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In this homework set, you may use all of the axioms of ZFC. If \(X\) is a set, the hereditary cardinality of \(X\) is the cardinality of the transitive closure of \(X\). If \(\theta\) is a cardinal, let \(H_\theta\) denote the class of all sets of hereditary cardinality less than \(\theta\).

Do any 3 problems.

1. Prove that \(H_\theta\) is a set and compute the cardinality of \(H_\theta\). Hint: the Mostowski collapse is helpful - see the construction of the Hartog ordinal.
2. Show that if \(\theta\) is a regular uncountable cardinal, then \((H_\theta,\in)\) satisfies all of the axioms of ZFC except possibly the Powerset Axiom. When does it satisfy the Powerset Axiom?
3. Suppose that \(\theta\) is a regular uncountable cardinal and \(A,B \in H_\theta\). Show that \(|A| \leq |B|\) if and only if \((H_\theta,\in) \models |A| \leq |B|\).
4. Suppose that \(\theta\) is a regular uncountable cardinal and \(M \prec H_\theta\) is countable. Show that \(M \cap \omega_1\) is an ordinal and for every \(X\) in \(M\), \(X \subseteq M\) if and only if \(X\) is countable.
5. Suppose that \(\theta\) is a regular uncountable cardinal and \(M \prec H_\theta\). If \(X \in M\) is uncountable and \(S \in M\) with \(\bigcup S = X\) and \(M \cap X \in S\), then \(S\) is stationary.
6. Suppose that \(\theta\) is a regular uncountable cardinal and \(M \prec H_\theta\). Show that if \(x \in H_\theta\) is definable from parameters in \(M\), then \(x \in M\). Here \(x\) is definable from parameters in \(M\) means that there is a formula \(\phi(u,\bar v)\) and a tuple \(\bar a\) of elements of \(M\) such that \(u=x\) is the unique solution to \(H_\theta \models \phi(u,\bar a)\).