Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 121 / 2

The definable power set isThus it contains exactly the subsets of $A$ definable over the structure $(A,\in )$ using finitely many parameters from $A$.

Fix $a\in L$. Inside the ambient transitive model $M$, the Axiom of power set makes the collection of constructible subsets of $a$ a set. For each such subset $x$, choose the least stage of the constructible hierarchy at which $x$ appears. The Axiom schema of replacement and the supremum of a set of ordinals give an ordinal $\beta$ bounding all these stages; enlarge $\beta$ so that $a\in L_{\beta }$.

Nowis definable over $L_{\beta }$ with parameter $a$. It therefore belongs to the definable power set $\mathcal{D}(L_{\beta })=L_{\beta +1}\subseteq L$. This set contains exactly the subsets of $a$ that belong to the constructible universe, so it witnesses the Axiom of power set in $L$. Therefore $L\models$ Power Set.

The definable power set $\mathcal{D}(A)$ performs one definability step over the single structure $(A,\in )$, whereas the constructible power set ${\mathcal{P}}^L(A)$ contains subsets of $A$ created at arbitrarily late stages of the constructible hierarchy.

For the concrete case $A=\omega$, there are only countably many first-order formulas and finite tuples of natural-number parameters, so $\mathcal{D}(\omega )$ is a countable set. In contrast, the constructible universe satisfies ZFC, and Cantor theorem makes its full power set ${\mathcal{P}}^L(\omega )$ uncountable inside $L$. Consequentlyso the two notions do not agree in general.

Take a transitive model $M\models \mathsf{ZFC}+2^{{\aleph }_0}={\aleph }_2$. In $M$, choose a bijection $e:{\omega }_2\to \mathcal{P}(\omega )$ and encode its graph by a set $A\subseteq {\omega }_2$, using a fixed bijection between ${\omega }_2\times \omega$ and ${\omega }_2$. The relative constructible universe $L(A)$ can decode $e$, and therefore contains every real number of $M$; being an inner model of $M$, it has no additional reals.

Models with the same reals have the same first uncountable ordinal, because their reals code exactly the same countable well-orders. If $L(A)$ satisfied the Continuum hypothesis, its bijection between ${\omega }_1$ and the reals would also belong to $M$, contradicting $M\models 2^{{\aleph }_0}={\aleph }_2$. This is the construction in relative constructible universe can violate the continuum hypothesis, and it gives