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Single Idea 13651

[filed under theme 4. Formal Logic / F. Set Theory ST / 3. Types of Set / a. Types of set ]

Full Idea

If, for every b∈d, a∈b entails that a∈d, the d is said to be 'transitive'. In other words, d is transitive if it contains every member of each of its members.

Gist of Idea

A set is 'transitive' if contains every member of each of its members

Source

Stewart Shapiro (Foundations without Foundationalism [1991], 4.2)

Book Ref

Shapiro,Stewart: 'Foundations without Foundationalism' [OUP 1991], p.85

A Reaction

The alternative would be that the members of the set are subsets, but the members of those subsets are not themselves members of the higher-level set.

The 4 ideas with the same theme [general ways of categorising types of set]:

18958 In type theory, 'x ∈ y' is well defined only if x and y are of the appropriate type [Putnam]
18107 A 'proper class' cannot be a member of anything [Bostock]
13651 A set is 'transitive' if contains every member of each of its members [Shapiro]
15921 Collections of things can't be too big, but collections by a rule seem unlimited in size [Lavine]