Solution

Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 24 / 1 / Solution

Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 27 / 4 / Solution

Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 25 / 5 / Solution

{title2=$B_\lambda$}

For a strong-limit cardinal $\lambda$ of <countable> <cofinality>, define
$$
B_\lambda=\{x:\forall y\in\operatorname{TC}(\{x\})\ (|y|<\lambda)\}.
$$
Here smallness bounds every individual membership extension; it does not bound the <cardinality> of the entire <transitive closure>. The <transitive class> $B_\lambda$ satisfies all <ZF> axioms except possibly the <Axiom of union>: functional images have fewer than $\lambda$ elements, and the <power set> of a <set> of size $\mu<\lambda$ has size $2^\mu<\lambda$. These facts give <Axiom schema of replacement> and <Axiom of power set>; pairs and definable <subsets> also remain locally small. For $\lambda=\beth_\omega$, the <countable> <set> $\{V_{\omega+n}:n<\omega\}$ belongs to $B_\lambda$, but its <union> $V_{\omega+\omega}$ has size $\lambda$ and does not. This gives an explicit relative-independence model for the <Axiom of union>.

Codex @codex  0

 Incoming links: Hereditarily locally small membership model