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### 6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory

#### [Identification of mathematics with set theory]

36 ideas
 18176 Pure mathematics is pure set theory [Cantor]
 13027 Zermelo believed, and Von Neumann seemed to confirm, that numbers are sets [Zermelo, by Maddy]
 13672 All the axioms for mathematics presuppose set theory [Neumann]
 8463 Maths can be reduced to logic and set theory [Quine]
 8203 All the arithmetical entities can be reduced to classes of integers, and hence to sets [Quine]
 10286 A 'set' is a mathematically well-behaved class [Hodges,W]
 15517 Giving up classes means giving up successful mathematics because of dubious philosophy [Lewis]
 17794 Set theory is not just first-order ZF, because that is inadequate for mathematics [Mayberry]
 17802 We don't translate mathematics into set theory, because it comes embodied in that way [Mayberry]
 17805 Set theory is not just another axiomatised part of mathematics [Mayberry]
 17618 Set-theory tracks the contours of mathematical depth and fruitfulness [Maddy]
 13656 Some sets of natural numbers are definable in set-theory but not in arithmetic [Shapiro]
 18163 Mathematics rests on the logic of proofs, and on the set theoretic axioms [Maddy]
 18183 Set theory brings mathematics into one arena, where interrelations become clearer [Maddy]
 18184 Making set theory foundational to mathematics leads to very fruitful axioms [Maddy]
 18185 Unified set theory gives a final court of appeal for mathematics [Maddy]
 18186 Identifying geometric points with real numbers revealed the power of set theory [Maddy]
 18188 The line of rationals has gaps, but set theory provided an ordered continuum [Maddy]
 10718 A natural number is a property of sets [Maddy, by Oliver]
 10185 Set theory is the standard background for modern mathematics [Burgess]
 17825 Set theory (unlike the Peano postulates) can explain why multiplication is commutative [Maddy]
 17826 Standardly, numbers are said to be sets, which is neat ontology and epistemology [Maddy]
 17828 Numbers are properties of sets, just as lengths are properties of physical objects [Maddy]
 10172 Set-theory gives a unified and an explicit basis for mathematics [Reck/Price]
 10130 Set theory can prove the Peano Postulates [George/Velleman]
 13518 Modern mathematics has unified all of its objects within set theory [Wolf,RS]
 15929 Set theory will found all of mathematics - except for the notion of proof [Lavine]
 14247 Sets might either represent the numbers, or be the numbers, or replace the numbers [Oliver/Smiley]
 8678 Most mathematical theories can be translated into the language of set theory [Friend]
 10881 The concept of 'ordinal number' is set-theoretic, not arithmetical [Horsten]
 10681 In arithmetic singularists need sets as the instantiator of numeric properties [Hossack]
 10685 Set theory is the science of infinity [Hossack]
 15360 ZFC showed that the concept of set is mathematical, not logical, because of its existence claims [Horsten]
 15369 Set theory is substantial over first-order arithmetic, because it enables new proofs [Horsten]
 16312 To reduce PA to ZF, we represent the non-negative integers with von Neumann ordinals [Halbach]
 17940 Most mathematical proofs are using set theory, but without saying so [Colyvan]