Suslin hypothesis

A hypothesis that states that every linearly ordered set without first and last elements that is moreover complete, dense and satisfies the Suslin condition, is order-isomorphic to the real line $ \mathbb{R} $. Here, completeness signifies the existence of a least upper bound for every non-empty bounded subset, density denotes the non-emptiness of any interval $ (a,b) $, where $ a < b $, and the Suslin condition requires that every family of pairwise-disjoint intervals is countable. The real line possesses all the properties that appear in the formulation of the Suslin hypothesis. The Suslin hypothesis thus states that the above properties of an ordered set define it completely. This hypothesis was formulated in 1920 by M.Ya. Suslin [1].

Within the framework of the system $ \mathsf{ZFC} $ (the system $ \mathsf{ZF} $ with the axiom of choice), it is impossible to prove or disprove the Suslin hypothesis, assuming that $ \mathsf{ZF} $ is consistent. It follows from Gödel’s axiom of constructibility, $ V = L $ (see Gödel constructive set), that the negation of the Suslin hypothesis holds. The consistency of the Suslin hypothesis with the axioms of $ \mathsf{ZFC} $ is proved by the construction of a corresponding model, using a variant of the forcing method, namely, iterated forcing. The addition of the continuum hypothesis ($ \mathsf{CH} $) to $ \mathsf{ZFC} $ does not give either a positive or negative solution to the Suslin hypothesis.

The Suslin hypothesis and its generalizations have had a great influence on the development of axiomatic set theory. Many ideas and methods have been worked out in conjunction with it. These include Jensen’s combinatorial principles $ \diamondsuit_{\kappa} $ and $ \Box_{\kappa} $ ([4]), the theory of the fine structure of the constructible hierarchy ([5]), Martin’s axiom ([7]), and the iterated forcing method ([2]).

Let $ \kappa $ be a limit ordinal. A subset $ A $ of $ \kappa $ is said to be closed unbounded (or club for short) if and only if it contains all of its limit points $ < \kappa $ and if, for any $ \alpha < \kappa $, there is a $ \beta \in A $ such that $ \alpha < \beta $. A subset $ A $ of $ \kappa $ is said to be stationary if and only if its intersection with every club subset of $ \kappa $ is non-empty. Jensen’s principles, for $ \kappa $ an uncountable cardinal, can then be stated as follows:

• Jensen’s $ \diamondsuit_{\kappa} $: There exists a sequence $ \langle S_{\alpha} \mid \alpha < \kappa \rangle $, where $ S_{\alpha} \subseteq \alpha $, such that for every $ X \subseteq \kappa $, the set $ \{ \alpha < \kappa \mid S_{\alpha} = X \cap \alpha \} $ is a stationary subset of $ \kappa $.

• Jensen’s $ \Box_{\kappa} $: There exists a function $ C $, with domain $ \{ \alpha < \kappa^{+} \mid \alpha \text{ is a limit ordinal} \} $, such that for all $ \alpha \in \operatorname{Dom}(C) $, the following hold:

1. $ C(\alpha) $ is a club subset of $ \alpha $.
2. If $ \mathsf{cf}(\alpha) < \kappa $, then $ \mathsf{card}(C(\alpha)) < \kappa $.
3. If $ \beta $ is a limit point in $ C(\alpha) $, i.e., $ \beta < \alpha $ and $ C(\alpha) \cap \beta $ is cofinal in $ \beta $, then $ C(\beta) = C(\alpha) \cap \beta $.

For every regular cardinal $ \kappa $, the principle $ \diamondsuit_{\kappa} $ follows from the axiom of constructibility, while the negation of the Suslin hypothesis follows from $ \diamondsuit_{\omega_{1}} $. Jensen’s combinatorial principles, as well as Martin’s axiom (see below), have been successfully used in topology ([4], [6], [8]).

Let $ \langle \mathbb{P},\leq \rangle $ be a partially ordered set. A set $ D \subseteq \mathbb{P} $ is called dense if and only if for every $ p \in \mathbb{P} $, there is a $ d \in D $ such that $ d \leq p $. A set $ Q \subseteq \mathbb{P} $ is said to be compatible if and only if for any finite subset $ F \subseteq Q $, there is a $ p \in \mathbb{P} $ such that $ p \leq r $ for every $ r \in F $. Two elements, $ p_{1} $ and $ p_{2} $, of $ \mathbb{P} $ are said to be incompatible if and only if the set $ \{ p_{1},p_{2} \} $ is not compatible. It is said that a partially ordered set $ \langle \mathbb{P},\leq \rangle $ satisfies the countable anti-chain condition if and only if every set that consists of pairwise-incompatible elements is countable. Martin's axiom ($ \mathsf{MA} $) states the following: If a partially ordered set $ \langle \mathbb{P},\leq \rangle $ satisfies the countable anti-chain condition and if $ \mathcal{P} $ is a family of dense subsets of cardinality less than $ 2^{\aleph_{0}} $, then there is a compatible set $ Q \subseteq \mathbb{P} $ such that for every $ D \in \mathcal{P} $, the intersection $ D \cap Q $ is non-empty.

In the presence of the continuum hypothesis, Martin’s axiom can be proved. The most interesting results are obtained by a combination of Martin’s axiom and the negation of the continuum hypothesis ($ \neg \mathsf{CH} $). The principle $ \diamondsuit_{\omega_{1}} $ contradicts the combination $ \mathsf{MA} + \neg \mathsf{CH} $, since $ \diamondsuit_{\omega_{1}} $ implies $ \mathsf{CH} $. It often turns out that a result that can be inferred from $ \diamondsuit_{\omega_{1}} $ will be disproved under the assumption $ \mathsf{MA} + \neg \mathsf{CH} $. This is, for example, the case with the Suslin hypothesis. Indeed, $ \mathsf{MA} + \neg \mathsf{CH} $ implies the Suslin hypothesis, while $ \diamondsuit_{\omega_{1}} $ implies the negation of the Suslin hypothesis.

The combination $ \mathsf{MA} + \neg \mathsf{CH} $ is compatible with $ \mathsf{ZFC} $ if $ \mathsf{ZF} $ is consistent.

References

[1]	M. [M.Ya. Suslin] Souslin, “Problème 3”, Fundam. Math., 1 (1920), pp. 223.
[2]	K.J. Devlin, H. Johnsbråten, “The Souslin problem”, Lect. notes in math., 405, Springer (1974).
[3]	T.J. Jech, “Lectures in set theory: with particular emphasis on the method of forcing”, Lect. notes in math., 217, Springer (1971).
[4]	J. Barwise (ed.), Handbook of mathematical logic, North-Holland (1977), Chapts. B4-B7.
[5]	K.J. Devlin, “Aspects of constructibility”, Lect. notes in math., 354, Springer (1973).
[6]	V.V. Fedorchuk, “Completely closed mappings and the compatibility of certain general topology theorems with the axioms of set theory”, Mat. Sb., 99: 1 (1976), pp. 3–33. (In Russian)
[7]	D.A. Martin, R. Solovay, “Internal Cohen extensions”, Ann. Math. Logic, 2 (1970), pp. 143–178.
[8]	V.I. Malykhin, “Topology and forcing”, Russian Math. Surveys, 38: 1 (1983), pp. 77–136; Uspekhi Mat. Nauk, 38: 1 (1983), pp. 69–118.

Comments

Jensen’s combinatorial principles $ \diamondsuit $ and $ \Box $ are called diamond and square respectively. The countable anti-chain condition is sometimes called the countable chain condition, which is then abbreviated to c.c.c.

References

[a1]	K.J. Devlin, “Constructibility”, Springer (1984).
[a2]	D.H. Fremlin, “Consequences of Martin’s axiom”, Cambridge Univ. Press (1984).
[a3]	T.J. Jech, “Multiple forcing”, Cambridge Univ. Press (1986).
[a4]	T.J. Jech, “Set theory”, Acad. Press (1978). (Translated from German)
[a5]	K. Kunen, “Set theory, an introduction to independence proofs”, North-Holland (1980).