did not give 2^{aleph0} the name aleph_{1}. Cantor's Continuum Hypothesis is exactly the question about whether 2^{aleph0} equals aleph_{1}. Also note that aleph_{1} is not the first transfinite number, aleph_{0} is.Guillen continues his confusion on page 50:
To this day we still don't know exactly how many irrationals there are, although it has been established that the total number cannot be more than aleph_{1}. Cantor himself guessed that the total number of irrationals is exactly aleph_{1}, mainly since aleph_{1} is the next largest infinity after aleph_{0} defined by set theory. His guess came to be known as the Continuum Hypothesis. But there is still the uneliminated possibility that the number of irrationals actually lies between aleph_{0} and aleph_{1}.This quote includes three confusions:
 " it has been established that the total number [of irrationals] cannot be more than aleph_{1}. "
Wrong. It has been established that the number of irrationals is the same as the number of reals, which is 2^{aleph0}, which is >= aleph_{1}. What we don't know about the irrationals (and the reals, which have the same cardinality as the irrationals) is where their cardinality lies in the sequence of transfinite numbers:
aleph_{0} aleph_{1} aleph_{2} aleph_{3} ...
 "Cantor himself guessed that the total number of irrationals is exactly aleph_{1}"
This is a correct statement of CH but since Guillen is assuming 2^{aleph0} = aleph_{1}, Guillen can restate this as:
Cantor himself guessed that the total number of irrationals is exactly 2^{aleph0}Cantor did not guess this, he proved it.
 "But there is still the uneliminated possibility that the number of irrationals actually lies between aleph_{0} and aleph_{1}."
Since aleph_{1} is defined to be the "second transfinite number," as Guillen himself said in the first quote above, it doesn't make sense to say there's a cardinal that lies between aleph_{0} and aleph_{1}.
Go to the main Continuum Hypothesis article.
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Infinite Ink
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Nancy McGough Last significant update ... a long time ago! Last tweak on May 23, 1997 
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