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Single Idea 15499
[filed under theme 4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
]
Full Idea
Just as a part of a part is itself a part, so a subclass of a subclass is itself a subclass; whereas a member of a member is not in general a member.
Gist of Idea
A subclass of a subclass is itself a subclass; a member of a member is not in general a member
Source
David Lewis (Parts of Classes [1991], 1.2)
Book Ref
Lewis,David: 'Parts of Classes' [Blackwell 1991], p.5
A Reaction
Lewis is showing the mereological character of sets, but this is a key distinction in basic set theory. When the members of members are themselves members, the set is said to be 'transitive'.
Related Ideas
Idea 13443
∈ relates across layers, while ⊆ relates within layers [Hart,WD]
Idea 13201
∈ says the whole set is in the other; ⊆ says the members of the subset are in the other [Enderton]
Idea 12337
There is 'transivity' iff membership ∈ also means inclusion ⊆ [Badiou]
Idea 15500
Classes divide into subclasses in many ways, but into members in only one way [Lewis]
The
35 ideas
from 'Parts of Classes'
18395
|
Sets are mereological sums of the singletons of their members
[Lewis, by Armstrong]
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10191
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Set theory reduces to a mereological theory with singletons as the only atoms
[Lewis, by MacBride]
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10566
|
Lewis prefers giving up singletons to giving up sums
[Lewis, by Fine,K]
|
14244
|
Lewis only uses fusions to create unities, but fusions notoriously flatten our distinctions
[Oliver/Smiley on Lewis]
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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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15497
|
We can replace the membership relation with the member-singleton relation (plus mereology)
[Lewis]
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15498
|
We can accept the null set, but there is no null class of anything
[Lewis]
|
15500
|
Classes divide into subclasses in many ways, but into members in only one way
[Lewis]
|
15499
|
A subclass of a subclass is itself a subclass; a member of a member is not in general a member
[Lewis]
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15501
|
We have no idea of a third sort of thing, that isn't an individual, a class, or their mixture
[Lewis]
|
15502
|
There are four main reasons for asserting that there is an empty set
[Lewis]
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15503
|
We needn't accept this speck of nothingness, this black hole in the fabric of Reality!
[Lewis]
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15504
|
Atomless gunk is an individual whose parts all have further proper parts
[Lewis]
|
15507
|
Set theory has some unofficial axioms, generalisations about how to understand it
[Lewis]
|
15506
|
If we don't understand the singleton, then we don't understand classes
[Lewis]
|
15508
|
If singletons are where their members are, then so are all sets
[Lewis]
|
15509
|
Some say qualities are parts of things - as repeatable universals, or as particulars
[Lewis]
|
15511
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If singleton membership is external, why is an object a member of one rather than another?
[Lewis]
|
15512
|
In mereology no two things consist of the same atoms
[Lewis]
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15513
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Maybe singletons have a structure, of a thing and a lasso?
[Lewis]
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15514
|
A huge part of Reality is only accepted as existing if you have accepted set theory
[Lewis]
|
15515
|
To be a structuralist, you quantify over relations
[Lewis]
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15516
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A property is any class of possibilia
[Lewis]
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15517
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Giving up classes means giving up successful mathematics because of dubious philosophy
[Lewis]
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15518
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I like plural quantification, but am not convinced of its connection with second-order logic
[Lewis]
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15520
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Existence doesn't come in degrees; once asserted, it can't then be qualified
[Lewis]
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15519
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Trout-turkeys exist, despite lacking cohesion, natural joints and united causal power
[Lewis]
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15521
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Given cats, a fusion of cats adds nothing further to reality
[Lewis]
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15522
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The one has different truths from the many; it is one rather than many, one rather than six
[Lewis]
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14748
|
The many are many and the one is one, so they can't be identical
[Lewis]
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15523
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Set theory isn't innocent; it generates infinities from a single thing; but mathematics needs it
[Lewis]
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15524
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Zermelo's model of arithmetic is distinctive because it rests on a primitive of set theory
[Lewis]
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15525
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Plural quantification lacks a complete axiom system
[Lewis]
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10660
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A commitment to cat-fusions is not a further commitment; it is them and they are it
[Lewis]
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6129
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Lewis affirms 'composition as identity' - that an object is no more than its parts
[Lewis, by Merricks]
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