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Single Idea 13656
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
]
Full Idea
There are sets of natural numbers definable in set-theory but not in arithmetic.
Gist of Idea
Some sets of natural numbers are definable in set-theory but not in arithmetic
Source
Stewart Shapiro (Foundations without Foundationalism [1991], 5.3.3)
Book Ref
Shapiro,Stewart: 'Foundations without Foundationalism' [OUP 1991], p.123
Related Idea
Idea 13657
First-order arithmetic can't even represent basic number theory [Shapiro]
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]
|
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]
|
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]
|