 The difficulty here, if not of the whole proof altogether, lies in showing that every real is constructed already on a countable level of the L-hierarchy. To show this Gödel argued as follows: Suppose A is a real number thought of as a set of natural numbers. By a combination of the Levy Reflection principle and the Löwenheim-Skolem Theorem there is a countable submodel &lt; M, ∈ &gt; of &lt; L, ∈ &gt; satisfying a sufficiently large finite part of the ZF axioms + V = L, such that A belongs to M. By a simple procedure &lt; M, ∈ &gt; can be converted into a transitive model &lt; N, ∈ &gt;. This procedure, used by Gödel already in 1937, was explicitly isolated by Mostowski (Mostowski 1949). The resulting model is referred to as the Mostowski Collapse.