Axiom of constructibility

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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, first investigated by Kurt Gödel, is inconsistent with the proposition that zero sharp exists and stronger large cardinal axioms (see List of large cardinal properties). Generalizations of this axiom are explored in inner model theory.

Implications

The axiom of constructibility implies the axiom of choice over Zermelo–Fraenkel set theory. 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, the negation of Suslin's hypothesis, and the existence of an analytical (in fact, ${\displaystyle \Delta _{2}^{1}}$) 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.

Significance

The major significance of the axiom of constructibility is in Kurt Gödel's 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 ${\displaystyle V=L}$ is relatively consistent, (i.e. set theory would be inconsistent if it could prove ${\displaystyle V\neq L}$,) and that

${\displaystyle V=L\implies AC\land GCH,}$

thereby establishing that AC and GCH are also relatively consistent.

Gödel's proof was complemented in later years by Paul Cohen's result that both AC and GCH are independent, i.e. that the negations of these axioms (${\displaystyle \lnot AC}$ and ${\displaystyle \lnot GCH}$) are also relatively consistent to ZF set theory.