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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.
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] |