Supertransitive class¶
A transitive class A that also contains every subset of each member: whenever x belongs to A, the entire power set P(x) is included in A.
Core Idea¶
A supertransitive class strengthens ordinary transitivity. Besides requiring every element of a member to lie in the class, it requires every subset of a member to lie in the class: if x∈A, then P(x)⊆A. The extra power-set closure is substantial because a subset of x need not itself be an element of x. The extra power-set closure is substantial because a subset of x need not itself be an element of x.
Scope of Application¶
Use supertransitive class only after checking both membership transitivity and full subset closure for every member. Use supertransitive class only after checking both membership transitivity and full subset closure for every member.
- Set-theoretic hierarchies. Describes strong downward closure.
- Inner-model arguments. Checks available subsets.
- Rank analysis. Compares closure with V-alpha.
- Class theory. Allows proper-class carriers.
- Definition comparison. Separates transitive and supertransitive.
Clarity¶
Transitivity follows elements down one membership edge; supertransitivity additionally admits every possible selection of those elements. The closest near miss sets the boundary: A transitive class is closest: it contains each member's elements but can omit subsets assembled from those elements.
Manages Complexity¶
The concise formula hides nested quantifiers over members and subsets. One counterexample subset is enough to refute the property, while examples require global closure proof. The central compact formula–strong closure tradeoff is this: One short implication demands membership for every subset of every member. A second class generality–set-sized examples tension matters because The definition covers proper classes and sets, while familiar rank examples may suggest more structure than required.
Abstract Reasoning¶
Use three linked moves: establish that A is a class under the working theory; verify membership transitivity; choose an arbitrary x in A. As a collapse test, the case exits when one member x has a subset y that is absent from A, or ordinary transitivity fails. A fourth check is to choose an arbitrary subset y of x and prove y in A. A final check is to keep stronger axioms and rank claims separate.
Knowledge Transfer¶
Downward closure plus subset completion transfers to hereditary structures, but membership and power sets delimit the set-theoretic property. The nearest stopping boundary is explicit: A transitive class is closest: it contains each member's elements but can omit subsets assembled from those elements. The inclusion test remains: A class is supertransitive exactly when it is transitive and, for every x in A, every subset of x also belongs to A. The structure no longer applies when the case exits when one member x has a subset y that is absent from A, or ordinary transitivity fails. No canonical parent prime is currently asserted; broader structural comparisons remain related-prime analogies until separately adjudicated in the DAG. Specified operations stay inside the class. Elements of members are included.
Neighborhood in Abstraction Space¶
Supertransitive class sits in a moderately populated region (46th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Abstract Algebra & Category Theory (24 abstractions)
Nearest neighbors
- Elementary Amenable Group — 0.89
- Well-founded set — 0.88
- Filtration (algebra) — 0.87
- Constructional System — 0.86
- Indiscrete space — 0.86
Computed from structural-signature embeddings · 2026-10-08