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Single Idea 18395
[filed under theme 4. Formal Logic / F. Set Theory ST / 1. Set Theory
]
Full Idea
Lewis pointed out that many-membered classes are nothing more than the mereological wholes of the classes formed by taking the singleton of each member.
Gist of Idea
Sets are mereological sums of the singletons of their members
Source
report of David Lewis (Parts of Classes [1991]) by David M. Armstrong - Truth and Truthmakers 09.4
Book Ref
Armstrong,D.M.: 'Truth and Truthmakers' [CUP 2004], p.118
A Reaction
You can't combine members to make the class, because the whole and the parts are of different type, but here the parts and whole are both sets, so they combine like waterdrops.
The
30 ideas
with the same theme
[general ideas concerning the theory of sets]:
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9987
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An aggregate in which order does not matter I call a 'set'
[Bolzano]
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15946
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Cantor developed sets from a progression into infinity by addition, multiplication and exponentiation
[Cantor, by Lavine]
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9616
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A set is a collection into a whole of distinct objects of our intuition or thought
[Cantor]
|
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15901
|
Trying to represent curves, we study arbitrary functions, leading to the ordinals, which produces set theory
[Cantor, by Lavine]
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13455
|
Frege did not think of himself as working with sets
[Frege, by Hart,WD]
|
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17608
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We take set theory as given, and retain everything valuable, while avoiding contradictions
[Zermelo]
|
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17607
|
Set theory investigates number, order and function, showing logical foundations for mathematics
[Zermelo]
|
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3302
|
Set theory is full of Platonist metaphysics, so Quine aimed to keep it separate from logic
[Quine, by Benardete,JA]
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18396
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The set theory brackets { } assert that the member is a unit
[Armstrong]
|
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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]
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10807
|
Mathematics reduces to set theory, which reduces, with some mereology, to the singleton function
[Lewis]
|
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18395
|
Sets are mereological sums of the singletons of their members
[Lewis, by Armstrong]
|
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15496
|
We can build set theory on singletons: classes are then fusions of subclasses, membership is the singleton
[Lewis]
|
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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]
|
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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]
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17884
|
Mathematical set theory has many plausible stopping points, such as finitism, and predicativism
[Koellner]
|
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17893
|
'Reflection principles' say the whole truth about sets can't be captured
[Koellner]
|
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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]
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