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

[general ideas concerning the theory of sets]

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