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Reflection principles for the continuum. (English)
Zhang, Yi (ed.), Logic and algebra. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 302, 59-84 (2002).
Summary: Let $\text{HC}’$ denote the set of sets of hereditary cardinality less than $2^ω$. We consider reflection principles for $\text{HC}’$ in analogy with the Levy reflection principle for HC. Let ${\cal B}$ be a class of complete Boolean algebras. The principle $\text{Max}({\cal B})$ says: If $R(x_1,\dots, x_n)$ is a property which is provably persistent in extensions by elements of ${\cal B}$, then $R(a_1,\dots, a_n)$ holds whenever $a_1,\dots, a_n\in \text{HC}’$ and $R(a_1,\dots, a_n)$ has a positive ${\bbfB}$-value for some ${\bbfB}\in{\cal B}$. Suppose ${\cal C}$ is the class of Cohen algebras. We prove that Con(ZF) implies $\text{Con}(\text{ZFC}+ \text{Max}({\cal C}))$. For a different principle, let ${\cal C}{\cal C}{\cal C}$ be the class of all CCC algebras. We prove that $\text{ZF}+ \text{Levy}$ schema, and $\text{ZFC}+\text{Max}({\cal C}{\cal C}{\cal C})$ are equiconsistent. $\text{Max}({\cal C}{\cal C}{\cal C})$ implies MA, while $\text{Max}({\cal C})$ implies $\neg\text{MA}$. We give applications of these reflection principles to Löwenheim-Skolem theorems of extensions of first-order logic. For example, $\text{Max}({\cal C})$ implies that the Löwenheim number of the extension of first-order logic by the Härtig quantifier is less than $2^ω$.