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Single Idea 8203

[filed under theme 6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory ]

Full Idea

The arithmetic of ratios and irrational and imaginary numbers can all be reduced by definition to the theory of classes of positive integers, and this can in turn be reduced to pure set theory.

Gist of Idea

All the arithmetical entities can be reduced to classes of integers, and hence to sets

Source

Willard Quine (Vagaries of Definition [1972], p.53)

Book Ref

Quine,Willard: 'Ways of Paradox and other essays' [Harvard 1976], p.53

A Reaction

This summarises Quine's ontology of mathematics, which tries to eliminate virtually everything, but has to affirm the existence of sets. Can you count sets and their members, if the sets are used to define the numbers?

The 36 ideas with the same theme [Identification of mathematics with set theory]:

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]
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]
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]
18183 Set theory brings mathematics into one arena, where interrelations become clearer [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]