Proving open properties by induction (original) (raw)
Related papers
Definitions of finiteness based on order properties
Fundamenta Mathematicae, 2006
A definition of finiteness is a set-theoretical property of a set that, if the Axiom of Choice (AC) is assumed, is equivalent to stating that the set is finite; several such definitions have been studied over the years. In this article we introduce a framework for generating definitions of finiteness in a systematical way: basic definitions are obtained from properties of certain classes of binary relations, and further definitions are obtained from the basic ones by closing them under subsets or under quotients.
Firstly, we show that the NEUMANN ordinals can be defined and understood in set theory without fixing a special theory of sets and classes such as NEUMANN-BERNAYS-GÖDEL, MORSE-KELLEY, or QUINE's ML, and without any axioms, but the Axiom of Extensionality. Especially, no axioms of choice, foundation, infinity, subset, or power are required. Secondly, for general monotonic functors we present KNASTER-TARSKI and well-ordered fixpoint construction. For set-continuous monotonic class operators we present least and greatest fixpoint construction in set theory. For algebraic class operators there is a special of construction of the elements of the least fixpoint with labeled well-founded rooted graphs. As special monotonic class operators we discuss closure operators and their relation to complete lattices, as well as algebraic closure operators and their relation to algebraic lattices. Finally, we show how to construct two monotonic closure operators from a monotonic functor, namely the least-fixpoint and the greatest-fixpoint operator.
A Note on Induction, Abstraction, and Dedekind-Finiteness
2011
The purpose of this note is to present a simplification of the system of arithmetical axioms given in previous work; specifically, it is shown how the induction principle can in fact be obtained from the remaining axioms, without the need of explicit postulation. The argument might be of more general interest, beyond the specifics of the proposed axiomatization, as it highlights the interaction of the notion of Dedekind-finiteness and the induction principle.
Electronic Proceedings in Theoretical Computer Science, 2016
In constructive mathematics, several nonequivalent notions of finiteness exist. In this paper, we continue the study of Noetherian sets in the dependently typed setting of the Agda programming language. We want to say that a set is Noetherian, if, when we are shown elements from it one after another, we will sooner or later have seen some element twice. This idea can be made precise in a number of ways. We explore the properties and connections of some of the possible encodings. In particular, we show that certain implementations imply decidable equality while others do not, and we construct counterexamples in the latter case. Additionally, we explore the relation between Noetherianness and other notions of finiteness.
Equivalence of the Induction Schema and the Least Number Principle for Open Formulas
SUT Journal of Mathematics
Let LA be the usual language for arithmetic. Let ϕ(x) be an LAformula. ϕ(x) may contain free variables distinct from x as parameters. We consider the following two schemata. (I ϕ(x)) ϕ(0) ∧ ∀x(ϕ(x) → ϕ(x + 1)) → ∀xϕ(x), (L ϕ(x)) ∃xϕ(x) → ∃x(∀y < x ¬ϕ(y) ∧ ϕ(x)). They are called the induction schema and the least number principle, respectively. IOpen, LOpen will denote the theory P A − ∪ {I ϕ(x) | ϕ(x) : open}, P A − ∪ {L ϕ(x) | ϕ(x) : open}, respectively. In this paper we prove the equivalence of IOpen and LOpen. Van den Dries [v.d.D] noted that this can be proven model theoretically by using ideas in the proof of Shepherdson's theorem in [S1]. Our proof is syntactical and not model theoretical.
Well Ordering Principles and Bar Induction
In this paper we show that the existence of ω-models of bar induction is equivalent to the principle saying that applying the Howard-Bachmann operation to any well-ordering yields again a well-ordering.
Noetherian Spaces in Verification
Lecture Notes in Computer Science, 2010
Noetherian spaces are a topological concept that generalizes well quasiorderings. We explore applications to infinite-state verification problems, and show how this stimulated the search for infinite procedures à la Karp-Miller.
A note on theories for quasi-inductive definitions
2009
This paper introduces theories for arithmetical quasi-inductive definitions (Burgess, 1986) as it has been done for first-order monotone and nonmonotone inductive ones. After displaying the basic axiomatic framework, we provide some initial result in the proof theoretic bounds line of research (the upper one being given in terms of a theory of sets extending Kripke–Platek set theory).(Received May 04 2009)