The Many Worlds Interpretation of Set Theory

Abstract
Standard presentations of axioms for set theory as truths simpliciter about actual-objects the sets-confront a number of puzzles associated with platonism and foundationalism. In his classic (1930), Zermelo suggested an alternative "many worlds" view. Independently, Putnam (1967) proposed something similar, explicitly incorporating modality. A modal-structural synthesis of these ideas is sketched in which obstacles to their formalization are overcome. Extendability principles are formulated and used to motivate many small large cardinals. The use of second-order logic as a coherent and clear framework for set theory is supported.
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