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