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.

Version: v1 · 2026-09-28 · History
Domain-specific #: 12382
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Set Theory → Mathematics

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 definition is universal over all members and all their subsets. It does not by itself assert that A is a set, a rank segment, or a model satisfying other set-theoretic axioms.

Structural Signature¶

Sig role-phrases:

class A. Supplies the collection whose closure is tested. Constitutive carrier. If altered: A single set x is not the supertransitive class unless treated as A.
membership transitivity. Requires y in x in A to imply y in A. Constitutive baseline closure. If altered: Subset closure alone is stated only for members and does not replace transitivity by name.
member x. Ranges over every object contained in A. Universal test input. If altered: Checking one favored member cannot establish the class property.
power-set inclusion. Requires every subset y of x to belong to A. Identity-bearing strengthening. If altered: Containing x's elements but not all subsets gives mere transitivity.
rank-style closure. Places the property near cumulative-hierarchy constructions. Diagnostic structural consequence. If altered: Supertransitivity alone does not make A a model of set theory.

What It Is Not¶

Transitive class. Are all subsets, not only elements, included?
Power set. Is one P(x) being confused with a class property?
Rank segment. Are extra hierarchy features assumed?
Model of set theory. Does the class satisfy axioms beyond closure?

Scope of Application¶

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.

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.

Abstract Reasoning¶

Establish that A is a class under the working theory.
Verify membership transitivity.
Choose an arbitrary x in A.
Choose an arbitrary subset y of x and prove y in A.
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.

Examples¶

Canonical¶

A rank segment V_alpha at an appropriate limit stage contains the elements and all subsets of each earlier-rank member, illustrating the required closure when the rank conditions are met.

Mapped back: class A → rank segment; membership transitivity → rank decreases along membership; member x → arbitrary earlier-rank set; power-set inclusion → subsets remain below stage; rank-style closure → explicit hierarchy.

Applied / In Practice¶

A class containing x and every element of x but omitting a subset assembled from several elements is transitive around x yet not supertransitive.

Mapped back: class A → candidate collection; membership transitivity → locally satisfied; member x → included; power-set inclusion → one subset omitted; rank-style closure → fails.

Structural Tensions¶

T1: compact formula vs. strong closure. One short implication demands membership for every subset of every member. Diagnostic: Has arbitrary subset closure really been proved?

T2: class generality vs. set-sized examples. The definition covers proper classes and sets, while familiar rank examples may suggest more structure than required. Diagnostic: Which extra axioms are actually used?

Structural–Framed Character¶

Description turns on class A, membership transitivity, member x, power-set inclusion, rank-style closure. Skeletal core. A collection is downward closed both through internal membership and every subcollection of each member. Domain-bound accent. Classes, membership, transitivity, subsets, power sets, and rank define the property. Transfer remains bounded because Why not prime. Layered closure is portable; this is a specialized set-theoretic condition. The negative boundary is concrete: Any transitive set, transitive class, power set, ordinal, rank segment, subset-closed family, or model of set theory is not automatically supertransitive. Supertransitivity is structural-formal: quantified membership and subset relations determine it exactly. Its character: transitivity strengthened to complete subset closure of every member.

Structural Core vs. Domain Accent¶

Skeletal core. A collection is downward closed both through internal membership and every subcollection of each member.

Domain-bound accent. Classes, membership, transitivity, subsets, power sets, and rank define the property.

Why not prime. Layered closure is portable; this is a specialized set-theoretic condition.

Closure. Specified operations stay inside the class.
Transitivity. Elements of members are included.
No strict parent is asserted.

Neighborhood in Abstraction Space¶

Supertransitive class sits in a moderately populated region (46^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Abstract Algebra & Category Theory (24 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

Transitive class. Tell: Are all subsets, not only elements, included?
Power set. Tell: Is one P(x) being confused with a class property?
Rank segment. Tell: Are extra hierarchy features assumed?
Model of set theory. Tell: Does the class satisfy axioms beyond closure?

References¶

Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Supertransitive_class (revision 1157996081).

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.