Lévy hierarchy in weak set theories

Journal of Philosophical Logic 37 (2):121 - 140 (2008)
Abstract
We investigate the interactions of formula complexity in weak set theories with the axioms available there. In particular, we show that swapping bounded and unbounded quantification preserves formula complexity in presence of the axiom of foundation weakened to an arbitrary set base, while it does not if the axiom of foundation is further weakened to a proper class base. More attention is being paid to the necessary axioms employed in the positive results, than to the combinatorial strength of the positive results themselves.
Keywords axiom of foundation  quantifier complexity  weak set theory
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