continuum-hypothesislo.logicra.rings-and-algebrasset-theory
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.
Since you have already linked to some of the contemporary primary sources, where of course the full accounts of those views can be found, let me interpret your question as a request for summary accounts of the various views on CH. I'll just describe in a few sentences each of what I find to be the main issues surrounding CH, beginning with some historical views. Please forgive the necessary simplifications.
Cantor. Cantor introduced the Continuum Hypothesis when he discovered the transfinite numbers and proved that the reals are uncountable. It was quite natural to inquire whether the continuum was the same as the first uncountable cardinal. He became obsessed with this question, working on it from various angles and sometimes switching opinion as to the likely outcome. Giving birth to the field of descriptive set theory, he settled the CH question for closed sets of reals, by proving (the Cantor-Bendixon theorem) that every closed set is the union of a countable set and a perfect set. Sets with this perfect set property cannot be counterexamples to CH, and Cantor hoped to extend this method to additional larger classes of sets.
Hilbert. Hilbert thought the CH question so important that he listed it as the first on his famous list of problems at the opening of the 20th century.
Goedel. Goedel proved that CH holds in the constructible universe $L$, and so is relatively consistent with ZFC. Goedel viewed $L$ as a device for establishing consistency, rather than as a description of our (Platonic) mathematical world, and so he did not take this result to settle CH. He hoped that the emerging large cardinal concepts, such as measurable cardinals, would settle the CH question, and as you mentioned, favored a solution of the form $2^\omega=\aleph_2$.
Cohen. Cohen introduced the method of forcing and used it to prove that $\neg$CH is relatively consistent with ZFC. Every model of ZFC has a forcing extension with $\neg$CH. Thus, the CH question is independent of ZFC, neither provable nor refutable. Solovay observed that CH also is forceable over any model of ZFC.
Large cardinals. Goedel's expectation that large cardinals might settle CH was decisively refuted by the Levy-Solovay theorem, which showed that one can force either CH or $\neg$CH while preserving all known large cardinals. Thus, there can be no direct implication from large cardinals to either CH or $\neg$CH. At the same time, Solovay extended Cantor's original strategy by proving that if there are large cardinals, then increasing levels of the projective hierarchy have the perfect set property, and therefore do not admit counterexamples to CH. All of the strongest large cardinal axioms considered today imply that there are no projective counterexamples to CH. This can be seen as a complete affirmation of Cantor's original strategy.
Basic Platonic position. This is the realist view that there is Platonic universe of sets that our axioms are attempting to describe, in which every set-theoretic question such as CH has a truth value. In my experience, this is the most common or orthodox view in the set-theoretic community. Several of the later more subtle views rest solidly upon the idea that there is a fact of the matter to be determined.
Old-school dream solution of CH. The hope was that we might settle CH by finding a new set-theoretic principle that we all agreed was obviously true for the intended interpretation of sets (in the way that many find AC to be obviously true, for example) and which also settled the CH question. Then, we would extend ZFC to include this new principle and thereby have an answer to CH. Unfortunately, no such conclusive principles were found, although there have been some proposals in this vein, such as Freilings axiom of symmetry.
Formalist view. Rarely held by mathematicians, although occasionally held by philosophers, this is the anti-realist view that there is no truth of the matter of CH, and that mathematics consists of (perhaps meaningless) manipulations of strings of symbols in a formal system. The formalist view can be taken to hold that the independence result itself settles CH, since CH is neither provable nor refutable in ZFC. One can have either CH or $\neg$CH as axioms and form the new formal systems ZFC+CH or ZFC+$\neg$CH. This view is often mocked in straw-man form, suggesting that the formalist can have no preference for CH or $\neg$CH, but philosophers defend more subtle versions, where there can be reason to prefer one formal system to another.
Pragmatic view. This is the view one finds in practice, where mathematicians do not take a position on CH, but feel free to use CH or $\neg$CH if it helps their argument, keeping careful track of where it is used. Usually, when either CH or $\neg$CH is used, then one naturally inquires about the situation under the alternative hypothesis, and this leads to numerous consistency or independence results.
Cardinal invariants. Exemplifying the pragmatic view, this is a very rich subject studying various cardinal characteristics of the continuum, such as the size of the smallest unbounded family of functions $f:\omega\to\omega$, the additivity of the ideal of measure-zero sets, or the smallest size family of functions $f:\omega\to\omega$ that dominate all other such functions. Since these characteristics are all uncountable and at most the continuum, the entire theory trivializes under CH, but under $\neg$CH is a rich, fascinating subject.
Canonical Inner models. The paradigmatic canonical inner model is Goedel's constructible universe $L$, which satisfies CH and indeed, the Generalized Continuum Hypothesis, as well as many other regularity properties. Larger but still canonical inner models have been built by Silver, Jensen, Mitchell, Steel and others that share the GCH and these regularity properties, while also satisfying larger large cardinal axioms than are possible in $L$. Most set-theorists do not view these inner models as likely to be the "real" universe, for similar reasons that they reject $V=L$, but as the models accommodate larger and larger large cardinals, it becomes increasingly difficult to make this case. Even $V=L$ is compatible with the existence of transitive set models of the very largest large cardinals (since the assertion that such sets exist is $\Sigma^1_2$ and hence absolute to $L$). In this sense, the canonical inner models are fundamentally compatible with whatever kind of set theory we are imagining.
Woodin. In contrast to the Old-School Dream Solution, Woodin has advanced a more technical argument in favor of $\neg$CH. The main concepts include $\Omega$-logic and the $\Omega$-conjecture, concerning the limits of forcing-invariant assertions, particularly those expressible in the structure $H_{\omega_2}$, where CH is expressible. Woodin's is a decidedly Platonist position, but from what I have seen, he has remained guarded in his presentations, describing the argument as a proposal or possible solution, despite the fact that others sometimes characterize his position as more definitive.
Foreman. Foreman, who also comes from a strong Platonist position, argues against Woodin's view. He writes supremely well, and I recommend following the links to his articles.
Multiverse view. This is the view, offered in opposition to the Basic Platonist Position above, that we do not have just one concept of set leading to a unique set-theoretic universe, but rather a complex variety of set concepts leading to many different set-theoretic worlds. Indeed, the view is that much of set-theoretic research in the past half-century has been about constructing these various alternative worlds. Many of the alternative set concepts, such as those arising by forcing or by large cardinal embeddings are closely enough related to each other that they can be compared from the perspective of each other. The multiverse view of CH is that the CH question is largely settled by the fact that we know precisely how to build CH or $\neg$CH worlds close to any given set-theoretic universe---the CH and $\neg$CH worlds are in a sense dense among the set-theoretic universes. The multiverse view is realist as opposed to formalist, since it affirms the real nature of the set-theoretic worlds to which the various set concepts give rise. On the Multiverse view, the Old-School Dream Solution is impossible, since our experience in the CH and $\neg$CH worlds will prevent us from accepting any principle $\Phi$ that settles CH as "obviously true". Rather, on the multiverse view we are to study all the possible set-theoretic worlds and especially how they relate to each other.
I should stop now, and I apologize for the length of this answer.
Here is a little progress towards AC.
Theorem. ICF implies the dual Cantor-Schröder-Bernstein theorem, that is $X$ surjects onto $Y$ and $Y$ surjects onto $X$, then they are bijective.
Proof. You explain this in the edit to the question. If $X\twoheadrightarrow Y$, then $2^Y\leq 2^X$ by taking pre-images, and so if also $Y\twoheadrightarrow X$, then $2^X\leq 2^Y$ and so $X\sim Y$ by ICF. QED
Theorem. ICF implies that there are no infinite D-finite sets.
Proof. (This is a solution to the exercise that you mention.) If $A$ is infinite and Dedekind-finite, then let $B$ be the set of all finite non-repeating finite sequences from $A$. This is also D-finite, since a countably infinite subset of $B$ easily gives rise to a countably infinite subset of $A$. But meanwhile, $B$ surjects onto $B+1$, since we can map the empty sequence to the new point, and apply the shift map to chop off the first element of any sequence. So $B$ and $B+1$ surject onto each other, and so by the dual Cantor-Schöder-Bernstein result, they are bijective, contradicting the fact that $B$ is D-finite. QED
Here is the new part:
Theorem. ICF implies that $\kappa^+$ injects into $2^\kappa$ for every ordinal $\kappa$.
Proof. We may assume $\kappa$ is infinite. Notice that $2^{\kappa^2}$ surjects onto $\kappa^+$, since every $\alpha<\kappa$ is coded by a relation on $\kappa$. Since $\kappa^2\sim\kappa$, this means $2^\kappa\twoheadrightarrow\kappa^+$ and consequently $2^{\kappa^+}\leq 2^{2^\kappa}$, by taking pre-images. It follows that $2^{2^\kappa}=2^{2^\kappa}\cdot 2^{\kappa^+}=2^{2^\kappa+\kappa^+}$ and so by ICF we get $2^\kappa+\kappa^+=2^\kappa$, which implies $\kappa^+\leq 2^\kappa$, as desired. QED
This conclusion already contradicts AD, for example, since AD implies that there is no $\omega_1$ sequence of distinct reals, which violates the conclusion when $\kappa=\omega$. In particular, this shows that ICF implies $\neg$AD, and so in every AD model, there are sets of different cardinalities, whose power sets are equinumerous.
Best Answer
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:
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. (Wayback Machine)
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 $\aleph_2$.