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In the
mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
field of
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
, Martin's axiom, introduced by
Donald A. Martin Donald Anthony Martin (born December 24, 1940), also known as Tony Martin, is an American set theorist and philosopher of mathematics at UCLA, where he is an emeritus professor of mathematics and philosophy. Education and career Martin receiv ...
and
Robert M. Solovay Robert Martin Solovay (born December 15, 1938) is an American mathematician specializing in set theory. Biography Solovay earned his Ph.D. from the University of Chicago in 1964 under the direction of Saunders Mac Lane, with a dissertation on '' ...
, is a statement that is independent of the usual axioms of ZFC set theory. It is implied by the
continuum hypothesis In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that or equivalently, that In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to ...
, 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 In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers \mathbb R, sometimes called the continuum. It is an infinite cardinal number and is denoted by \mathfrak c (lowercase fraktur "c") or , \math ...
, \mathfrak c, behave roughly like \aleph_0. 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 any cardinal 𝛋, we define a statement, denoted by MA(𝛋):
For any
partial order In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
''P'' satisfying the countable chain condition (hereafter ccc) and any family ''D'' of dense sets in ''P'' such that '', D, '' ≤ 𝛋, there is a filter ''F'' on ''P'' such that ''F'' ∩ ''d'' is non-
empty Empty may refer to: ‍ Music Albums * ''Empty'' (God Lives Underwater album) or the title song, 1995 * ''Empty'' (Nils Frahm album), 2020 * ''Empty'' (Tait album) or the title song, 2001 Songs * "Empty" (The Click Five song), 2007 * ...
for every ''d'' in ''D''.
\operatorname(\aleph_0) is simply true — this is known as the Rasiowa–Sikorski lemma. \operatorname(2^) is false: , 1is a
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Britis ...
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the ma ...
, which is separable and so ccc. It has no isolated points, so points in it are nowhere dense, but it is the union of 2^=\mathfrak c many points. (See the condition equivalent to \operatorname(\mathfrak c) below.) Since it is a theorem of ZFC that \operatorname(\mathfrak c) fails, Martin's axiom is stated as:
Martin's axiom (MA): For every 𝛋 < \mathfrak c, MA(𝛋) holds.
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 In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are u ...
.


Equivalent forms of MA(𝛋)

The following statements are equivalent to MA(𝛋): * If ''X'' is a compact Hausdorff
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
that satisfies the ccc then ''X'' is not the union of 𝛋 or fewer
nowhere dense In mathematics, a subset of a topological space is called nowhere dense or rare if its closure has empty interior. In a very loose sense, it is a set whose elements are not tightly clustered (as defined by the topology on the space) anywher ...
subsets. * If ''P'' is a non-empty upwards ccc
poset In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary r ...
and ''Y'' is a family 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 In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas i ...
and ''F'' a family of subsets of ''A'' with '', F, '' ≤ 𝛋. Then there is a boolean homomorphism φ: ''A'' → Z/2Z such that for every ''X'' in ''F'' either there is an ''a'' in ''X'' with φ(''a'') = 1 or there is an upper bound ''b'' for ''X'' with φ(''b'') = 0.


Consequences

Martin's axiom has a number of other interesting combinatorial,
analytic Generally speaking, analytic (from el, ἀναλυτικός, ''analytikos'') refers to the "having the ability to analyze" or "division into elements or principles". Analytic or analytical can also have the following meanings: Chemistry * ...
and topological consequences: * The union of 𝛋 or fewer
null set In mathematical analysis, a null set N \subset \mathbb is a measurable set that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length. The notion of null ...
s in an atomless σ-finite
Borel measure In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below. ...
on a
Polish space In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named be ...
is null. In particular, the union of 𝛋 or fewer subsets of R of
Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wi ...
0 also has Lebesgue measure 0. * A compact Hausdorff space ''X'' with '', X, '' < 2𝛋 is sequentially compact, i.e., every sequence has a convergent subsequence. * No non-principal
ultrafilter In the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P is a certain subset of P, namely a maximal filter on P; that is, a proper filter on P that cannot be enlarged to a bigger proper filter o ...
on N has a base of cardinality < 𝛋. * Equivalently for any ''x'' in βN\N we have 𝜒(''x'') ≥ 𝛋, where 𝜒 is the
character Character or Characters may refer to: Arts, entertainment, and media Literature * ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk * ''Characters'' (Theophrastus), a classical Greek set of character sketches attributed to The ...
of ''x'', and so 𝜒(βN) ≥ 𝛋. * \operatorname(\aleph_1) implies that a product of ccc topological spaces is ccc (this in turn implies there are no
Suslin line In mathematics, Suslin's problem is a question about totally ordered sets posed by and published posthumously. It has been shown to be independent of the standard axiomatic system of set theory known as ZFC: showed that the statement can neit ...
s). * MA + ¬CH implies that there exists a Whitehead group that is not free;
Shelah Shelah may refer to: * Shelah (son of Judah), a son of Judah according to the Bible * Shelah (name), a Hebrew personal name * Shlach, the 37th weekly Torah portion (parshah) in the annual Jewish cycle of Torah reading * Salih, a prophet described ...
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 In set theory, a branch of mathematical logic, Martin's maximum, introduced by and named after Donald Martin, is a generalization of the proper forcing axiom, itself a generalization of Martin's axiom. It represents the broadest class of forcings ...
. *Sheldon W. Davis has suggested in his book that Martin's axiom is motivated by the
Baire category theorem The Baire category theorem (BCT) is an important result in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space (a topological space such that the ...
.


References


Further reading

* * Jech, Thomas, 2003. ''Set Theory: The Third Millennium Edition, Revised and Expanded''. Springer. . * Kunen, Kenneth, 1980. ''Set Theory: An Introduction to Independence Proofs''. Elsevier. . {{Set theory Axioms of set theory Independence results Set theory