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Notion of self-reference in mathematics and philosophy
means to be predicative or impredicative. Authors have given different but related definitions. The opposite of impredicativity is predicativity, which essentially
Impredicativity
Basis of generic programming
including polymorphic types. An example of a system supporting full impredicativity is System F, which allows instantiating ∀ α . α → α {\displaystyle
Parametric_polymorphism
School of thought in philosophy of mathematics
totality I, Ά, β, and Γ. As noted below, some commentators insist that impredicativity in commonsense versions is harmless, but as the examples show below
Logicism
Inconsistent pure type systems related to Girard's paradox
important because it is strong enough to express a form of "type-in-type"/impredicativity that leads to Girard's paradox. Girard proved System U inconsistent
System_U
Type theory created by Thierry Coquand
predicative calculus of inductive constructions (which removes some impredicativity).[citation needed] The CoC is a higher-order typed lambda calculus
Calculus_of_constructions
Axiomatic set theories based on the principles of mathematical constructivism
axioms to be set bounded. The latter is motivated by results tied to impredicativity. In constructive mathematical theories, one is commonly unable to prove
Constructive_set_theory
Set-theoretic function
For this reason, ordinal collapsing functions are described as an impredicative manner of naming ordinals. The details of the definition of ordinal
Ordinal_collapsing_function
Paradox in set theory
In the section before this he objects strenuously to the notion of impredicativity as defined by Poincaré (and soon to be taken by Russell, too, in his
Russell's_paradox
Concept in programming language design
first-class data type Generic Haskell, C++ first-class polymorphism impredicative polymorphism first-class message dynamic messages (method calls) Smalltalk
First-class_citizen
Sentence, idea or formula that refers to itself
mathematics and computability theory, self-reference (also known as impredicativity) is the key concept in proving limitations of many systems. Gödel's
Self-reference
Logic formula
this makes its definition impredicative. Kleene asserts that attempts to argue this away can be used to uphold the impredicative definitions in the paradoxes
Propositional_formula
Alternative foundation of mathematics
proposed both intensional and extensional variants of the theory and early impredicative versions, shown to be inconsistent by Girard's paradox, gave way to
Intuitionistic_type_theory
Branch of mathematics that studies sets
specification and replacement, as well as the axiom of power set, introduces impredicativity, a type of circularity, into the definitions of mathematical objects
Set_theory
Well-quasi-ordering of finite trees
thus giving the first example of a predicative result with a provably impredicative proof. This case of the theorem is still provable by Π1 1-CA0, but by
Kruskal's_tree_theorem
Type of propositional logic
functions. The most widely known formalism is the intuitionistic logic with impredicative quantification, System F. Parigot (1997) showed how this calculus can
Second-order propositional logic
Second-order_propositional_logic
Ordinals in mathematics and set theory
theory, Springer 1989 ISBN 0-387-51842-8 (for Veblen hierarchy and some impredicative ordinals). This is probably the most readable book on large countable
Large_countable_ordinal
Mathematical concept for comparing objects
not y ≤ x . {\displaystyle y\leq x.} Towsner, Henry (2013). "Partial impredicativity in reverse mathematics". The Journal of Symbolic Logic. 78 (2): 459–488
Well-quasi-ordering
French mathematician, physicist and engineer (1854–1912)
He strongly opposed Cantorian set theory, objecting to its use of impredicative definitions. Moreover, in La logique de l'infini, Poincaré extended
Henri_Poincaré
Austrian-German philosopher (1859–1938)
"bracketing" or "epoché". Hermann Weyl's interest in intuitionistic logic and impredicativity appears to have resulted from his reading of Husserl. He was introduced
Edmund_Husserl
Type of types in a type system
of Leivant's stratified system F, a predicative variant of Girard's impredicative system F. Since higher-order type operators are uncommon in programming
Kind_(type_theory)
Topics referred to by the same term
a grammatical adjective In mathematics and logic something without impredicativity, without a self-referencing definition Predicative programming, a methodology
Predicative
Mathematical-logical tenet purposed to disallow circular reasoning
rather than the theory of simple types. (See "Ramified Hierarchy and Impredicative Principles".) An analysis of the paradoxes to be avoided shows that
Vicious_circle_principle
Branch of mathematical logic
theory whose proofs make use of strongly impredicative arguments; this equivalence shows that these impredicative arguments cannot be removed. The following
Reverse_mathematics
Axiomatic set theory devised by W.V.O. Quine
restriction, though NFI is not a predicative theory: it admits enough impredicativity to define the set of natural numbers (defined as the intersection of
New_Foundations
Branch of mathematical logic
and set theory. One major challenge has been the ordinal analysis of impredicative theories. The first breakthrough in this direction was Takeuti's proof
Proof_theory
Logical principle
excluded middle was to assert that objections against "the use of 'impredicative definitions'" had "carried more weight" than "the law of excluded middle
Law_of_excluded_middle
Mathematical technique used in proof theory
Analysis: Recent Proof-Theoretical Studies W. Buchholz, Proof Theory of Impredicative Subsystems of Analysis (Studies in Proof Theory, Monographs, Vol 2 (1988)
Ordinal_analysis
Web development programming language
development UrWeb is out of beta "The Ur Programming Language Family". Impredicative.com/ur. Retrieved 3 April 2016. Chlipala, Adam (January 2015). "Ur/Web:
Ur_(programming_language)
System of mathematical set theory
ISBN 978-3-540-22525-6. Chuaqui, Rolando (1981), Axiomatic Set Theory: Impredicative Theories of Classes, North-Holland, ISBN 0-444-86178-5. Cohen, Paul
Von Neumann–Bernays–Gödel set theory
Von_Neumann–Bernays–Gödel_set_theory
Axiom in Russell's ramified theory of types
the familiar analysis, which [see Kleene's example at Impredicativity] contains impredicative definitions. To escape this outcome, Russell postulated
Axiom_of_reducibility
About mathematical functions
Zermelo rails against Poincaré's (and therefore Russell's) notion of impredicative definition. ——; Zermelo, Ernst (1967) [1908a]. "Zermelo (1908a) Investigations
History of the function concept
History_of_the_function_concept
History of maths
different kinds (classical or constructive, bounded, predicative or impredicative, well-founded or non-well-founded, ...), the various constructions of
Timeline of category theory and related mathematics
Timeline_of_category_theory_and_related_mathematics
Compiler for Haskell programming language
FC. Major extensions to the type system include: Arbitrary-rank and impredicative polymorphism. Essentially, a polymorphic function or datatype constructor
Glasgow_Haskell_Compiler
Concept in mathematics
{\displaystyle n} . Wolfram Pohlers, Proof Theory: The First Step Into Impredicativity (2009) (p.247) K. Devlin, An introduction to the fine structure of
Jensen_hierarchy
Framework in lambda calculus
principle of explosion. In general, λ2 adds the possibility to have impredicative types such as ⊥ {\textstyle \bot } , that is terms quantifying over
Lambda_cube
Large countable ordinal
ISSN 0168-0072. Buchholz, Wilfried; Schütte, Kurt (1988). Proof Theory of Impredicative Subsystems of Analysis. Studies in Proof Theory, Monographs. Vol. 2
Takeuti–Feferman–Buchholz ordinal
Takeuti–Feferman–Buchholz_ordinal
substitution of arguments for the apparent variable. By this method impredicativity could be avoided. In 1921, Emil Post would develop a theory of "truth
History_of_type_theory
Apparent contradiction in metamathematics
ZFC are not based on this sort of predicative framework, and allow impredicative definitions. Richard (1905) presented a solution to the paradox from
Richard's_paradox
System of mathematical set theory
theory to include impredicative class comprehension was Quine's ML, that built on New Foundations rather than on ZFC. Impredicative class comprehension
Morse–Kelley_set_theory
Swedish logician, philosopher, and mathematical statistician
Martin-Löf's first draft article on type theory dates back to 1971. This impredicative theory generalized Girard's System F. However, this system turned out
Per_Martin-Löf
possibly be, used to explore concepts of necessity and possibility. impredicative definition A definition that refers to a set that includes the entity
Glossary_of_logic
Large countable ordinal
such that φα(0) = α. This ordinal is sometimes said to be the first impredicative ordinal, though this is controversial, partly because there is no generally
Feferman–Schütte_ordinal
Kind of proof calculus
obtains the type-theoretic analogue of higher-order logic, also known as impredicative polymorphism. Various combinations of dependency and polymorphism have
Natural_deduction
Brazilian philosopher and activist
philosophy of the second Wittgenstein" and her doctoral thesis was called "Impredicativity, Generality and the Development of Wittgenstein's Thought". She began
Camila_Jourdan
mathematics Formal language Principia Mathematica Hilbert's program Impredicative Definable real number Algebraic logic Boolean algebra (logic) Dialectica
List of mathematical logic topics
List_of_mathematical_logic_topics
Foundational controversy in twentieth-century mathematics
discusses §11. The paradoxes, §12. First inferences from the paradoxes [impredicative definitions, Logicism etc.], §13. Intuitionism, §14. Formalism, §15
Brouwer–Hilbert_controversy
Branch of logic using category theory to study mathematical structures
generality and/or is inconvenient. R.A.G. Seely's modeling of various impredicative theories, such as System F, is an example of the usefulness of categorical
Categorical_logic
Polish mathematician and computer scientist
hierarchy of constructible sets, models of second-order arithmetic, the impredicative theory of Kelley–Morse classes. He proved that the so-called Fraïssé
Victor_W._Marek
Hierarchy of formulas in set theory
(1991), pp.439--457. W. Pohlers, Proof Theory: The First Step into Impredicativity (2009) (p.245) Jon Barwise, Admissible Sets and Structures. Perspectives
Lévy_hierarchy
German mathematician
intuitionistischer Logik, Springer 1968 with Wilfried Buchholz: Proof Theory of Impredicative Subsystems of Analysis, Bibliopolis, Naples 1988 with Helmut Schwichtenberg:
Kurt_Schütte
Using positional determinacy of parity games (which is where we need impredicative comprehension), non-existence of player 1 winning strategy gives a player
S2S_(mathematics)
ordinal The Feferman–Schütte ordinal Γ0 is in some sense the smallest impredicative ordinal filter A filter is a non-empty subset of a poset that is downward-directed
Glossary_of_set_theory
IMPREDICATIVITY
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Boy/Male
British, English
From the Hills
Girl/Female
German
Mighty with a spear.
Girl/Female
Indian, Punjabi, Sikh
The Immortal Love of Lord; Immortal Love of God
Surname or Lastname
English
English : variant spelling of Beeman.Americanized spelling of German Biemann, a habitational name for someone from Biene, Bien, or Bienen, all places in the Rhine-Ems area.
Girl/Female
Arabic, Bengali, Gujarati, Hindu, Indian, Jain, Kannada, Malayalam, Marathi, Muslim, Sindhi, Tamil, Telugu
Wish; A Bird; Desire; Hope
Female
Scandinavian
 Scandinavian form of Latin Clara, KLARA means "clear, bright." Compare with other forms of Klara.
Girl/Female
Arabic
Brings Rain
Girl/Female
Native American
Child of importance.
Boy/Male
Indian, Muslim
Big
Surname or Lastname
English
English : variant spelling of Bobbitt.
IMPREDICATIVITY
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IMPREDICATIVITY