[Math] The Continuum Hypothesis and Countable Unions

axiom-of-choicecontinuum-hypothesislo.logicset-theory

I recently edited an answer of mine on math.SE which discussed the implication of the two assertions:

$AH(0)$ which is $2^{\aleph_0}=\aleph_1$, and
$CH$ which says that if $A\subseteq 2^{\omega}$ and $\aleph_0<|A|$ then $|A|=2^{\aleph_0}$.

We know they are indeed equivalent under the axiom of choice (and actually much less). It is also trivial to see that $AH(0)\Rightarrow CH$. However the converse is not true, indeed in Solovay's model (or in models of AD) there are no $\aleph_1$ many reals, but $CH$ holds since every uncountable set of reals has a perfect subset.

While revising my answer I tried to find a reference whether or not in the Feferman-Levy model, in which the continuum is a countable union of countable sets, satisfies the continuum hypothesis (we already know that it does not satisfy $AH(0)$).

To my surprise the answer is negative. There exists a set whose cardinality is strictly between the continuum and $\omega$, the construction is described in A. Miller's paper [1] in which he remarks that in the Feferman-Levy the constructed set cannot be put in bijection with the continuum.

I was wondering whether or not this is always true in models in which the continuum is a countable union of countable sets, or is this just one of the peculiarities of the Feferman-Levy model.

Questions:

Let $V$ be a model of $ZF$ in which $2^{\omega}$ can be written as the countable union of countable sets. Does $CH$ fail in $V$?

Suppose that $V$ is a model of $\omega_1\nleq2^\omega$ and $CH$, does this imply that $\omega_1$ is regular (which means inaccessible in $L$)?

Bibliography:

Miller, A. A Dedekind Finite Borel Set. Arch. Math. Logic 50 (2011), no. 1-2, 1–17.

Best Answer

The answer for the second question is no. Truss proved in [1] that if we repeat Solovay's construction from a limit cardinal $\kappa$, we obtain a model in which the following properties:

Countable unions of countable sets of real numbers are countable;
Every well-orderable set of real numbers is countable;
Every uncountable set of reals has a perfect subset;
DC holds iff $\omega_1$ is regular iff $\kappa$ is inaccessible in the ground model;
Every set of real numbers is Borel.

This shows that it is possible to have $CH+\aleph_1\nleq2^{\aleph_0}+\operatorname{cf}(\omega_1)=\omega$. However it does not answer the original (first) question.

Bibliography:

Truss, John, Models of set theory containing many perfect sets. Ann. Math. Logic 7 (1974), 197–219.

Related Solutions

Solutions to the Continuum Hypothesis

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.

[Math] Countable unions and the axiom of countable choice

The implication $ACC \implies UCC$ is irreversible in $ZF$. This follows again from the transfer theorem of Pincus:

$UCC$ is an injectively boundable statement, see note 103 in "Consequences of the axiom of choice" by Howard & Rubin. Right after the theorem in pp. 285 and its corollary, there are examples of statements of this kind, one of which is form 31 (which is precisely $UCC$). The fact that this is the case follows in turn from the application of lemma 3.5 in Howard, P.-Solski, J.: "The strength of the $\Delta$-system lemma", Notre Dame J. Formal Logic vol 34, pp. 100-106 - 1993
It is known that $¬ACC$ is boundable, and hence injectively boundable.

Then we can apply the transfer theorem of Pincus to the conjunction $UCC \wedge ¬ACC$ and we are done.

SECOND PROOF: Browsing through the "Consequences..." book I've just found another less direct proof of the same fact. I'm adding it here to avoid the interested person from looking it up for himself (especially since the AC website is not working these days). It involves form 9, known as $W_{\aleph_0}$: a set is finite if and only if it is Dedekind finite. Now, $ACC \implies W_{\aleph_0}$ is provable in $ZF$ (in fact this was already proved by Dedekind), while $UCC$ does not imply $W_{\aleph_0}$. The latter follows from the fact that in the basic Fraenkel model, $\mathcal{N}1$, $UCC$ is valid while $W_{\aleph_0}$ is not, and such a result is transferable by considerations of Pincus that can be found at Pincus, D.: "Zermelo-Fraenkel consistency results by Fraenkel Mostowski methods", J. of Symbolic Logic vol 37, pp. 721-743 - 1972 (doi:10.2307/2272420)

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Solutions to the Continuum Hypothesis

[Math] Countable unions and the axiom of countable choice

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