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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]
10191 Set theory reduces to a mereological theory with singletons as the only atoms [Lewis, by MacBride]
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]
15496 We can build set theory on singletons: classes are then fusions of subclasses, membership is the singleton [Lewis]
15497 We can replace the membership relation with the member-singleton relation (plus mereology) [Lewis]
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]
15501 We have no idea of a third sort of thing, that isn't an individual, a class, or their mixture [Lewis]
15503 We needn't accept this speck of nothingness, this black hole in the fabric of Reality! [Lewis]
15502 There are four main reasons for asserting that there is an empty set [Lewis]
15504 Atomless gunk is an individual whose parts all have further proper parts [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]
15507 Set theory has some unofficial axioms, generalisations about how to understand it [Lewis]
15509 Some say qualities are parts of things - as repeatable universals, or as particulars [Lewis]
15511 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]
15513 Maybe singletons have a structure, of a thing and a lasso? [Lewis]
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]
15516 A property is any class of possibilia [Lewis]
15517 Giving up classes means giving up successful mathematics because of dubious philosophy [Lewis]
15518 I like plural quantification, but am not convinced of its connection with second-order logic [Lewis]
15520 Existence doesn't come in degrees; once asserted, it can't then be qualified [Lewis]
15519 Trout-turkeys exist, despite lacking cohesion, natural joints and united causal power [Lewis]
15521 Given cats, a fusion of cats adds nothing further to reality [Lewis]
15522 The one has different truths from the many; it is one rather than many, one rather than six [Lewis]
14748 The many are many and the one is one, so they can't be identical [Lewis]
15523 Set theory isn't innocent; it generates infinities from a single thing; but mathematics needs it [Lewis]
15524 Zermelo's model of arithmetic is distinctive because it rests on a primitive of set theory [Lewis]
15525 Plural quantification lacks a complete axiom system [Lewis]
10660 A commitment to cat-fusions is not a further commitment; it is them and they are it [Lewis]
6129 Lewis affirms 'composition as identity' - that an object is no more than its parts [Lewis, by Merricks]