measurable cardinal
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mathematics 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 ...
, a measurable cardinal is a certain kind of
large cardinal In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers. Cardinals with such properties are, as the name suggests, generally very "large" (for example, bigger than the least Î ...
number. In order to define the concept, one introduces a two-valued
measure Measure may refer to: * Measurement, the assignment of a number to a characteristic of an object or event Law * Ballot measure, proposed legislation in the United States * Church of England Measure, legislation of the Church of England * Mea ...
on a cardinal , or more generally on any set. For a cardinal , it can be described as a subdivision of all of its
subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
s into large and small sets such that itself is large, and all
singleton Singleton may refer to: Sciences, technology Mathematics * Singleton (mathematics), a set with exactly one element * Singleton field, used in conformal field theory Computing * Singleton pattern, a design pattern that allows only one instance ...
s are small,
complement A complement is something that completes something else. Complement may refer specifically to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-clas ...
s of small sets are large and vice versa. The intersection of fewer than large sets is again large. It turns out that
uncountable In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal num ...
cardinals endowed with a two-valued measure are large cardinals whose existence cannot be proved from ZFC. The concept of a measurable cardinal was introduced by Stanislaw Ulam in 1930.


Definition

Formally, a measurable cardinal is an uncountable
cardinal number In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. Th ...
κ such that there exists a κ-additive, non-trivial, 0-1-valued
measure Measure may refer to: * Measurement, the assignment of a number to a characteristic of an object or event Law * Ballot measure, proposed legislation in the United States * Church of England Measure, legislation of the Church of England * Mea ...
on the
power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...
of ''κ''. (Here the term ''κ-additive'' means that, for any sequence ''A''''α'', α<λ of cardinality ''λ'' < ''κ'', ''A''''α'' being pairwise disjoint sets of ordinals less than κ, the measure of the union of the ''A''''α'' equals the sum of the measures of the individual ''A''''α''.) Equivalently, ''κ'' is measurable means that it is the critical point of a non-trivial
elementary embedding In model theory, a branch of mathematical logic, two structures ''M'' and ''N'' of the same signature ''σ'' are called elementarily equivalent if they satisfy the same first-order ''σ''-sentences. If ''N'' is a substructure of ''M'', one often ...
of the
universe The universe is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the universe. Acc ...
''V'' into a
transitive class In set theory, a branch of mathematics, a set A is called transitive if either of the following equivalent conditions hold: * whenever x \in A, and y \in x, then y \in A. * whenever x \in A, and x is not an urelement, then x is a subset of A. Sim ...
''M''. This equivalence is due to Jerome Keisler and
Dana Scott Dana Stewart Scott (born October 11, 1932) is an American logician who is the emeritus Hillman University Professor of Computer Science, Philosophy, and Mathematical Logic at Carnegie Mellon University; he is now retired and lives in Berkeley, Ca ...
, and uses the
ultrapower The ultraproduct is a mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is a quotient of the direct product of a family of structures. All factor ...
construction from
model theory In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the s ...
. Since ''V'' is a
proper class Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map for ...
, a technical problem that is not usually present when considering ultrapowers needs to be addressed, by what is now called
Scott's trick In set theory, Scott's trick is a method for giving a definition of equivalence classes for equivalence relations on a proper class (Jech 2003:65) by referring to levels of the cumulative hierarchy. The method relies on the axiom of regularity but ...
. Equivalently, ''κ'' is a measurable cardinal if and only if it is an uncountable cardinal with a -complete, 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 ...
. Again, this means that the intersection of any ''strictly less than'' ''κ''-many sets in the ultrafilter, is also in the ultrafilter.


Properties

Although it follows from ZFC that every measurable cardinal is inaccessible (and is
ineffable Ineffability is the quality of something that surpasses the capacity of language to express it, often being in the form of a taboo or incomprehensible term. This property is commonly associated with philosophy, aspects of existence, and similar ...
,
Ramsey Ramsey may refer to: Geography British Isles * Ramsey, Cambridgeshire, a small market town in England * Ramsey, Essex, a village near Harwich, England ** Ramsey and Parkeston, a civil parish formerly called just "Ramsey" * Ramsey, Isle of Man, t ...
, etc.), it is consistent with ZF that a measurable cardinal can be a successor cardinal. It follows from ZF +
axiom of determinacy In mathematics, the axiom of determinacy (abbreviated as AD) is a possible axiom for set theory introduced by Jan Mycielski and Hugo Steinhaus in 1962. It refers to certain two-person topological games of length ω. AD states that every game of ...
that ω1 is measurable, and that every subset of ω1 contains or is disjoint from a closed and unbounded subset. Ulam showed that the smallest cardinal κ that admits a non-trivial countably-additive two-valued measure must in fact admit a κ-additive measure. (If there were some collection of fewer than κ measure-0 subsets whose union was κ, then the induced measure on this collection would be a counterexample to the minimality of κ.) From there, one can prove (with the Axiom of Choice) that the least such cardinal must be inaccessible. It is trivial to note that if κ admits a non-trivial κ-additive measure, then κ must be regular. (By non-triviality and κ-additivity, any subset of cardinality less than κ must have measure 0, and then by κ-additivity again, this means that the entire set must not be a union of fewer than κ sets of cardinality less than κ.) Finally, if λ < κ, then it can't be the case that κ ≤ 2λ. If this were the case, then we could identify ''κ'' with some collection of 0-1 sequences of length ''λ''. For each position in the sequence, either the subset of sequences with 1 in that position or the subset with 0 in that position would have to have measure 1. The intersection of these ''λ''-many measure 1 subsets would thus also have to have measure 1, but it would contain exactly one sequence, which would contradict the non-triviality of the measure. Thus, assuming the Axiom of Choice, we can infer that ''κ'' is a strong limit cardinal, which completes the proof of its inaccessibility. If κ is measurable and ''p''∈''V''''κ'' and ''M'' (the ultrapower of ''V'') satisfies ψ(κ,''p''), then the set of ''α'' < ''κ'' such that ''V'' satisfies ''ψ''(''α'',''p'') is stationary in κ (actually a set of measure 1). In particular if ''ψ'' is a Π1 formula and ''V'' satisfies ψ(κ,''p''), then ''M'' satisfies it and thus ''V'' satisfies ''ψ''(''α'',''p'') for a stationary set of ''α'' < ''κ''. This property can be used to show that ''κ'' is a limit of most types of large cardinals that are weaker than measurable. Notice that the ultrafilter or measure witnessing that ''κ'' is measurable cannot be in ''M'' since the smallest such measurable cardinal would have to have another such below it, which is impossible. If one starts with an elementary embedding ''j''1 of ''V'' into ''M''1 with critical point κ, then one can define an ultrafilter ''U'' on κ as . Then taking an ultrapower of ''V'' over ''U'' we can get another elementary embedding ''j''2 of ''V'' into ''M''2. However, it is important to remember that ''j''2 ≠ ''j''1. Thus other types of large cardinals such as
strong cardinal In set theory, a strong cardinal is a type of large cardinal. It is a weakening of the notion of a supercompact cardinal. Formal definition If λ is any ordinal, κ is λ-strong means that κ is a cardinal number and ther ...
s may also be measurable, but not using the same embedding. It can be shown that a strong cardinal κ is measurable and also has κ-many measurable cardinals below it. Every measurable cardinal κ is a 0-
huge cardinal In mathematics, a cardinal number κ is called huge if there exists an elementary embedding ''j'' : ''V'' → ''M'' from ''V'' into a transitive inner model ''M'' with critical point κ and :^M \subset M.\! Here, ''αM'' is the class of al ...
because ''κ''''M''⊆''M'', that is, every function from κ to ''M'' is in ''M''. Consequently, ''V''''κ''+1⊆''M''.


Real-valued measurable

A cardinal κ is called real-valued measurable if there is a κ-additive probability measure on the power set of κ that vanishes on singletons. Real-valued measurable cardinals were introduced by . showed that 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 ...
implies that is not real-valued measurable. showed (see below for parts of Ulam's proof) that real valued measurable cardinals are weakly inaccessible (they are in fact
weakly Mahlo In mathematics, a Mahlo cardinal is a certain kind of large cardinal number. Mahlo cardinals were first described by . As with all large cardinals, none of these varieties of Mahlo cardinals can be proven to exist by ZFC (assuming ZFC is consist ...
). All measurable cardinals are real-valued measurable, and a real-valued measurable cardinal κ is measurable if and only if κ is greater than . Thus a cardinal is measurable if and only if it is real-valued measurable and strongly inaccessible. A real valued measurable cardinal less than or equal to exists if and only if there is a
countably additive In mathematics, an additive set function is a function mapping sets to numbers, with the property that its value on a union of two disjoint sets equals the sum of its values on these sets, namely, \mu(A \cup B) = \mu(A) + \mu(B). If this additivit ...
extension of the Lebesgue measure to all sets of real numbers if and only if there is an atomless probability measure on the power set of some non-empty set. showed that existence of measurable cardinals in ZFC, real valued measurable cardinals in ZFC, and measurable cardinals in ZF, are
equiconsistent In mathematical logic, two theories are equiconsistent if the consistency of one theory implies the consistency of the other theory, and vice versa. In this case, they are, roughly speaking, "as consistent as each other". In general, it is not p ...
.


Weak inaccessibility of real-valued measurable cardinals

Say that a cardinal number is an ''Ulam number'' ifThe notion in the article
Ulam number In mathematics, the Ulam numbers comprise an integer sequence devised by and named after Stanislaw Ulam, who introduced it in 1964. The standard Ulam sequence (the (1, 2)-Ulam sequence) starts with ''U''1 = 1 and ''U''2 =&nbs ...
is different.
whenever then ::\operatorname X \le \alpha\Rightarrow\mu(X) = 0. Equivalently, a cardinal number is an Ulam number if whenever # is an outer measure on a set , and a disjoint family of subsets of , # \nu\left(\bigcup F\right) < \infty, # \nu(A) = 0 for A \in F, # \bigcup G is -measurable for every G \subset F then ::\operatorname F \le \alpha\Rightarrow\nu\left(\bigcup F\right) = 0. The smallest infinite cardinal is an Ulam number. The class of Ulam numbers is closed under the cardinal successor operation. If an infinite cardinal has an immediate predecessor that is an Ulam number, assume satisfies properties ()–() with X = \beta. In the
von Neumann model The von Neumann architecture — also known as the von Neumann model or Princeton architecture — is a computer architecture based on a 1945 description by John von Neumann, and by others, in the ''First Draft of a Report on the EDVAC''. The ...
of ordinals and cardinals, choose
injective function In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
s :f_x:x \rightarrow \alpha, \quad \forall x \in \beta, and define the sets :U(b, a) = \, \quad a \in \alpha, b \in \beta. Since the are one-to-one, the sets :\left\ \text a \text, :\left\ \text b \text are disjoint. By property () of , the set :\left\ is
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...
, and hence :\operatorname\left\ \le \aleph_0 \cdot \alpha = \alpha. Thus there is a such that :\mu(U(b_0, a)) = 0 \quad \forall a \in \alpha implying, since is an Ulam number and using the second definition (with \nu = \mu and conditions ()–() fulfilled), :\mu\left(\bigcup_U(b_0, a)\right) = 0. If b_0 < x < \beta, then f_x(b_0) = a_x \Rightarrow x\in U(b_0, a_x). Thus :\beta = b_0 \cup \ \cup \bigcup_U(b_0, a), By property (), \mu\ = 0, and since \operatorname b_0 \le \alpha, by (), () and (), \mu(b_0) = 0. It follows that \mu(\beta) = 0. The conclusion is that is an Ulam number. There is a similar proof that the supremum of a set of Ulam numbers with \operatorname S an Ulam number is again a Ulam number. Together with the previous result, this implies that a cardinal that is not an Ulam number is weakly inaccessible.


See also

*
Normal measure In set theory, a normal measure is a measure on a measurable cardinal κ such that the equivalence class of the identity function on κ maps to κ itself in the ultrapower construction. Equivalently, if f:κ→κ is such that f(α)<α for most α<κ ...
*
Mitchell order In mathematical set theory, the Mitchell order is a well-founded preorder on the set of normal measures on a measurable cardinal ''κ''. It is named for William Mitchell. We say that ''M'' ◅ ''N'' (this is a strict order) if ''M'' is in the ultr ...
* List of large cardinal properties


Notes


Citations


References

*. *. *. *. *. *. *. A copy of parts I and II of this article with corrections is available at th
author's web page
*. *{{Citation , last1=Ulam , first1=Stanislaw , authorlink = Stanislaw Ulam, title=Zur Masstheorie in der allgemeinen Mengenlehre , url=https://eudml.org/doc/212487 , year=1930 , journal=
Fundamenta Mathematicae ''Fundamenta Mathematicae'' is a peer-reviewed scientific journal of mathematics with a special focus on the foundations of mathematics, concentrating on set theory, mathematical logic, topology and its interactions with algebra, and dynamical sys ...
, issn=0016-2736 , volume=16 , pages=140–150, doi=10.4064/fm-16-1-140-150 , doi-access=free . Large cardinals Determinacy