more on this theme     |     more from this thinker

Single Idea 15500

[filed under theme 4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST ]

Full Idea

A class divides exhaustively into subclasses in many different ways; whereas a class divides exhaustively into members in only one way.

Gist of Idea

Classes divide into subclasses in many ways, but into members in only one way

Source

David Lewis (Parts of Classes [1991], 1.2)

Book Ref

Lewis,David: 'Parts of Classes' [Blackwell 1991], p.5

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 15499 A subclass of a subclass is itself a subclass; a member of a member is not in general a member [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]