# When was the continuum hypothesis born?

The question Solutions to the Continuum Hypothesis states that the continuum hypothesis was posed by Cantor in 1890. In http://en.wikipedia.org/wiki/Continuum_hypothesis the year 1878 is quoted without source. What is the correct date?

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## 3 Answers

Finally, a good use for the newly purchased copy of "Zermelo's Axiom of Choice".

Moore writes that Cantor formulated the following problem in 1878:

Every infinite subset of $\Bbb R$ is either denumerable or has the power of the continuum.

The given reference is:

Cantor G. "Ein Beitrag zur Mannigfaltigkeitslehre." Journal für die reine und angewandte Mathematik 84 (1878), pp. 242-258.

Although searching for the reference, it seems that it may have published in 1877, so it's unclear which one is the correct date. The paper was written in 1877, though. Moore points that the hypothesis was given on page 257, but I don't read German very well (or at all, for that matter), so I can't tell.

Moore also adds that in 1895 Cantor pointed out that in $\aleph$ notation, which introduced in the paper below, that this is equal to $2^{\aleph_0}=\aleph_1$. Cantor already made this observation in 1882 in a letter to Dedekind, where he used the terminology "numbers of the second number-class" to talk about $\aleph_1$.

Cantor G. "Beiträge zur Begründung der transfiniten Mengenlehre." Mathematische Annalen 46 pp. 481-512.

However it is good to note that this equivalence requires the axiom of choice. The question you cite refers to this statement, and not the first statement.

(Both the statements appear on page 41 of the book "Zermelo's Axiom of Choice: Its Origins, Developments & Influence" by Gregory H. Moore)

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I'm very impressed with myself. I managed to type in the name for the paper and the journal (except the umlaut) without any mistakes! –  Asaf Karagila Jul 6 '13 at 9:03
(The second paper is a copy-paste from the internet, but I also typed it without mistakes into Google!) –  Asaf Karagila Jul 6 '13 at 11:00
Very impressive. :-) –  Andres Caicedo Jul 6 '13 at 15:18

The main theorem of Cantor's "Beitrag" (contribution) is a bijection between $\mathbb R$ and $\mathbb R^n$, as well as a bijection between $\mathbb R$ and $\mathbb R^{\aleph_0}$. He uses (in modern language) Hilbert's Hotel to find a bijection between the real numbers and the irrational numbers (in the unit interval), and then uses continued fractions to find a bijection to Baire space $\mathbb N^{\mathbb N}$. The bijection between Baire space and its square is then easy.

At the end of the paper he introduces what we now call the continuum hypothesis:

Cantor's original text, dated July 11, 1877: (from the link quoted by Asaf)

[...] Verstehen wir unter einer linearen Mannigfaltigkeit jeden denkbaren Inbegriff unendlich vieler, voneinander verschiedener reeller Zahlen, fragt es sich, in wie viel und in welche Klassen die linearen Mannigfaltigkeiten zerfallen, wenn Mannigfaltigkeiten von gleicher Mächtigkeit in eine und dieselbe Klasse, Mannigfaltigkeiten von verschiedener Mächtigkeit in verschiedene Klassen gebracht werden. [...]

Durch ein Inductionsverfahren, auf dessen Darstellung wir hier nicht näher eingehen, wird der Satz nahe gebracht, dass die Anzahl der nach diesem Eintheilungsprinzip sich ergebenden Klassen eine endliche, und zwar, dass sie gleich zwei ist. [...]

Eine genaue Untersuchung dieser Frage verschieben wir auf eine spätere Gelegenheit.

My translation

...If we define a linear manifold to be any embodiment of infinitely many different real numbers, the question appears, in how many (and which) classes the linear manifolds can be divided, if manifolds of the same power are placed in the same class, manifolds of different power in different classes. [...]

Using an inductive method, the presentation of which we will not address here, the theorem suggests itself that the number of classes resulting from this division is finite, and in fact two. [namely, the countable sets and those which can be bijected onto the unit interval]

We defer a closer investigation of this question to a later occasion.

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Thank you very much! –  Asaf Karagila Jul 6 '13 at 9:48
what is Cantor's meaning of "Mannigfaltigkeiten" in this setting? surely it doesn't mean what we mean when we talk about manifolds. –  Toink Jul 6 '13 at 12:08
He had not invented the term "set" ("Menge") yet. He uses the term "Mannigfaltigkeit" throughout the paper for what we call "set" now. In several places he feels the need to point out that the elements of his "manifolds" are different from each other. I have added this definition to my answer now. (Not sure what a good translation of "Inbegriff" is.) –  Goldstern Jul 6 '13 at 13:19
This is a transcript of a comment by Peter Luschny, on Google+: The remarks of Goldstern to the terms 'Menge', 'Inbegriff' and 'Begriff' are somewhat unreliable if not misleading. Goldstern writes: "He had not invented the term "set" ("Menge") yet." But it was not Cantor who introduced the term 'Menge', it was Bernard Bolzano and Cantor adopted the term from Bolzano. –  Andres Caicedo Jul 8 '13 at 15:28
(Cont.) The disambiguation took place more than 25 years before Cantor's 1877 'Beitrag'. The often overlooked work of Bolzano is: Paradoxien des Unendlichen, Reclam Leipzig, 1851. Bolzano writes: "Es gibt Inbegriffe, die, obgleich dieselben Teile A,B,C,D ... enthaltend, doch nach dem Gesichtspunkte (Begriffe), unter dem wir sie so eben auffassen, sich als verschieden darstellen. [...] Einen Inbegriff, den wir einem solchen Begriff unterstellen, bei dem die Anordnung seiner Teile gleichgültig ist, nenne ich eine Menge." –  Andres Caicedo Jul 8 '13 at 15:29

Since your question includes the reference-request tag, I would point you towards the following two sources as well (also written by Moore):

Moore, G. H. (1988). The origins of forcing, Logic Colloquium ’86 (Frank R. Drake and John K. Truss, editors). Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 143-173.

Moore, G. H. (1989). Towards a history of Cantor’s continuum problem. The history of modern mathematics, 1, 79-121.

The latter is probably most hopeful for the question you have asked here, though I personally enjoyed the former, which I read through carefully when thinking about some of the issues outlined in this MO post. (I have listed two other sources on Cohen's related work and forcing there.)

Another nice source on CH and its "solution" is:

Yandell, B. (2002). The honors class: Hilbert's problems and their solvers.

I use scare quotes because the Continuum Hypothesis was ultimately shown to be independent from ZF(C), which means, in some sense, you need Z and F (and C) before you can talk about the Hypothesis in its resolved-form.

The Z part probably dates back to around 1908. See:

Zermelo, E. (1908). Untersuchungen über die Grundlagen der Mengenlehre. I. Mathematische Annalen, 65(2), 261-281.

The F part probably dates back to around 1922.

For other dates and references, I think the Axiom of Choice book Asaf mentions is a nice source.

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Actually, I was the one who added the reference-request tag. But this is a good contribution to the discussion nonetheless! –  Asaf Karagila Jul 6 '13 at 9:59
I suppose I should have also noted that, consistent with Moore's other writing, these referenced texts claim that Cantor formulated CH in 1878. –  Benjamin Dickman Jul 6 '13 at 10:03
Of course. But the thing is that asserting there is no intermediate cardinal and that $2^{\aleph_0}=\aleph_1$ are two different things if you don't have have a full agreement about the axiom of choice yet. The question cited by the OP talks about the latter, rather than the former. –  Asaf Karagila Jul 6 '13 at 10:05
I suppose, as long as we're adding references, we could include Cantor's collected works, edited by Zermelo and published in 1932. The title is "Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, mit erläuternden Anmerkungen sowie mit Ergänzungen aus dem Briefwechsel Cantor-Dedekind." –  Andreas Blass Jul 6 '13 at 16:43
@Andreas: I thought that Noether and Cavallies edited the "Briefwechsel Cantor-Dedekind", or is that another version of letters between the two? –  Asaf Karagila Jul 6 '13 at 21:07