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I am new here, so forgive me if this question does not satisfy the protocols of the site. I know there are so many equivalents to the AC (axiom of choice) and there are books that lists this equivalent forms of AC.

But how can I find equivalent forms of the continuum hypothesis ?

Specially I am interested in Algebraic equivalent form.

What references do we have in the literature !?

Thanks in Advance.

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3 Answers 3

up vote 23 down vote accepted

A classical reference is Hypothèse du Continu by Waclaw Sierpiński (1934), available through the Virtual Library of Science as part of the series Mathematical Monographs of the Institute of Mathematics of the Polish Academy of Sciences.

Sierpiński discusses equivalences and consequences. The statements covered include examples from set theory, combinatorics, analysis, and algebra. Most of the consequences he did not show equivalent were found later (mainly by Martin, Solovay, and Kunen) to be strictly weaker in that they follow from Martin's Axiom, and some are discussed in the original Martin-Solovay paper.

(In fact, the discovery of Martin's Axiom and the subsequent research on cardinal characteristics of the continuum helped clarify what the role of CH is in many classical arguments, and nowadays results that classically would be stated as consequences of CH are stated as consequences of some equality between cardinal characteristics. See the articles by Blass and Bartoszyński on the Handbook of Set Theory.)

Of course, many equivalents were found after 1934. For example:

  • Around 1943, Erdős and Kakutani proved that CH is equivalent to there being countably many Hamel bases whose union is $\mathbb R\setminus\{0\}$.
  • In the early 60s, Erdős found a nice equivalent in terms of analytic functions (see Chapter 17 in Aigner-Ziegler Proofs from THE BOOK).
  • Quite recently, Zoli proved that CH is equivalent to the transcendental reals being the union of countably many transcendence bases.

I do not know of an encyclopedic work updating Sierpiński's monograph. Most recent work on CH centers on what Stevo Todorcevic calls Combinatorial Dichotomies in Set Theory. It turns out that for quite a few statements, CH proves a "nonclassification" result, while strong forcing axioms (such as PFA) prove strong "classifications". For example, J. Moore proved that there is a 5-element basis for the uncountable linear orders if PFA holds, while Sierpiński showed that CH gives us $2^{\aleph_1}$ non-isomorphic uncountable dense sets of reals, none of which embeds into another in an order-preserving fashion.

Though not specifically concerned with CH and its equivalences, you may find interesting Steprans's History of the Continuum in the Twentieth Century.

Another recent line of study on CH centers on the role of choice. Propositions equivalent to CH in ZFC may have wildly different truth values if choice is not assumed. For example, under determinacy, CH is true in the sense that every set of reals is either countable or of the same size as the reals. However, it is also false in the sense that $\aleph_1\not\le|\mathbb R|$, and that there is a surjection from $\mathbb R$ onto $\aleph_2$.

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Hi Andres. Some minor points: 1) The question asked especially about algebraic equivalents of CH. Do you happen to know one beyond the one mentioned by Mariano in another thread? 2) I think the original article by Erdős is very readable and crystal clear. 3) I would add Gödel's Monthly article to recommended reading. 4) Since you brought up MA: As a non-set theorist who happened to need to understand it I can recommend Fremlin's book Happy New Year! –  Theo Buehler Dec 30 '12 at 18:02
Hi Theo: I do not know of "natural" algebraic characterizations, beyond Mariano's example. The third example I list using transcendence bases is not quite natural, for instance, in the sense that it is not something I would expect an algebraist would wonder about. There are a few additional natural examples from Ramsey theory (in partition calculus, and also generalizations of Schur's theorem by Jacob Fox and his collaborators), and from discrete geometry (de la Vega, and Schmerl, they have a couple of papers in Fundamenta), but purely algebraic ones seem missing. –  Andrés Caicedo Dec 30 '12 at 18:43
The place to look would be "Almost Free Modules: Set-theoretic methods", Revised edition, by Eklof and Mekler, North-Holland Mathematical Library vol. 65 (2002), but I am not familiar with equivalences of CH discussed there. –  Andrés Caicedo Dec 30 '12 at 18:45
(And yes, Fremlin's book is an excellent reference for early work on Martin's axiom and cardinal characteristics.) –  Andrés Caicedo Dec 30 '12 at 18:46
Thanks for the comments. But it seems that there is not much well-known equivalent algebraic form for CH (compared to the long list of algebraic forms of AC). am I right !? And Happy New Year to all. –  user30300 Dec 30 '12 at 19:03

Following Hausdorff, an ordered set A is said to be an $\eta_{1}$-ordering if for all subsets X and Y of A where X < Y (every member of X is less than every member of Y) and |X|, |Y| < $\aleph_{1}$, there is a $z \in A$ such that X < {z} < Y.

While I do not know if it has ever been stated as such, the following six assertions, the first of which is CH and the last five of which are of an ordered algebraic nature, are all equivalent.

(0) CH (i) There is a real-closed ordered field that is an $\eta_{1}$-ordering of power $\aleph_{1}$. (ii) There is an ordered field that is an $\eta_{1}$-ordering of power $\aleph_{1}$, (iii) There is a divisible ordered abelian group that is an $\eta_{1}$-ordering of power $\aleph_{1}$. (iv) There is an ordered abelian group that is an $\eta_{1}$-ordering of power $\aleph_{1}$. (v) There is an ordered group that is an $\eta_{1}$-ordering of power $\aleph_{1}$.

Proof. By a result essentially due to Hausdorff [1909], which was rediscovered and made famous by Erdös, Gillman and Henriksen [1955], (0) implies (i). But (i) implies (ii), (ii) implies (iii) and so on. However (v) implies there is an $\eta_{1}$-ordering of power $\aleph_{1}$, which, by a result of Sierpinski [1949], implies (0).

It is worth noting that analogs of the above readily extend to the GCH!

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Hi Philip. Yes, these statements are discussed in a few places. Most recently, the book on "Super-real fields" by Dales and Woodin. Note that they discuss appropriate versions of (ii)-(v) that do not require CH, but then assume CH (and therefore obtain the equivalences above) to study the properties of the structures they investigate. –  Andrés Caicedo Jan 3 '13 at 23:20
The general statement is that assuming CH, every countable theory with infinite models has a saturated model of cardinality $\aleph_1$. The converse holds for any theory with no $\omega$-stable completion (such as any theory involving a linear order). –  Emil Jeřábek Jan 4 '13 at 12:36

A very simple algebraic statement equivalent to CH is the assertion that the Baer-Specker group $\mathbb{Z}^{\omega}$ is almost free, i.e. all its subgroups of cardinality less than $\mid \mathbb{Z}^{\omega} \mid$ are free. (This is somewhat unsatisfactory in that almost freenees refers to cardinality.)

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