formal theories of ν-times iterated inductive definitions". IDν extends PA by ν iterated least fixed points of monotone operators. The formal theory IDω...

early theories by generalizing the collected data inductively, and Newton was able to unify prior theory and measurements into the consequences of his laws...

x is a transitive set of transitive sets. These definitions cannot be used in non-well-founded set theories. In set theories with urelements, one has...

is Thierry Coquand's Calculus of Inductive Constructions. Type theory was created to avoid paradoxes in naive set theory and formal logic, such as Russell's...

greater reward than mutual cooperation. The iterated prisoner's dilemma is fundamental to some theories of human cooperation and trust. Assuming that the...

-comprehension + transfinite induction IDω, the system of ω-times iterated inductive definitions Let Ω α {\displaystyle \Omega _{\alpha }} represent the...

ordinal of several mathematical theories, such as arithmetical transfinite recursion. It is named after Solomon Feferman and Kurt Schütte, the former of whom...

two large categories: intensional definitions (which try to give the sense of a term), and extensional definitions (which try to list the objects that...

Madore, A Zoo of Ordinals (2017). Accessed September 2021. W. Richter, P. Aczel, Inductive Definitions and Reflecting Properties of Admissible Ordinals...

well as sets of natural numbers, and I D < ω {\displaystyle ID_{<\omega }} , the "formal theory of finitely iterated inductive definitions". Since the...

properties of sets arising from the inductive and recursive properties of higher inductive types. Principles such as the axiom of choice and the law of the excluded...

}}}} , the theory of finitely iterated inductive definitions, and of K P ℓ 0 {\displaystyle KP\ell _{0}} , a fragment of Kripke-Platek set theory extended...

background knowledge and hypotheses. The term "inductive" here refers to philosophical (i.e. suggesting a theory to explain observed facts) rather than mathematical...

on ν-times iterated generalized inductive definitions. A u t ( U ( I D ) ) {\displaystyle {\mathsf {Aut(U(ID))}}} is autonomously iterated U ( I D ν )...

sets are said to be inductive. The intersection of all inductive sets is still an inductive set. This intersection is the set of the natural numbers....

the non-constructive equivalent of the constructive definition using iterated exponentiation; but the two definitions are equally non-constructive at...

ordinal of several mathematical theories, such as Kripke–Platek set theory (with the axiom of infinity) and the system CZF of constructive set theory. It...

Wayback Machine, PhD thesis, University of Edinburgh, 1970. Shapiro, Ehud Y. Inductive inference of theories from facts Archived 21 August 2021 at the...

}(\beta )} , some of which are modifications of the Veblen functions to produce countable ordinals even for uncountable arguments, and some of which are "collapsing...

notation coincides with iterated base ε 0 {\displaystyle \varepsilon _{0}} notation (the pieces being themselves written in iterated Cantor normal form):...

Feferman–Schütte ordinal Γ0. Most systems of notation use symbols such as ψ(α), θ(α), ψα(β), some of which are modifications of the Veblen functions to produce...

short descriptions of redirect targets Infinite compositions of analytic functions – Mathematical theory about infinitely iterated function composition...

Inductive programming (IP) is a special area of automatic programming, covering research from artificial intelligence and programming, which addresses...

relationships between these different definitions is one motivation for the study of hyperarithmetical theory. The first definition of the hyperarithmetic sets uses...

the effects of iterating the limiting procedure; this allows any arithmetic predicate to be computed. Schubert wrote, "Intuitively, iterated limiting identification...

ordinal of the theory ID<ω of arbitrary finite iterations of an inductive definition. (Wainer 1989) The functions at finite levels (α < ω) of any fast-growing...

not an assertion of its probability. In 370 BC, Plato's Parmenides may have contained traces of an early example of an implicit inductive proof, however...

as inductively defined data structures such as lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of a...

Feferman–Schütte ordinal Γ0. Most systems of notation use symbols such as ψ(α), θ(α), ψα(β), some of which are modifications of the Veblen functions to produce...

Recursion Theory (1989), p.378. North-Holland, 0-444-87295-7 P. D. Welch, "Weak Systems of Determinacy and Arithmetical Quasi-Inductive Definitions" (2010...