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### 4. Formal Logic / F. Set Theory ST / 1. Set Theory

#### [general ideas concerning the theory of sets]

30 ideas
 9987 An aggregate in which order does not matter I call a 'set' [Bolzano]
 15946 Cantor developed sets from a progression into infinity by addition, multiplication and exponentiation [Cantor, by Lavine]
 9616 A set is a collection into a whole of distinct objects of our intuition or thought [Cantor]
 15901 Trying to represent curves, we study arbitrary functions, leading to the ordinals, which produces set theory [Cantor, by Lavine]
 13455 Frege did not think of himself as working with sets [Frege, by Hart,WD]
 17608 We take set theory as given, and retain everything valuable, while avoiding contradictions [Zermelo]
 17607 Set theory investigates number, order and function, showing logical foundations for mathematics [Zermelo]
 3302 Set theory is full of Platonist metaphysics, so Quine aimed to keep it separate from logic [Quine, by Benardete,JA]
 18396 The set theory brackets { } assert that the member is a unit [Armstrong]
 10482 The logic of ZF is classical first-order predicate logic with identity [Boolos]
 18114 There is no single agreed structure for set theory [Bostock]
 10807 Mathematics reduces to set theory, which reduces, with some mereology, to the singleton function [Lewis]
 18395 Sets are mereological sums of the singletons of their members [Lewis, by Armstrong]
 15496 We can build set theory on singletons: classes are then fusions of subclasses, membership is the singleton [Lewis]
 3326 Set theory attempts to reduce the 'is' of predication to mathematics [Benardete,JA]
 3327 The set of Greeks is included in the set of men, but isn't a member of it [Benardete,JA]
 9967 'Impure' sets have a concrete member, while 'pure' (abstract) sets do not [Jubien]
 13456 Set theory articulates the concept of order (through relations) [Hart,WD]
 13497 Nowadays ZFC and NBG are the set theories; types are dead, and NF is only useful for the whole universe [Hart,WD]
 8309 A set is a 'number of things', not a 'collection', because nothing actually collects the members [Lowe]
 10888 Sets can be defined by 'enumeration', or by 'abstraction' (based on a property) [Zalabardo]
 8474 Unlike elementary logic, set theory is not complete [Orenstein]
 10702 Set theory's three roles: taming the infinite, subject-matter of mathematics, and modes of reasoning [Potter]
 13451 The two best understood conceptions of set are the Iterative and the Limitation of Size [Rayo/Uzquiano]
 17884 Mathematical set theory has many plausible stopping points, such as finitism, and predicativism [Koellner]
 17893 'Reflection principles' say the whole truth about sets can't be captured [Koellner]
 15945 Second-order set theory just adds a version of Replacement that quantifies over functions [Lavine]
 16309 Every attempt at formal rigour uses some set theory [Halbach]
 15657 To prove the consistency of set theory, we must go beyond set theory [Halbach]
 18830 Most set theorists doubt bivalence for the Continuum Hypothesis, but still use classical logic [Rumfitt]