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CONTINUUM HYPOTHESIS
CONFUSION

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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:

    1 2 3 4 5 ... aleph1 aleph2 aleph3 ...

and we say "there are aleph1 points on a line" or "there are aleph2 different curves" ...

First note that Gamow's sequence of finite and transfinite numbers is not correct. It should be:

    1 2 3 4 5 ... aleph0 aleph1 aleph2 aleph3 ...

Also note that he is assuming CH when he says "there are aleph1 points on a line" and GCH when he says "there are aleph2 different curves."


Another example of confusion is on page 46 of Michael Guillen's Bridges to Infinity:

an aleph0 set ... has precisely 2aleph0 conceivable subsets ... It is the first stepping stone beyond infinity, the first transfinite number, which Cantor named aleph1 ... A set with aleph1 elements in turn, has precisely 2aleph1 conceivable subsets. This is the second stepping stone, the second transfinite number aleph2, and so forth.
Note that Cantor did not give 2aleph0 the name aleph1. Cantor's Continuum Hypothesis is exactly the question about whether 2aleph0 equals aleph1. Also note that aleph1 is not the first transfinite number, aleph0 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 aleph1. Cantor himself guessed that the total number of irrationals is exactly aleph1, mainly since aleph1 is the next largest infinity after aleph0 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 aleph0 and aleph1.
This quote includes three confusions:
  1. " it has been established that the total number [of irrationals] cannot be more than aleph1. "

    Wrong. It has been established that the number of irrationals is the same as the number of reals, which is 2aleph0, which is >= aleph1. 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:

        aleph0 aleph1 aleph2 aleph3 ...

  2. "Cantor himself guessed that the total number of irrationals is exactly aleph1"

    This is a correct statement of CH but since Guillen is assuming 2aleph0 = aleph1, Guillen can restate this as:

    Cantor himself guessed that the total number of irrationals is exactly 2aleph0
    Cantor did not guess this, he proved it.

  3. "But there is still the uneliminated possibility that the number of irrationals actually lies between aleph0 and aleph1."

    Since aleph1 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 aleph0 and aleph1.


Go to the main Continuum Hypothesis article.


 
 

 

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