more from this thinker     |     more from this text

Single Idea 17607

[filed under theme 4. Formal Logic / F. Set Theory ST / 1. Set Theory ]

Full Idea

Set theory is that branch whose task is to investigate mathematically the fundamental notions 'number', 'order', and 'function', taking them in their pristine, simple form, and to develop thereby the logical foundations of all of arithmetic and analysis.

Gist of Idea

Set theory investigates number, order and function, showing logical foundations for mathematics

Source

Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908], Intro)

Book Ref

'From Frege to Gödel 1879-1931', ed/tr. Heijenoort,Jean van [Harvard 1967], p.200

A Reaction

At this point Zermelo seems to be a logicist. Right from the start set theory was meant to be foundational to mathematics, and not just a study of the logic of collections.

The 30 ideas with the same theme [general ideas concerning the theory of sets]:

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]