Axiom of constructibility
Encyclopedia
The axiom of constructibility is a possible axiom
for set theory
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
and the constructible universe
, respectively.
. It also settles many natural mathematical questions independent of Zermelo–Fraenkel set theory with the axiom of choice (ZFC). For example, the axiom of constructibility implies the generalized continuum hypothesis, the negation of Suslin's hypothesis, and the existence of an analytical
(in fact,
) non-measurable set of real numbers, all of which are independent of ZFC.
The axiom of constructibility implies the non-existence of those large cardinals with consistency strength greater or equal to 0#
, which includes some "relatively small" large cardinals. Thus, no cardinal can be ω1-Erdős
in L. While L does contain the initial ordinals 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
) 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, 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 axioms. This point of view is especially associated with the Cabal
, or the "California school" as Saharon Shelah
would have it.
Axiom
In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...
for set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
in mathematics that asserts that every set is constructible
Constructible universe
In mathematics, the constructible universe , denoted L, is a particular class of sets which can be described entirely in terms of simpler sets. It was introduced by Kurt Gödel in his 1938 paper "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis"...
. The axiom is usually written as V = L, where V and L denote the von Neumann universe
Von Neumann universe
In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted V, is the class of hereditary well-founded sets...
and the constructible universe
Constructible universe
In mathematics, the constructible universe , denoted L, is a particular class of sets which can be described entirely in terms of simpler sets. It was introduced by Kurt Gödel in his 1938 paper "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis"...
, respectively.
Implications
The axiom of constructibility implies the axiom of choice over Zermelo–Fraenkel set theoryZermelo–Fraenkel set theory
In mathematics, Zermelo–Fraenkel set theory with the axiom of choice, named after mathematicians Ernst Zermelo and Abraham Fraenkel and commonly abbreviated ZFC, is one of several axiomatic systems that were proposed in the early twentieth century to formulate a theory of sets without the paradoxes...
. It also settles many natural mathematical questions independent of Zermelo–Fraenkel set theory with the axiom of choice (ZFC). For example, the axiom of constructibility implies the generalized continuum hypothesis, the negation of Suslin's hypothesis, and the existence of an analytical
Analytical hierarchy
In mathematical logic and descriptive set theory, the analytical hierarchy is a higher type analogue of the arithmetical hierarchy. It thus continues the classification of sets by the formulas that define them.-The analytical hierarchy of formulas:...
(in fact,
) non-measurable set of real numbers, all of which are independent of ZFC.The axiom of constructibility implies the non-existence of those large cardinals with consistency strength greater or equal to 0#
Zero sharp
In the mathematical discipline of set theory, 0# is the set of true formulas about indiscernibles in the Gödel constructible universe. It is often encoded as a subset of the integers , or as a subset of the hereditarily finite sets, or as a real number...
, which includes some "relatively small" large cardinals. Thus, no cardinal can be ω1-Erdős
Erdos cardinal
In mathematics, an Erdős cardinal, also called a partition cardinal is a certain kind of large cardinal number introduced by .The Erdős cardinal κ is defined to be the least cardinal such that for every function...
in L. While L does contain the initial ordinals 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
Measurable cardinal
- Measurable :Formally, a measurable cardinal is an uncountable cardinal number κ such that there exists a κ-additive, non-trivial, 0-1-valued measure on the power set of κ...
) 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, 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 axioms. This point of view is especially associated with the Cabal
Cabal (set theory)
The Cabal was, or perhaps is, a grouping of set theorists in Southern California, particularly at UCLA and Caltech, but also at UC Irvine. Organization and procedures range from informal to nonexistent, so it is difficult to say whether it still exists or exactly who has been a member, but it has...
, or the "California school" as Saharon Shelah
Saharon Shelah
Saharon Shelah is an Israeli mathematician. He is a professor of mathematics at the Hebrew University of Jerusalem and Rutgers University in New Jersey.-Biography:...
would have it.
External links
- How many real numbers are there?, Keith Devlin, Mathematical Association of AmericaMathematical Association of AmericaThe Mathematical Association of America 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 and applied mathematicians; computer scientists;...
, June 2001