more from William D. Hart

### Single Idea 13443

#### [catalogued under 4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / a. Symbols of ST]

Full Idea

∈ relates across layers (Plato is a member of his unit set and the set of people), while ⊆ relates within layers (the singleton of Plato is a subset of the set of people). This distinction only became clear in the 19th century.

Gist of Idea

∈ relates across layers, while ⊆ relates within layers

Source

William D. Hart (The Evolution of Logic [2010], 1)

Book Reference

Hart,W.D.: 'The Evolution of Logic' [CUP 2010], p.5

A Reaction

Getting these two clear may be the most important distinction needed to understand how set theory works.

Related Ideas

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]**

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