## Reasoning about Theoretical EntitiesReductionism is one of those philosophical myths that are either enthusiastically embraced or wholeheartedly rejected. And, like all other philosophical myths, it rarely gets serious consideration. Reasoning About Theoretical Entities strives to give reductionism its day in court, as it were, by explicitly developing several versions of the reductionist project and assessing their merits within the framework of modern symbolic logic. Not since the days of Carnap''s Aufbau has reductionism received such close attention (albeit in a necessarily restricted and regimented setting such as that of modern mathematical logic). As such this book fills a void in the philosophical literature and presents a challenge to every would-be (anti-)reductionist. It should be required reading for every first-year graduate student in philosophy. |

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### Contents

Virtual Objects | 32 |

Cardinal Arithmetic | 42 |

Iterated Virtuality in Cardinal Arithmetic | 53 |

Ordinals | 67 |

89 | |

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### Common terms and phrases

arise assertion axiom of choice bijection binary relation Burali-Forti paradox canonical bijection canonical simulation cardinal arithmetic child-tree clauses congruence relation context datatype defined denote disjoint domain element equipollent equivalence classes equivalence relation example expressions finite formula fraktur gives rise Greek variables hereditarily transitive higher-order identity implementation of ordinals implementation-insensitive induction initial segments injects interpretation irredundant isomorphism lemma logical mathematical multisets multisets of cardinals natural numbers Neumann implementation Neumann ordinal notation number-of-planets(a obvious ordered pairs pairs of cardinals partition permutation model predicate letter prefix proof prove quantifiers Quine's recursion Roman letters Roman variable set of wellorderings set theory setlike sets of ordinals singly virtual singular descriptions structure subscript subset suite of congruence syntactic theorem things tion transitive set tree truth-value type-theory variables ranging virtual entities virtual objects virtualist Von Neumann ordinals wellfounded wellorderings wellorderings of sets