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Theaxiom of constructibility is a possibleaxiom forset theory in mathematics that asserts that every set isconstructible. The axiom is usually written asV =L. The axiom, first investigated byKurt Gödel, is inconsistent with the proposition thatzero sharp exists and strongerlarge cardinal axioms (seelist of large cardinal properties). Generalizations of this axiom are explored ininner model theory.^[1]

The axiom of constructibility implies theaxiom of choice (AC), givenZermelo–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 thegeneralized continuum hypothesis, the negation ofSuslin's hypothesis, and the existence of ananalytical (in fact,${\displaystyle {\mathrm{\Delta }}_2^1}$)non-measurable set ofreal numbers, all of which are independent of ZFC.

The axiom of constructibility implies the non-existence of thoselarge cardinals withconsistency strength greater or equal to0^#, which includes some "relatively small" large cardinals. For example, no cardinal can be ω₁-Erdős inL. WhileL does contain theinitial ordinals of those large cardinals (when they exist in a supermodel ofL), and they are still initial ordinals inL, it excludes the auxiliary structures (e.g.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 arealist bent, who believe that the axiom of constructibility is either true or false, most believe that it is false.^[2] This is in part because it seems unnecessarily "restrictive", as it allows only certain subsets of a given set (for example,${\displaystyle 0^{\mathrm{♯}}\subseteq \omega }$ 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 stronglarge cardinal axioms. This point of view is especially associated with theCabal, or the "California school" asSaharon Shelah would have it.

Especially from the 1950s to the 1970s, there have been some investigations into formulating an analogue of the axiom of constructibility forsubsystems of second-order arithmetic. A few results stand out in the study of such analogues:

• John Addison's${\displaystyle {\mathrm{\Sigma }}_2^1}$ formula${\displaystyle \text{Constr}(X)}$ such that${\displaystyle \mathcal{P}(\omega )\models \text{Constr}(X)}$ iff${\displaystyle X\in \mathcal{P}(\omega )\cap L}$, i.e.${\displaystyle X}$ is a constructible real.^[3][4]

• There is a${\displaystyle {\mathrm{\Pi }}_3^1}$ formula known as the "analytical form of the axiom of constructibility" that has some associations to the set-theoretic axiom V=L.^[5] For example, some cases where${\displaystyle M\models \text{V=L}}$ iff${\displaystyle M\cap \mathcal{P}(\omega )\models \text{Analytical}\text{form}\text{of}\text{V=L}}$ have been given.^[5]

The major significance of the axiom of constructibility is inKurt Gödel's 1938 proof of the relativeconsistency of theaxiom of choice and thegeneralized continuum hypothesis toVon Neumann–Bernays–Gödel set theory. (The proof carries over toZermelo–Fraenkel set theory, which has become more prevalent in recent years.)

Namely Gödel proved that${\displaystyle V=L}$ is relatively consistent (i.e. if${\displaystyle ZFC+(V=L)}$ can prove a contradiction, then so can${\displaystyle ZF}$), and that in${\displaystyle ZF}$

${\displaystyle V=L⟹AC\wedge GCH,}$

thereby establishing that AC and GCH are also relatively consistent.

Gödel's proof was complemented in 1962 byPaul Cohen's result that both AC and GCH areindependent, i.e. that the negations of these axioms (${\displaystyle \mathrm{\neg }AC}$ and${\displaystyle \mathrm{\neg }GCH}$) are also relatively consistent to ZF set theory.

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Here is a list of propositions that hold in theconstructible universe (denoted byL):

• Thegeneralized continuum hypothesis and as a consequence
  • Theaxiom of choice
• Diamondsuit
  • Clubsuit
• Global square
• The existence ofmorasses
• The negation of theSuslin hypothesis
• The non-existence of0^# and as a consequence
  • The non existence of alllarge cardinals that imply the existence of ameasurable cardinal
• The existence of a${\displaystyle {\mathrm{\Delta }}_2^1}$ set of reals (in theanalytical hierarchy) that is notmeasurable.
• The truth ofWhitehead's conjecture that everyabelian groupA withExt¹(A,Z) = 0 is afree abelian group.
• The existence of a definablewell-order of all sets (the formula for which can be given explicitly). In particular,L satisfiesV=HOD.
• The existence of a primitive recursive class surjection${\displaystyle F:\text{Ord}\to \text{V}}$, i.e. a class function from Ord whose range contains all sets.^[6]

Accepting the axiom of constructibility (which asserts that every set isconstructible) these propositions also hold in thevon Neumann universe, resolving many propositions in set theory and some interesting questions inanalysis.

• How many real numbers are there?, Keith Devlin,Mathematical Association of America, June 2001

• Axioms of set theory
• Constructible universe

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