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The '''axiom of constructibility''' is a possible [[axiom]] for [[set theory]] in mathematics that asserts that every set is [[Constructible universe|constructible]]. The axiom is usually written as '''''V'' = ''L''''', where ''V'' and ''L'' denote the [[von Neumann universe]] and the [[constructible universe]], respectively. The axiom is inconsistent with the proposition that [[zero sharp]] exists and stronger [[large cardinal axiom]]s (see [[List of large cardinal properties]]). | |||
== Implications == | |||
The axiom of constructibility implies the [[axiom of choice]] over [[Zermelo–Fraenkel set theory]]. 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 [[Continuum hypothesis#The_generalized_continuum_hypothesis|generalized continuum hypothesis]], the negation of [[Suslin's hypothesis]], and the existence of an [[analytical hierarchy|analytical]] (in fact, <math>\Delta^1_2</math>) [[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 [[zero sharp|0#]], which includes some "relatively small" large cardinals. Thus, no cardinal can be ω<sub>1</sub>-[[Erdős cardinal|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. [[measurable cardinal|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 [[Philosophy of mathematics#Mathematical realism or Platonism|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 axiom]]s. This point of view is especially associated with the [[Cabal (set theory)|Cabal]], or the "California school" as [[Saharon Shelah]] would have it. | |||
==See also== | |||
*[[Statements true in L]] | |||
== References == | |||
*{{cite book | |||
| last=Devlin | |||
| first=Keith | |||
| authorlink=Keith Devlin | |||
| year = 1984 | |||
| title = Constructibility | |||
| publisher = [[Springer-Verlag]] | |||
| isbn = 3-540-13258-9 | |||
}} | |||
==External links== | |||
*[http://www.maa.org/devlin/devlin_6_01.html ''How many real numbers are there?''], Keith Devlin, [[Mathematical Association of America]], June 2001 | |||
[[Category:Axioms of set theory]] | |||
[[Category:Constructible universe]] | |||
Latest revision as of 07:35, 26 September 2013
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. The axiom is inconsistent with the proposition that zero sharp exists and stronger large cardinal axioms (see List of large cardinal properties).
Implications
The axiom of constructibility implies the axiom of choice over Zermelo–Fraenkel set theory. 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.
See also
References
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External links
- How many real numbers are there?, Keith Devlin, Mathematical Association of America, June 2001