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Axiomatic set theories based on the principles of mathematical constructivism
Axiomatic constructive set theory is an approach to mathematical constructivism following the program of axiomatic set theory. The same first-order language
Constructive_set_theory
Philosphical view that existence proofs must be constructive
Constructivism also includes the study of constructive set theories such as CZF and the study of topos theory. Constructivism is often identified with
Constructivism (philosophy of mathematics)
Constructivism_(philosophy_of_mathematics)
Branch of mathematics that studies sets
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any
Set_theory
Standard system of axiomatic set theory
set theories: Morse–Kelley set theory Von Neumann–Bernays–Gödel set theory Tarski–Grothendieck set theory Constructive set theory Internal set theory
Zermelo–Fraenkel_set_theory
First article on transfinite set theory
Cantor's first set theory article contains Georg Cantor's first theorems of transfinite set theory, which studies infinite sets and their properties. One
Cantor's first set theory article
Cantor's_first_set_theory_article
Theorem in mathematical logic
assumed. The proof below is therefore given using the means of a constructive set theory. It is evident from the proof how the theorem relies on the axiom
Diaconescu's_theorem
Alternative to the standard Zermelo–Fraenkel set theory
set theory Constructive set theory Zermelo set theory General set theory Mac Lane set theory Non-well-founded set theory List of first-order theories
List of alternative set theories
List_of_alternative_set_theories
Axiom of set theory
an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, one can identify another set containing one
Axiom_of_choice
Concept in axiomatic set theory
axiom of power set appears in most axiomatizations of set theory. It is generally considered uncontroversial, although constructive set theory prefers a weaker
Axiom_of_power_set
British mathematician and logician
Manchester. He is known for his work in non-well-founded set theory, constructive set theory, and Frege structures. Aczel completed his Bachelor of Arts
Peter_Aczel
Finite sets whose elements are all hereditarily finite sets
mathematics and set theory, hereditarily finite sets are defined as finite sets whose elements are all hereditarily finite sets. In other words, the set itself
Hereditarily_finite_set
Logical quantification that ranges over a subset of the universe of discourse
hierarchy. Bounded quantifiers are important in Kripke–Platek set theory and constructive set theory, where only Δ0 separation is included. That is, it includes
Bounded_quantifier
Method of proof in mathematics
proofs were essentially constructive. The first non-constructive constructions appeared with Georg Cantor’s theory of infinite sets, and the formal definition
Constructive_proof
Formalization of quantum field theory
In mathematical physics, constructive quantum field theory is the field devoted to showing that quantum field theory can be defined in terms of precise
Constructive quantum field theory
Constructive_quantum_field_theory
of constructive theories such as Heyting arithmetic and constructive set theories (Rathjen 2005). The disjunction property is satisfied by a theory if
Disjunction and existence properties
Disjunction_and_existence_properties
Constructive set theory (e.g., CZF — Constructive Zermelo–Fraenkel set theory): Builds sets constructively. Realizability Theory: Ties constructive logic
Constructive_logic
Set of elements common to all of some sets
In set theory, the intersection of two sets A {\displaystyle A} and B , {\displaystyle B,} denoted by A ∩ B , {\displaystyle A\cap B,} is the set containing
Intersection_(set_theory)
Proof in set theory
S\leq S} , also in constructive set theory. It is however harder or impossible to order ordinals and also cardinals, constructively. For example, the Schröder–Bernstein
Cantor's_diagonal_argument
Mathematical theory of data types
to set theory as a foundation of mathematics. Examples include Alonzo Church's simple theory of types and Per Martin-Löf's intuitionistic type theory. Many
Type_theory
Large countable ordinal
several mathematical theories, such as Kripke–Platek set theory (with the axiom of infinity) and the system CZF of constructive set theory. It was introduced
Bachmann–Howard_ordinal
System of mathematical set theory
connections between KP, computability theory, and the theory of admissible ordinals. KP can be studied as a constructive set theory by dropping the law of excluded
Kripke–Platek_set_theory
Set of the elements not in a given subset
In set theory, the complement of a set A, often denoted by A c {\displaystyle A^{c}} (or A′), is the set of elements not in A. When all elements in the
Complement_(set_theory)
Collection of sets in mathematics that can be defined based on a property of its members
In set theory and its applications throughout mathematics, a class is a collection of mathematical objects (often sets) that can be unambiguously defined
Class_(set_theory)
Any one of the distinct objects that make up a set in set theory
"Set Theory", Stanford Encyclopedia of Philosophy, Metaphysics Research Lab, Stanford University Suppes, Patrick (1972) [1960], Axiomatic Set Theory,
Element_of_a_set
Axiomatization of arithmetic
intuitionistic analogue of Boolean algebras. BHK interpretation Constructive analysis Constructive set theory Harrop formula Realizability Troelstra 1973:18 Sørenson
Heyting_arithmetic
Set of elements in any of some sets
In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection. It is one of the fundamental operations
Union_(set_theory)
Branch of mathematical logic
techniques from recursion theory as well as proof theory. Functional interpretations are interpretations of non-constructive theories in functional ones. Functional
Proof_theory
Set whose elements all belong to another set
of k {\displaystyle k} -subsets of an n {\displaystyle n} -element set. In set theory, the notation [ A ] k {\displaystyle [A]^{k}} is also common, especially
Subset
Theorem in set theory
In set theory, Kőnig's theorem states that if the axiom of choice holds, I is a set, κ i {\displaystyle \kappa _{i}} and λ i {\displaystyle \lambda _{i}}
Kőnig's_theorem_(set_theory)
Mathematica List of topics in set theory Set-builder notation P. Aczel, The Type Theoretic Interpretation of Constructive Set Theory (1978) Bostock, David (2012)
Glossary_of_set_theory
Alternative foundation of mathematics
Intuitionistic type theory (also known as constructive type theory, or Martin-Löf type theory (MLTT)) is a type theory and an alternative foundation of
Intuitionistic_type_theory
Property of sets used in constructive mathematics
is not valid in constructive or intuitionistic logic, and so this separate terminology is mostly used in the set theory of constructive mathematics. In
Inhabited_set
Study of computable functions and Turing degrees
computability theory overlaps with proof theory and effective descriptive set theory. Basic questions addressed by computability theory include: What
Computability_theory
Informal set theories
Naive set theory is any of several set theories used in the discussion of the foundations of mathematics. Unlike axiomatic set theories, which are defined
Naive_set_theory
Axiom of set theory
theories that have weak comprehension and the capability to encode functions. This is the case, for example, in some weak constructive set theories or
Axiom_of_non-choice
Logical principle
Consequentia mirabilis – Pattern of reasoning in propositional logic Constructive set theory Diaconescu's theorem – Theorem in mathematical logic Dichotomy –
Law_of_excluded_middle
nontrivial partitions of the continuum. In constructive set theory (CZF), it is consistent to assume the universe of all sets is indecomposable—so that any class
Indecomposability (intuitionistic logic)
Indecomposability_(intuitionistic_logic)
Type of infinite structure
underlying set. O-minimal structures originated in model theory and so have a simpler—but equivalent—definition using the language of model theory. Namely
O-minimal_theory
British mathematician
constructive set theory based on natural numbers, functions, and sets, rather than (as in many other foundational theories) basing it purely on sets.
John_Myhill
Collection of mathematical objects
of sets. Set theory studies possible axiom systems and their consequences. Since the first half of the 20th century, ZFC (Zermelo–Fraenkel set theory with
Set_(mathematics)
Branch of mathematical logic
previous work in constructive analysis and proof theory. The use of second-order arithmetic also allows many techniques from recursion theory to be employed;
Reverse_mathematics
special relativity. Constructive set theory an approach to mathematical constructivism following the program of axiomatic set theory, using the usual first-order
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
System of mathematical set theory
set theory (sometimes denoted by Z-), as set out in a seminal paper in 1908 by Ernst Zermelo, is the ancestor of modern Zermelo–Fraenkel set theory (ZF)
Zermelo_set_theory
Approach in philosophy of mathematics and logic
Cantor's set theory. Modern constructive set theory includes the axiom of infinity from ZFC (or a revised version of this axiom) and the set N {\displaystyle
Intuitionism
Mathematical property of sets
of surjections onto the set X {\displaystyle X} being characterized. The language here is common in constructive set theory texts, but the name subcountable
Subcountability
analysis. Constructivism also includes the study of constructive set theories such as Constructive Zermelo–Fraenkel and the study of philosophy. Some notable
Mathematical_object
Elements in exactly one of two sets
of sets Boolean function Complement (set theory) Difference (set theory) Exclusive or Fuzzy set Intersection (set theory) Jaccard index List of set identities
Symmetric_difference
Theories in mathematical logic
KP; Pocket set theory General set theory, GST Constructive set theory, CZF Mac Lane set theory and Elementary topos theory Zermelo set theory; Z Zermelo–Fraenkel
List_of_first-order_theories
Branch of mathematics
upon the concepts of set theory, arithmetic, and binary relations. Orders are special binary relations. Suppose that P is a set and that ≤ is a relation
Order_theory
Size of a set in mathematics
unprovable and undisprovable in standard set theories such as Zermelo–Fraenkel set theory. Alternative set theories and additional axioms give rise to different
Cardinality
System of mathematical set theory
mathematics, Morse–Kelley set theory (MK), Kelley–Morse set theory (KM), Morse–Tarski set theory (MT), Quine–Morse set theory (QM) or the system of Quine
Morse–Kelley_set_theory
Paradox in set theory
Russell's paradox. The term "naive set theory" is used in various ways. In one usage, naive set theory is a formal theory, that is formulated in a first-order
Russell's_paradox
Basic framework of mathematics
foundation of set theory. The intuitionistic school did not attract many adherents, and it was not until Bishop's work in 1967 that constructive mathematics
Foundations_of_mathematics
Formal language and associated computer program
from a constructive point of view, starting with the axioms of intuitionistic logic and continuing with axiom systems of constructive set theory. This
Metamath
Set with exactly one element
0} . Within the framework of Zermelo–Fraenkel set theory, the axiom of regularity guarantees that no set is an element of itself. This implies that a singleton
Singleton_(mathematics)
System of mathematical set theory
Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of Zermelo–Fraenkel–choice set theory (ZFC). NBG introduces
Von Neumann–Bernays–Gödel set theory
Von_Neumann–Bernays–Gödel_set_theory
Theorem in set theory
such, the above proof is not a constructive one. In fact, in a constructive set theory such as intuitionistic set theory I Z F {\displaystyle {\mathsf
Schröder–Bernstein_theorem
Mathematical construction of a set with an equivalence relation
theory of constructive mathematics based on the Curry–Howard correspondence, one often identifies a mathematical proposition with its set of proofs (if
Setoid
Set theory concept
In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted by V, is the class of hereditary
Von_Neumann_universe
Diagram that shows all possible logical relations between a collection of sets
between sets, popularized by John Venn (1834–1923) in the 1880s. The diagrams are used to teach elementary set theory, and to illustrate simple set relationships
Venn_diagram
German logician and mathematician (1871–1953)
mathematics. He is known for his role in developing Zermelo–Fraenkel axiomatic set theory and his proof of the well-ordering theorem. Furthermore, his 1929 work
Ernst_Zermelo
Type theory in logic and mathematics
theoretic aspects of constructive type theory" in 2008. At about the same time, Vladimir Voevodsky was independently investigating type theory in the context
Homotopy_type_theory
Falsifiable explanation of natural phenomena
theories: "Constructive theories" and "principle theories". Constructive theories are constructive models for phenomena: for example, kinetic theory.
Scientific_theory
Area of mathematical logic
the sets that can be defined in a model of a theory, and the relationship of such definable sets to each other. As a separate discipline, model theory goes
Model_theory
Mathematical set that can be enumerated
be sets which are incomparable to N {\displaystyle \mathbb {N} } , the so-called Dedekind finite infinite sets. In 1874, in his first set theory article
Countable_set
System of mathematical set theory
Tarski–Grothendieck set theory (TG, named after mathematicians Alfred Tarski and Alexander Grothendieck) is an axiomatic set theory. It is a non-conservative
Tarski–Grothendieck set theory
Tarski–Grothendieck_set_theory
Theorem in topology
intuitionism, with some modifications. For further details see constructive set theory. Milnor 1965, pp. 1–19 Teschl, Gerald (2019). "10. The Brouwer
Brouwer_fixed-point_theorem
Set of all things that may be the input of a mathematical function
1971). Axiomatic Set Theory, Part 1. American Mathematical Soc. ISBN 978-0-8218-0245-8. Sharma, A. K. (2010). Introduction To Set Theory. Discovery Publishing
Domain_of_a_function
3-volume treatise on mathematics, 1910–1913
that a satisfactory solution is yet obtainable. Dr Leon Chwistek [Theory of Constructive Types] took the heroic course of dispensing with the axiom without
Principia_Mathematica
Various systems of symbolic logic
calculus. BHK interpretation Computability logic Constructive analysis Constructive proof Constructive set theory Curry–Howard correspondence Game semantics
Intuitionistic_logic
Mathematical analysis
{\mathbb {N} }^{\mathbb {N} }} , constructive second-order arithmetic, or strong enough topos-, type- or constructive set theories such as C Z F {\displaystyle
Constructive_analysis
Kind of transfinite induction
set theory, ∈ {\displaystyle \in } -induction, also called epsilon-induction or set-induction, is a principle that can be used to prove that all sets
Epsilon-induction
Mathematical logic concept
In computability theory, a set S of natural numbers is called computably enumerable (c.e.), recursively enumerable (r.e.), semidecidable, partially decidable
Computably_enumerable_set
contradictions within modern axiomatic set theory. Set theory as conceived by Georg Cantor assumes the existence of infinite sets. As this assumption cannot be
Paradoxes_of_set_theory
Use of braces for specifying sets
{Z} ,n=2k\}} — The set of all even integers, expressed in set-builder notation. In mathematics and more specifically in set theory, set-builder notation
Set-builder_notation
Topics referred to by the same term
science, the name appears also in: Myhill congruence Myhill's constructive set theory Myhill graph Myhill isomorphism theorem Myhill–Nerode theorem Myhill's
Myhill
Any collection of sets, or subsets of a set
In set theory and related branches of mathematics, family or collection is used to mean set, indexed set, multiset, tuple, or class. It is usually used
Family_of_sets
Theory that allows sets to be elements of themselves
Non-well-founded set theories (sometimes unhyphenated, as nonwellfounded; or poorly founded) are variants of axiomatic set theory that allow sets to be elements
Non-well-founded_set_theory
Journalistic approach emphasising solutions, context and societal possibilities
roots in positive psychology, media effects, systems theory, and democratic theory. Constructive journalism is related to, but distinct from, solutions
Constructive_journalism
Mathematical set containing no elements
empty set or void set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure
Empty_set
On bipartite matching and vertex cover
In the mathematical area of graph theory, Kőnig's theorem, proved by Dénes Kőnig (1931), describes an equivalence between the maximum matching problem
Kőnig's theorem (graph theory)
Kőnig's_theorem_(graph_theory)
Subfield of mathematics
mathematical logic into four areas: set theory model theory recursion theory, and proof theory and constructive mathematics (considered as parts of a
Mathematical_logic
Mathematical set containing all objects
In set theory, a universal set is a set that contains all of the objects in the theory, including itself. In set theory as usually formulated, it can be
Universal_set
Non-contradiction of a theory
enough fragment of arithmetic—including set theories such as Zermelo–Fraenkel set theory (ZF). These set theories cannot prove their own Gödel sentence—provided
Consistency
Concept in axiomatic set theory
In many popular versions of axiomatic set theory, the axiom schema of specification, also known as the axiom schema of separation (Aussonderungsaxiom)
Axiom_schema_of_specification
Particular class of sets which can be described entirely in terms of simpler sets
in set theory, the constructible universe (or Gödel's constructible universe), denoted by L , {\displaystyle L,} is a particular class of sets that
Constructible_universe
Logical connective OR
disjunction to tagged union types. The membership of an element of a union set in set theory is defined in terms of a logical disjunction: x ∈ A ∪ B ⇔ ( x ∈ A
Logical_disjunction
System of mathematical set theory
General set theory (GST) is George Boolos's (1998) name for a fragment of the axiomatic set theory Z. GST is sufficient for all mathematics not requiring
General_set_theory
discussed below are provably independent of ZFC (the canonical axiomatic set theory of contemporary mathematics, consisting of the Zermelo–Fraenkel axioms
List of statements independent of ZFC
List_of_statements_independent_of_ZFC
Set of sentences in a formal language
In mathematical logic, a theory (also called a formal theory) is a set of sentences in a formal language. In most scenarios a deductive system is first
Theory_(mathematical_logic)
Maximal proper filter
In the mathematical field of set theory, an ultrafilter on a set X {\displaystyle X} is a maximal filter on the set X . {\displaystyle X.} In other words
Ultrafilter_on_a_set
Branch of mathematics
descriptive set theory and in constructive analysis. In particular, standard examples of Polish spaces such as the real line, the Cantor set and the Baire
Effective descriptive set theory
Effective_descriptive_set_theory
Mathematical technique used in proof theory
first theory of inductive definitions. KP, Kripke–Platek set theory with the axiom of infinity. CZF, Aczel's constructive Zermelo–Fraenkel set theory. EON
Ordinal_analysis
Set that is not a finite set
In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. The set of natural numbers (whose existence
Infinite_set
Family of subsets representing "large" sets
topology, including set theory, mathematical logic, model theory (ultraproducts for example), abstract algebra, and others. Filters on a set were later generalized
Filter_on_a_set
Identities and relationships involving sets
mathematics, particularly in the study of set theory, the algebra of sets defines the properties and laws of sets, the set-theoretic operations of union, intersection
Algebra_of_sets
In mathematics, operation on sets
family of pairwise disjoint sets is their union. In category theory, the disjoint union is the coproduct of the category of sets, and thus defined up to a
Disjoint_union
Sets whose elements have degrees of membership
does not belong to the set. By contrast, fuzzy set theory permits the gradual assessment of the membership of elements in a set; this is described with
Fuzzy_set
Concept in set theory
In set theory, the axiom schema of replacement is a schema of axioms in Zermelo–Fraenkel set theory (ZF) that asserts that the image of any set under any
Axiom_schema_of_replacement
Sequence of words formed by specific rules
computational complexity theory, decision problems are typically defined as formal languages, and complexity classes are defined as the sets of the formal languages
Formal_language
CONSTRUCTIVE SET-THEORY
CONSTRUCTIVE SET-THEORY
CONSTRUCTIVE SET-THEORY
CONSTRUCTIVE SET-THEORY
CONSTRUCTIVE SET-THEORY
CONSTRUCTIVE SET-THEORY
CONSTRUCTIVE SET-THEORY
CONSTRUCTIVE SET-THEORY
CONSTRUCTIVE SET-THEORY