What did Cantor take to be the relationship between the countable ordinals and the power set of the naturals? 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:


*

*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?

*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
 A: 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.
A: 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:


*

*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?

*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.
