Gödel’s constructible universe

Gödel's constructible universe, denoted as L, is a concept in set theory that addresses the continuum hypothesis. The continuum hypothesis, proposed by Georg Cantor, concerns the size of infinite sets. It states that there is no set with a cardinality (size) strictly between that of the natural numbers and that of the real numbers.

Kurt Gödel demonstrated in 1939 that the continuum hypothesis is consistent with the standard axioms of set theory (ZF). He achieved this by constructing a model of ZF, known as the "constructible universe" (L), in which the continuum hypothesis holds true. This means that if the standard axioms of set theory are consistent, then adding the continuum hypothesis to them does not create a contradiction. However, Gödel also showed that the negation of the continuum hypothesis cannot be proven within ZF, and later, Paul Cohen proved that the continuum hypothesis itself cannot be proven within ZF. This implies that the continuum hypothesis is independent of the standard axioms of set theory.12