The axiom of constructibility is a possible
axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
for
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 conce ...
in mathematics that asserts that every set is
constructible. The axiom is usually written as ''V'' = ''L'', where ''V'' and ''L'' denote the
von Neumann universe
In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted by ''V'', is the class of hereditary well-founded sets. This collection, which is formalized by Zermelo–Fraenkel set theory (Z ...
and the
constructible universe
In mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted by , is a particular class of sets that can be described entirely in terms of simpler sets. is the union of the constructible hierarchy . It w ...
, respectively. The axiom, first investigated by
Kurt Gödel
Kurt Friedrich Gödel ( , ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel had an imme ...
, is inconsistent with the proposition that
zero sharp
In the mathematical discipline of set theory, 0# (zero sharp, also 0#) is the set of true formulae about indiscernibles and order-indiscernibles in the Gödel constructible universe. It is often encoded as a subset of the integers (using Gödel nu ...
exists and stronger
large cardinal axiom
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 ...
s (see
list of large cardinal properties
This page includes a list of cardinals with large cardinal properties. It is arranged roughly in order of the consistency strength of the axiom asserting the existence of cardinals with the given property. Existence of a cardinal number κ of a g ...
). Generalizations of this axiom are explored in
inner model theory In set theory, inner model theory is the study of certain models of ZFC or some fragment or strengthening thereof. Ordinarily these models are transitive subsets or subclasses of the von Neumann universe ''V'', or sometimes of a generic extensio ...
.
Implications
The axiom of constructibility implies the
axiom of choice
In mathematics, the axiom of choice, or AC, 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 collectio ...
(AC), given
Zermelo–Fraenkel set theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as ...
without the axiom of choice (ZF). It also settles many natural mathematical questions that are independent of Zermelo–Fraenkel set theory with the axiom of choice (ZFC); for example, the axiom of constructibility implies the
generalized 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 ...
, the negation of
Suslin's hypothesis, and the existence of an
analytical (in fact,
)
non-measurable
In mathematics, a non-measurable set is a set which cannot be assigned a meaningful "volume". The mathematical existence of such sets is construed to provide information about the notions of length, area and volume in formal set theory. In Ze ...
set of
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s, all of which are independent of ZFC.
The axiom of constructibility implies the non-existence of those
large cardinals
In the mathematical field of set theory, a large cardinal property is a certain kind of property of Transfinite number, transfinite cardinal numbers. Cardinals with such properties are, as the name suggests, generally very "large" (for example, big ...
with
consistency strength
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 po ...
greater or equal to
0#, which includes some "relatively small" large cardinals. For example, no cardinal can be ω
1-
Erdős
Erdős, Erdos, or Erdoes is a Hungarian surname.
People with the surname include:
* Ágnes Erdős (born 1950), Hungarian politician
* Brad Erdos (born 1990), Canadian football player
* Éva Erdős (born 1964), Hungarian handball player
* Józse ...
in ''L''. While ''L'' does contain the
initial ordinal
In a written or published work, an initial capital, also referred to as a drop capital or simply an initial cap, initial, initcapital, initcap or init or a drop cap or drop, is a letter at the beginning of a word, a chapter, or a paragraph that ...
s of those large cardinals (when they exist in a supermodel of ''L''), and they are still initial ordinals in ''L'', it excludes the auxiliary structures (e.g.
measures
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
* Measu ...
) which endow those cardinals with their large cardinal properties.
Although the axiom of constructibility does resolve many set-theoretic questions, it is not typically accepted as an axiom for set theory in the same way as the ZFC axioms. Among set theorists of a
realist bent, who believe that the axiom of constructibility is either true or false, most believe that it is false. This is in part because it seems unnecessarily "restrictive", as it allows only certain subsets of a given set (for example,
can't exist), with no clear reason to believe that these are all of them. In part it is because the axiom is contradicted by sufficiently strong
large cardinal axiom
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 ...
s. This point of view is especially associated with the
Cabal
A cabal is a group of people who are united in some close design, usually to promote their private views or interests in an ideology, a state, or another community, often by intrigue and usually unbeknownst to those who are outside their group. Th ...
, or the "California school" as
Saharon Shelah
Saharon Shelah ( he, שהרן שלח; born July 3, 1945) is an Israeli mathematician. He is a professor of mathematics at the Hebrew University of Jerusalem and Rutgers University in New Jersey.
Biography
Shelah was born in Jerusalem on July 3, ...
would have it.
In arithmetic
Especially from the 1950s to the 1970s, there have been some investigations into formulating an analogue of the axiom of constructibility for
subsystems of second-order arithmetic. A few results stand out in the study of such analogues:
* Addison's
formula
such that
iff
, i.e.
is a constructible real
* There's a
formula known as the "analytical form of the axiom of constructibility" which has some associations to the set-theoretic axiom V=L.
[W. Marek]
sets, a characterization of β₂-models of full second-order arithmetic and some related facts
(pp.176--177). Accessed 2021 November 3. For example, some cases where
iff
have been given.
Significance
The major significance of the axiom of constructibility is in
Kurt Gödel
Kurt Friedrich Gödel ( , ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel had an imme ...
's proof of the relative
consistency
In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent ...
of the
axiom of choice
In mathematics, the axiom of choice, or AC, 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 collectio ...
and the
generalized 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 ...
to
Von Neumann–Bernays–Gödel set theory
In the foundations of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of Zermelo–Fraenkel–choice set theory (ZFC). NBG introduces the notion of class, which is a collect ...
. (The proof carries over to
Zermelo–Fraenkel set theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as ...
, which has become more prevalent in recent years.)
Namely Gödel proved that
is relatively consistent (i.e. if
can prove a contradiction, then so can
), and that in
:
thereby establishing that AC and GCH are also relatively consistent.
Gödel's proof was complemented in later years by
Paul Cohen
Paul Joseph Cohen (April 2, 1934 – March 23, 2007) was an American mathematician. He is best known for his proofs that the continuum hypothesis and the axiom of choice are independent from Zermelo–Fraenkel set theory, for which he was award ...
's result that both AC and GCH are ''independent'', i.e. that the negations of these axioms (
and
) are also relatively consistent to ZF set theory.
Statements true in ''L''
Here is a list of propositions that hold in the
constructible universe
In mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted by , is a particular class of sets that can be described entirely in terms of simpler sets. is the union of the constructible hierarchy . It w ...
(denoted by ''L''):
* The
generalized 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 ...
and as a consequence
** The
axiom of choice
In mathematics, the axiom of choice, or AC, 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 collectio ...
*
Diamondsuit In mathematics, and particularly in axiomatic set theory, the diamond principle is a combinatorial principle introduced by Ronald Jensen in that holds in the constructible universe () and that implies the continuum hypothesis. Jensen extracted t ...
**
Clubsuit In mathematics, and particularly in axiomatic set theory, ♣''S'' (clubsuit) is a family of combinatorial principles that are a weaker version of the corresponding ◊''S''; it was introduced in 1975 by Adam Ostaszewski.
Definition
For a given ca ...
*
Global square In mathematical set theory, a square principle is a combinatorial principle asserting the existence of a cohering sequence of
short closed unbounded (club) sets so that no one (long) club set coheres with them all. As such they may be viewed as a ...
* The existence of
morasses
* The negation of the
Suslin hypothesis
* The non-existence of
0# and as a consequence
** The non existence of all
large cardinals
In the mathematical field of set theory, a large cardinal property is a certain kind of property of Transfinite number, transfinite cardinal numbers. Cardinals with such properties are, as the name suggests, generally very "large" (for example, big ...
which imply the existence of a
measurable cardinal
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 , or more generally on any set. For a cardinal , it can be described as a subdivisio ...
* The truth of
Whitehead's conjecture that every
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
''A'' with
Ext
Ext, ext or EXT may refer to:
* Ext functor, used in the mathematical field of homological algebra
* Ext (JavaScript library), a programming library used to build interactive web applications
* Exeter Airport (IATA airport code), in Devon, England
...
1(''A'', Z) = 0 is a
free abelian group
In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subse ...
.
* The existence of a definable
well-order
In mathematics, a well-order (or well-ordering or well-order relation) on a set ''S'' is a total order on ''S'' with the property that every non-empty subset of ''S'' has a least element in this ordering. The set ''S'' together with the well-orde ...
of all sets (the formula for which can be given explicitly). In particular, ''L'' satisfies
V=HOD.
* The existence of a primitive recursive class surjection
, i.e. a class function from Ord whose range contains all sets.
[W. Richter, P. Aczel]
Inductive Definitions and Reflecting Properties of Admissible Ordinals
(1974, p.23). Accessed 30 August 2022.
Accepting the axiom of constructibility (which asserts that every set is
constructible) these propositions also hold in the
von Neumann universe
In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted by ''V'', is the class of hereditary well-founded sets. This collection, which is formalized by Zermelo–Fraenkel set theory (Z ...
, resolving many propositions in set theory and some interesting questions in analysis.
References
*
External links
''How many real numbers are there?'' Keith Devlin,
Mathematical Association of America
The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure a ...
, June 2001
{{Set theory
Axioms of set theory
Constructible universe