Axiom of constructibility

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The axiom of constructibility is a possible axiom for set theory in mathematics. It is simply the assertion that every set is constructible, i.e., the assertion that every set belongs to L, the constructible universe.

The axiom of constructiblity settles an impressive number of natural mathematical questions independent of the standard axiomatization of set theory, ZFC. For example, the axiom of constructibility implies the generalized continuum hypothesis (and also the axiom of choice), the negation of Suslin's hypothesis, and the existence of a simple (Δ¹₂) non-measurable set of real numbers.

For better or worse--many set theorists think "worse," the axiom of constructibility implies the non-existence of relatively small large cardinals. Thus, the ω₁-st Erdos cardinal, <math>\eta_{\omega_1}</math>, cannot exist in L.

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" (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

• Statements true in L

External links