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I've been told that Cantor sees a relationship between the countable ordinals (Cantor's second number class) and the powerset of the natural numbers.

I've read the "Grundlagen" a few times, but can't seem to locate what he takes this relation to be.


I'm suspecting this is related to the continuum hypothesis CH, though I do not know if my suspicion is correct. Here is why I suspect that this is related to CH. According to wikipedia,

There is no set whose cardinality is strictly between that of the integers and that of the real numbers.

The |powerset of integers| = |powerset of naturals| = |reals| = $2^{\aleph_0}$.

The cardinality of the countable ordinals (the second number class) is identified with $\aleph_1$.

Cantor took $2^{\aleph_0} = \aleph_1$. Hence my suspicion.

My questions:

  1. Am I right that Cantor "conjectured" that the relationship between the countable ordinals and the power set of the naturals is that they are of the same cardinality?
  2. Am I right in my suspicion that this is (or is related to) the continuum hypothesis?

A direct answer to these questions and some commentary would be most appreciated.

Thanks

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Sees? ${}{}{}{}$ –  Qiaochu Yuan Jun 12 '12 at 3:58
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Note that Cantor did not "take" $2^{\aleph_0}=\aleph_1$": he conjectured it, but did not assert it as true. –  Arturo Magidin Jun 12 '12 at 4:02
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@Arturo: Cantor did claim he could prove the continuum hypothesis on various occasions, e.g. in a letter to Mittag-Leffler dated August 26, 1884 (from G.H. Moore, Zermelo's axiom of choice). –  t.b. Jun 12 '12 at 4:34
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The BBC Documentary: Dangerous Knowlege might be of interest to the OP. –  Alexander Thumm Jun 12 '12 at 4:43
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@pichael: Yes on the first; the second is rather like saying that one suspects that water is sort of wet. That conjecture is the continuum hypothesis. "Continuum" in fact refers to the real line. Bold is fine, but large fonts and all caps tend to "raise the volume" rather than provide emphasis. –  Arturo Magidin Jun 12 '12 at 6:25
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2 Answers

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As Wikipedia says in article about Cantor:

Cantor was the first to formulate what later came to be known as the continuum hypothesis or CH: there exists no set whose power is greater than that of the naturals and less than that of the reals (or equivalently, the cardinality of the reals is exactly aleph-one, rather than just at least aleph-one). Cantor believed the continuum hypothesis to be true and tried for many years to prove it, in vain.

I'll add two quotes from the book Georg Cantor: His Mathematics and Philosophy of the Infinite by Joseph Warren Dauben,

Cantor's second number class is what we would call today $\omega_1$, the smallest uncountable ordinal, which might perhaps be interesting for you in connection with this question.

p.110

Despite all that the Grundlagen had accomplished, there was a serious lacuna. Though Cantor had made it clear that his introduction of the transfinite numbers, particularly those of the second number class, was essential to sharpening the concept of power, the question of the power of the continuum was still unanswered. He hoped a proof would be forthcoming, establishing his continuum hypothesis that the power of the continuum was none other than that of the second number class (II). The benefits of such a proof would be numerous. It would immediately follow that all infinite point sets were either of the power of the first or second number class, something Cantor had long claimed. It would also establish that the set of all functions of one or more variables represented by infinite series was necessarily equal in power to the second number class. Likewise, the set of all analytic functions or that of all functions represented in terms of trigonometric series would also be shown to have the power of the class (II).

The Grundlagen went no further in settling any of these issues. Instead, Cantor published a sequel in the following year as a sixth in the series of papers on the Punktmannigfaltigkeitslehre. Though it did not bear the title of its predecessor, its sections were continuously numbered, 15 through 19; it was clearly meant to be taken as a continuation of the earlier 14 sections of the Grundlagen itself. In searching for a still more comprehensive analysis of continuity, and in the hope of establishing his continuum hypothesis, he focused chiefly upon the properties of perfect sets and introduced as well an accompanying theory of content.

p.137

In the fall of 1884 he again took up the intricate problem of the continuum hypothesis. On August 26, 1884, little more than a week after his letter of reconciliation to Kronecker, Cantor had written to Mittag-Leffler announcing, at last, an extraordinarily simple proof that the continuum was equal in power to the second number class (II). The proof attempted to show that there were closed sets of the second power. Based upon straightforward decompositions and the fact that every perfect set was of power equal to that of the continuum, Cantor was certain that he had triumphed. He summarized the heart of his proposed proof in a single sentence: "Thus you see that everything comes down to defining a single closed set of the second power. When I've put it all in order, I will send you the details."

But on October 20 Cantor sent a lengthy letter to Mittag-Leffler followed three weeks later by another announcing the complete failure of the continuum hypothesis. On November 14 he wrote saying he had found a rigorous proof that the continuum did not have the power of the second number class or of any number class. He consoled himself by saying that "the eventual elimination of so fatal an error, which one has held for so long, ought to be all the greater an advance." Perhaps he was thinking back to the similar difficulties he had encountered in trying to decide whether or not the real numbers were denumerable, or how lines and planes might be corresponded. Cantor had come to learn that one should never be entirely surprised by the unexpected. Nevertheless, within twenty-four hours he had decided that his latest proof was wrong and that the continuum hypothesis was again an open question. It must have been embarrassing for him to have been compelled to reverse himself so often within such a short period of time in his correspondence with Mittag-Leffler. But even more discouraging must have been the realization that the simplicity of the continuum hypothesis concealed difficulties of a high order, ones that, despite all his efforts and increasingly sophisticated methods, he seemed no better able to resolve.

In attempting to find a solution for his continuum hypothesis, Cantor was led to introduce a number of new concepts enabling more sophisticated decompositions of point sets. These, he hoped, would eventually lead to a means of determining the power of the continuum. His attempt to publish these new methods and results marked the final and most devastating episode responsible for his disillusionment with mathematics and his discontent with colleagues both in Germany and abroad.

EDIT: You've added clarification to your question that you would like to know this:

  1. Am I right that Cantor "conjectured" that the relationship between the countable ordinals and the power set of the naturals is that they are of the same cardinality?
  2. Am I right in my suspicion that this is (or is related to) the continuum hypothesis?

(1) Yes, I believe that the excerpts I provided above give sufficient support for this claim.

(2) I am used to $\aleph_1=2^{\aleph_0}$ as the usual formulation of CH. And this is the same thing as you wrote in your Question 1. However, we should be careful if we want to avoid Axiom of Choice.

If we are working in ZF, i.e. without Axiom of Choice, this formulation is not equivalent to "There is no set whose cardinality is strictly between that of the integers and that of the real numbers." The reason is that $\aleph_1$ and $2^{\aleph_0}$ can be incomparable.

The relation between these two claims (which are in ZFC both equivalent formulations of CH) is explained in detail here.

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Thanks for these excerpts. A lot of background on Cantor's trying to come to grips with the well-ordering principle and CH is also given in chapter 1 (especially section 1.5 -- at least in my older edition) of G.H. Moore, Zermelo's Axiom of Choice: Its Origins, Development, and Influence. –  t.b. Jun 12 '12 at 6:32
    
@Martin: Your post did provide sufficient support. I was hasty and didn't read but a couple of comments before I figured that my question wasn't clear enough for my anal-retentiveness. Thanks for the great post! –  pichael Jun 12 '12 at 6:39
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It should be mentioned that Cantor implicitly used the axiom of choice in his work. In fact he proved the famous Cantor-Bernstein theorem as a very simple corollary from the fact that every two cardinalities are comparable, which we know is an equivalent of choice. –  Asaf Karagila Jun 12 '12 at 7:31
    
This is like good all times: t.b. adding interesting historical remarks and plenty of references; Asaf stressing the use of Axiom of Choice. There are so many questions where I have seen this. –  Martin Sleziak Jun 12 '12 at 8:02
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The only clarification I would make is the following: Cantor didn't see how to show that there was a subset of the reals of cardinality strictly between $\aleph_0$ and $\mathfrak{c}:=2^{\aleph_0}$--in fact, as Arturo points out in the comments below, every infinite subset of the reals he tried had cardinality either $\aleph_0$ or $\mathfrak{c}$--and so conjectured (not the same as concluded, as Arturo points out in the comments above) that $\aleph_1=\mathfrak{c}$.

One can show that the set of countable ordinals is not "bigger" than the power set of the natural numbers. Of this, Cantor was aware.

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More than just not seeing how to show there was such a set, every set he tried he could show it was either equipollent to $\mathbb{R}$ or to $\mathbb{N}$. (A bit more, since the way you phrased it, it might have been that he had a set he could not tell whether it as bijectable with $\mathbb{N}$, with $\mathbb{R}$, or with neither...) –  Arturo Magidin Jun 12 '12 at 4:11
    
Thanks, Arturo. A useful clarification. –  Cameron Buie Jun 12 '12 at 4:12
    
@Arturo: Is there a reference for "every set he tried he could show it was either equipollent to ℝ or to ℕ"? –  pichael Jun 12 '12 at 9:28
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