Martin's axiom

Last updated April 24, 2024

In the mathematical field of set theory, Martin's axiom, introduced by Donald A. Martin and Robert M. Solovay,^[1] is a statement that is independent of the usual axioms of ZFC set theory. It is implied by the continuum hypothesis, but it is consistent with ZFC and the negation of the continuum hypothesis. Informally, it says that all cardinals less than the cardinality of the continuum, 𝔠, behave roughly like ℵ₀. The intuition behind this can be understood by studying the proof of the Rasiowa–Sikorski lemma. It is a principle that is used to control certain forcing arguments.

Statement

For a cardinal number κ, define the following statement:

MA(κ): For any partial order P satisfying the countable chain condition (hereafter ccc) and any set D = {D_i}_i∈I of dense subsets of P such that |D| ≤ κ, there is a filter F on P such that F ∩ D_i is non-empty for every D_i ∈ D.

In this case (for application of ccc), an antichain is a subset A of P such that any two distinct members of A are incompatible (two elements are said to be compatible if there exists a common element below both of them in the partial order). This differs from, for example, the notion of antichain in the context of trees.

MA(ℵ₀) is provable in ZFC and known as the Rasiowa–Sikorski lemma.

MA(2^ℵ₀) is false: [0, 1] is a separable compact Hausdorff space, and so (P, the poset of open subsets under inclusion, is) ccc. But now consider the following two 𝔠-size sets of dense sets in P: no x ∈ [0, 1] is isolated, and so each x defines the dense subset { S | x ∉ S }. And each r ∈ (0, 1], defines the dense subset { S | diam(S) < r }. The two sets combined are also of size 𝔠, and a filter meeting both must simultaneously avoid all points of [0, 1] while containing sets of arbitrarily small diameter. But a filter F containing sets of arbitrarily small diameter must contain a point in ⋂F by compactness. (See also § Equivalent forms of MA(κ).)

Martin's axiom is then that MA(κ) holds for every κ for which it could:

Martin's axiom (MA): MA(κ) holds for every κ < 𝔠.

Equivalent forms of MA(κ)

The following statements are equivalent to MA(κ):

If X is a compact Hausdorff topological space that satisfies the ccc then X is not the union of κ or fewer nowhere dense subsets.
If P is a non-empty upwards ccc poset and Y is a set of cofinal subsets of P with |Y| ≤ κ then there is an upwards-directed set A such that A meets every element of Y.
Let A be a non-zero ccc Boolean algebra and F a set of subsets of A with |F| ≤ κ. Then there is a boolean homomorphism φ: A → Z/2Z such that for every X ∈ F, there is either an a ∈ X with φ(a) = 1 or there is an upper bound b ∈ X with φ(b) = 0.

Consequences

Martin's axiom has a number of other interesting combinatorial, analytic and topological consequences:

The union of κ or fewer null sets in an atomless σ-finite Borel measure on a Polish space is null. In particular, the union of κ or fewer subsets of R of Lebesgue measure 0 also has Lebesgue measure 0.
A compact Hausdorff space X with |X| < 2^κ is sequentially compact, i.e., every sequence has a convergent subsequence.
No non-principal ultrafilter on N has a base of cardinality less than κ.
Equivalently for any x ∈ βN\N we have 𝜒(x) ≥ κ, where 𝜒 is the character of x, and so 𝜒(βN) ≥ κ.
MA(ℵ₁) implies that a product of ccc topological spaces is ccc (this in turn implies there are no Suslin lines).
MA + ¬CH implies that there exists a Whitehead group that is not free; Shelah used this to show that the Whitehead problem is independent of ZFC.

Further development

Martin's axiom has generalizations called the proper forcing axiom and Martin's maximum.
Sheldon W. Davis has suggested in his book that Martin's axiom is motivated by the Baire category theorem.^[2]

Related Research Articles

In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory equivalent to the statement that a Cartesian product of a collection of non-empty sets is non-empty. Informally put, the axiom of choice says that given any collection of sets, each containing at least one element, it is possible to construct a new set by choosing one element from each set, even if the collection is infinite. Formally, it states that for every indexed family $of nonempty sets, there exists an indexed set such that for every . The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem.$

In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence $of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence.$

This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric topology.

In the mathematical field of order theory, an ultrafilter on a given partially ordered set $is a certain subset of namely a maximal filter on that is, a proper filter on that cannot be enlarged to a bigger proper filter on$

In the mathematical discipline of general topology, Stone–Čech compactification is a technique for constructing a universal map from a topological space X to a compact Hausdorff space βX. The Stone–Čech compactification βX of a topological space X is the largest, most general compact Hausdorff space "generated" by X, in the sense that any continuous map from X to a compact Hausdorff space factors through βX. If X is a Tychonoff space then the map from X to its image in βX is a homeomorphism, so X can be thought of as a (dense) subspace of βX; every other compact Hausdorff space that densely contains X is a quotient of βX. For general topological spaces X, the map from X to βX need not be injective.

In the mathematical discipline of set theory, forcing is a technique for proving consistency and independence results. Intuitively, forcing can be thought of as a technique to expand the set theoretical universe $to a larger universe by introducing a new "generic" object .$

In mathematics, Suslin's problem is a question about totally ordered sets posed by Mikhail Yakovlevich Suslin and published posthumously. It has been shown to be independent of the standard axiomatic system of set theory known as ZFC; Solovay & Tennenbaum (1971) showed that the statement can neither be proven nor disproven from those axioms, assuming ZF is consistent.

In set theory, an uncountable cardinal is inaccessible if it cannot be obtained from smaller cardinals by the usual operations of cardinal arithmetic. More precisely, a cardinal $κ$ is strongly inaccessible if it satisfies the following three conditions: it is uncountable, it is not a sum of fewer than $κ$ cardinals smaller than $κ$ , and $implies .$

In mathematics, a measurable cardinal is a certain kind of large cardinal number. In order to define the concept, one introduces a two-valued measure on a cardinal κ, or more generally on any set. For a cardinal κ, it can be described as a subdivision of all of its subsets into large and small sets such that κ itself is large, ∅ and all singletons {α} are small, complements of small sets are large and vice versa. The intersection of fewer than κ large sets is again large.

<span class="mw-page-title-main">Aleph number</span> Infinite cardinal number

In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality of infinite sets that can be well-ordered. They were introduced by the mathematician Georg Cantor and are named after the symbol he used to denote them, the Hebrew letter aleph (ℵ).

In mathematics, a Grothendieck universe is a set U with the following properties:

If x is an element of U and if y is an element of x, then y is also an element of U. (U is a transitive set.)
If x and y are both elements of U, then $is an element of U .$
If x is an element of U, then P(x), the power set of x, is also an element of U.
If $is a family of elements of U, and if I is an element of U, then the union is an element of U .$

In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological spaces which is closed under homeomorphisms. That is, a property of spaces is a topological property if whenever a space X possesses that property every space homeomorphic to X possesses that property. Informally, a topological property is a property of the space that can be expressed using open sets.

In order theory, a partially ordered set X is said to satisfy the countable chain condition, or to be ccc, if every strong antichain in X is countable.

In mathematics, set-theoretic topology is a subject that combines set theory and general topology. It focuses on topological questions that are independent of Zermelo–Fraenkel set theory (ZFC).

<span class="mw-page-title-main">Axiom of limitation of size</span>

In set theory, the axiom of limitation of size was proposed by John von Neumann in his 1925 axiom system for sets and classes. It formalizes the limitation of size principle, which avoids the paradoxes encountered in earlier formulations of set theory by recognizing that some classes are too big to be sets. Von Neumann realized that the paradoxes are caused by permitting these big classes to be members of a class. A class that is a member of a class is a set; a class that is not a set is a proper class. Every class is a subclass of V, the class of all sets. The axiom of limitation of size says that a class is a set if and only if it is smaller than V—that is, there is no function mapping it onto V. Usually, this axiom is stated in the equivalent form: A class is a proper class if and only if there is a function that maps it onto V.

In the mathematical field of set theory, the proper forcing axiom (PFA) is a significant strengthening of Martin's axiom, where forcings with the countable chain condition (ccc) are replaced by proper forcings.

In axiomatic set theory, the Rasiowa–Sikorski lemma named after Helena Rasiowa and Roman Sikorski is one of the most fundamental facts used in the technique of forcing. In the area of forcing, a subset E of a poset (P, ≤) is called dense in P if for any p ∈ P there is e ∈ E with e ≤ p. If D is a set of dense subsets of P, then a filter F in P is called D-generic if

This is a glossary of set theory.

References

↑ Martin, Donald A.; Solovay, Robert M. (1970). "Internal Cohen extensions". Ann. Math. Logic. 2 (2): 143–178. doi: 10.1016/0003-4843(70)90009-4 . MR 0270904.
↑ Davis, Sheldon W. (2005). Topology. McGraw Hill. p. 29. ISBN 0-07-291006-2.

v t e Set theory
Overview	Set (mathematics)
Axioms	Adjunction Choice countable dependent global Constructibility (V=L) Determinacy Extensionality Infinity Limitation of size Pairing Power set Regularity Union Martin's axiom Axiom schema replacement specification
Operations	Cartesian product Complement (i.e. set difference) De Morgan's laws Disjoint union Identities Intersection Power set Symmetric difference Union
Concepts Methods	Almost Cardinality Cardinal number (large) Class Constructible universe Continuum hypothesis Diagonal argument Element ordered pair tuple Family Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram
Set types	Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset · Superset Transitive Uncountable Universal
Theories	Alternative Axiomatic Naive Cantor's theorem Zermelo General Principia Mathematica New Foundations Zermelo–Fraenkel von Neumann–Bernays–Gödel Morse–Kelley Kripke–Platek Tarski–Grothendieck
Paradoxes Problems	Russell's paradox Suslin's problem Burali-Forti paradox
Set theorists	Paul Bernays Georg Cantor Paul Cohen Richard Dedekind Abraham Fraenkel Kurt Gödel Thomas Jech John von Neumann Willard Quine Bertrand Russell Thoralf Skolem Ernst Zermelo