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Single Idea 15945
[filed under theme 4. Formal Logic / F. Set Theory ST / 1. Set Theory
]
Full Idea
Second-order set theory is just like first-order set-theory, except that we use the version of Replacement with a universal second-order quantifier over functions from set to sets.
Gist of Idea
Second-order set theory just adds a version of Replacement that quantifies over functions
Source
Shaughan Lavine (Understanding the Infinite [1994], VII.4)
Book Ref
Lavine,Shaughan: 'Understanding the Infinite' [Harvard 1994], p.226
Related Idea
Idea 13034
Replacement: ∀x∈A ∃!y φ(x,y) → ∃Y ∀X∈A ∃y∈Y φ(x,y) [Kunen]
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]
|