Set Theory & Constructive Foundations

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Abstractions about axiomatic and constructive set theories, finiteness, choice principles, foundational identifications, and alternative universes of sets.

15 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Axiom of infinity — Assert in Zermelo–Fraenkel set theory that an inductive set exists—one containing the empty set and closed under the successor x mapped to x union singleton x—thereby supplying a set from which omega and the natural-number sequence can be isolated.
• Benacerraf's identification problem — The philosophical problem that many equally adequate set-theoretic constructions realize the natural numbers, so arithmetic does not determine which particular sets the numbers intrinsically are.
• Category of sets — The category Set whose objects are sets and morphisms are total functions, with function composition, serving as a foundational categorical universe in which products, coproducts, limits, exponentials, and element-based intuitions are realized.
• Diaconescu's theorem — The constructive-logic result that a sufficiently strong axiom of choice entails the law of excluded middle.
• Finite set — A set equipotent with the natural numbers below some n, equivalently one whose elements can be completely counted and assigned a natural-number cardinality.
• Heyting arithmetic — A first-order theory of natural-number arithmetic using intuitionistic rather than classical logic while retaining arithmetic axioms and induction.
• Inhabited set — A set for which an element can be constructively exhibited or otherwise supplied as a witness, a stronger datum than double-negated nonemptiness in intuitionistic logic.
• Markov's Principle — A constructive-logical schema allowing double-negated existence of a witness for a decidable predicate on the natural numbers to be converted into positive existence, reflecting the legitimacy of unbounded search.
• Non-well-founded set theory — Study axiomatic set universes in which membership may contain infinite descent or cycles because Foundation is omitted or replaced by a declared anti-foundation principle, with graph decoration and bisimulation specifying which circular presentations denote equal sets.
• Scott–Potter set theory — Build set theory from a cumulative hierarchy of levels and histories, treating sets as subcollections of levels while allowing urelements and avoiding a primitive iterative-stage ontology.
• Tarski–Grothendieck set theory — An axiomatic set theory extending ZFC with an axiom that places every set inside a Grothendieck-style universe, thereby implying unbounded inaccessible cardinals.
• Ugly duckling theorem — A formal result showing that, when every logically definable predicate is weighted equally, any two distinct objects share the same number of properties, so similarity requires inductive bias.
• Von Neumann–Bernays–Gödel set theory — A finitely axiomatizable two-sorted set theory with sets and classes that conservatively extends ZFC for statements about sets.
• Whitehead problem — The question whether every abelian group A with Ext-one of A and the integers equal to zero must be free, a statement independent of ZFC.
• Zermelo set theory — The original axiomatic set theory built from extensionality, elementary sets, separation, power set, union, choice, and infinity without the later replacement axiom.