Single Idea 15499

[catalogued under 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 Reference

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]