Editing Axiom of constructibility

{{short description|Possible axiom for set theory in mathematics}}
{{no footnotes|date=May 2017}}
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'''''. The axiom, first investigated by [[Kurt Gödel]], is inconsistent with the proposition that [[zero sharp]] exists and stronger [[large cardinal axiom]]s (see [[list of large cardinal properties]]). Generalizations of this axiom are explored in [[inner model theory]].<ref>{{Cite web |last=Hamkins |first=Joel David |date=February 27, 2015 |title=Embeddings of the universe into the constructible universe, current state of knowledge, CUNY Set Theory Seminar, March 2015 |url=https://jdh.hamkins.org/tag/constructible-universe/ |url-status=live |archive-url=https://web.archive.org/web/20240423205201/https://jdh.hamkins.org/tag/constructible-universe/ |archive-date=April 23, 2024 |access-date=September 22, 2024 |website=[[Joel David Hamkins|jdh.hamkins.org]] |language=en-US}}</ref>

== Implications ==
The axiom of constructibility implies the [[axiom of choice]] (AC), given [[Zermelo–Fraenkel set theory]] 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 [[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 number]]s, 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<sup>#</sup>]], which includes some "relatively small" large cardinals. For example, no cardinal can be ω<sub>1</sub>-[[Erdős cardinal|Erdős]] in ''L''.  While ''L'' does contain the [[initial ordinal]]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. [[measurable cardinal|measures]]) that 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|realist]] bent, who believe that the axiom of constructibility is either true or false, most believe that it is false.<ref>"Before Silver, many mathematicians believed that <math>V\neq L</math>, but after Silver they knew why." - from {{cite|author=P. Maddy|journal=The Journal of Symbolic Logic|title=Believing the Axioms. I|url=https://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms1.pdf|volume=53|year=1988}}, p. 506</ref> This is in part because it seems unnecessarily "restrictive", as it allows only certain subsets of a given set (for example, <math>0^\sharp\subseteq \omega</math> 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]]s. This point of view is especially associated with the [[Cabal (set theory)|Cabal]], or the "California school" as [[Saharon Shelah]] 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 [[Second-order_arithmetic#Subsystems|subsystems of second-order arithmetic]]. A few results stand out in the study of such analogues:

* John Addison's <math>\Sigma_2^1</math> formula <math>\textrm{Constr}(X)</math> such that <math>\mathcal P(\omega)\vDash\textrm{Constr}(X)</math> iff <math>X\in\mathcal P(\omega)\cap L</math>, i.e. <math>X</math> is a constructible real.<ref>[[Victor W. Marek|W. Marek]], Observations Concerning Elementary Extensions of ω-models. II (1973, p.227). Accessed 2021 November 3.</ref><ref>W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm98/fm9818.pdf ω-models of second-order arithmetic and admissible sets] (1975, p.105). Accessed 2021 November 3.</ref>

* There is a <math>\Pi_3^1</math> formula known as the "analytical form of the axiom of constructibility" that has some associations to the set-theoretic axiom V=L.<ref name="beta2models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of β₂-models of full second-order arithmetic and some related facts] (pp.176--177). Accessed 2021 November 3.</ref> For example, some cases where <math>M\vDash\textrm{V=L}</math> iff <math>M\cap\mathcal P(\omega)\vDash\textrm{Analytical}\;\textrm{form}\;\textrm{of}\;\textrm{V=L}</math> have been given.<ref name="beta2models" />

== Significance ==
The major significance of the axiom of constructibility is in [[Kurt Gödel]]'s 1938 proof of the relative [[consistency]] of the [[axiom of choice]] and the [[generalized continuum hypothesis]] to [[Von Neumann–Bernays–Gödel set theory]].  (The proof carries over to [[Zermelo–Fraenkel set theory]], which has become more prevalent in recent years.)

Namely Gödel proved that <math>V=L</math> is relatively consistent (i.e. if <math>ZFC + (V=L)</math> can prove a contradiction, then so can <math>ZF</math>), and that in <math>ZF</math>

:<math>V=L\implies AC\land GCH,</math>

thereby establishing that AC and GCH are also relatively consistent.

Gödel's proof was complemented in 1962 by [[Paul Cohen]]'s result that both AC and GCH are ''independent'', i.e. that the negations of these axioms (<math>\lnot AC</math> and <math>\lnot GCH</math>) are also relatively consistent to ZF set theory.

== Statements true in ''L'' ==
{{Unreferenced section|date=November 2017}}
Here is a list of propositions that hold in the [[constructible universe]] (denoted by ''L''):

* The [[Continuum hypothesis#The generalized continuum hypothesis|generalized continuum hypothesis]] and as a consequence
** The [[axiom of choice]]
* [[Diamondsuit]]
** [[Clubsuit]]
* [[Global square]]
* The existence of [[Morass (set theory)|morasses]]
* The negation of the [[Suslin hypothesis]]
* The non-existence of [[zero sharp|0<sup>#</sup>]] and as a consequence
** The non existence of all [[large cardinals]] that imply the existence of a [[measurable cardinal]]
* The existence of a <math>\Delta_2^1</math> set of reals (in the [[analytical hierarchy]]) that is not [[Lebesgue measure|measurable]].
* The truth of [[Whitehead problem|Whitehead's conjecture]] that every [[abelian group]] ''A'' with [[Ext functor|Ext]]<sup>1</sup>(''A'', '''Z''') = 0 is a [[free abelian group]].
* The existence of a definable [[well-order]] of all sets (the formula for which can be given explicitly).  In particular, ''L'' satisfies [[ordinal definable set|V=HOD]].
* The existence of a primitive recursive class surjection <math>F:\textrm{Ord}\to\textrm{V}</math>, i.e. a class function from Ord whose range contains all sets. <ref>W. Richter, [[Peter Aczel|P. Aczel]], [https://www.duo.uio.no/bitstream/handle/10852/44063/1973-13.pdf Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1974, p.23). Accessed 30 August 2022.</ref>

Accepting the axiom of constructibility (which asserts that every set is [[constructible universe|constructible]]) these propositions also hold in the [[von Neumann universe]], resolving many propositions in set theory and some interesting questions in [[mathematical analysis|analysis]].


== References ==
{{Reflist}}
{{Refbegin}}
* {{cite book|last=Devlin|first=Keith|author-link=Keith Devlin|year=1984|title=Constructibility|publisher=[[Springer-Verlag]]|isbn=3-540-13258-9}}
{{Refend}}

== External links ==
* [https://web.archive.org/web/20140903070752/https://www.maa.org/external_archive/devlin/devlin_6_01.html ''How many real numbers are there?''], Keith Devlin, [[Mathematical Association of America]], June 2001

{{Mathematical logic}}
{{Set theory}}

[[Category:Axioms of set theory]]
[[Category:Constructible universe]]