Using Continuity Induction

Abstract

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A Local-Global Principle for the Real Continuum

“…Our conclusion about classical approaches is that several classical proofs are quite difficult and subtle for the beginner (e.g., proofs of EVT or Heine using BW). Recent attempts to improve this situation in the literature advocate the use of Cousin [12] or Ind [5,17], but this can also be cumbersome at times (although we agree that this is a matter of opinion).…”

“…[8, pp.1-2] Felix Klein coined the phrase "the arithmetizing of mathematics" [22], a classic of the era of rigor. Until today, the foundation has not been put into question; it is still recognized as satisfactory in all classical textbooks which define R as any ordered field satisfying one of the equivalent completeness axioms listed below 1 : Sup (Least Upper Bound Property) Any set of real numbers has a supremum (and an infimum) 2 ; Cut (Dedekind's Completeness) Any cut defines a (unique) real number; Nest+Arch (Cantor's Property) Any sequence of nested closed intervals has a common point + Archimedean property; Cauchy+Arch (Cauchy's Completeness) Any Cauchy sequence converges + Archimedean property; Mono (Monotone Convergence) Any monotonic sequence has a limit 2 ; BW (Bolzano-Weierstrass) Any infinite set of real numbers (or any sequence) has a limit point 2 ; BL (Borel-Lebesgue) Any cover of a closed interval by open intervals has a finite subcover 3 ; Cousin (Cousin's partition [12]) Any gauge defined on a closed interval admits a fine tagged partition of this interval; Ind (Continuous Induction [5,17,20]).…”

The Instructor’s Guide to Real Induction

“…Theorem 2 is due to D. Hathaway [Ha11] and, independently, to me. But mathematically equivalent ideas have been around in the literature for a long time, some of which are much closer to our formulation than the one of Chao given above.…”

“…But this is just the first -if it actually is the first -of many similar formulations of "continuous induction". A literature search turned up the following papers, each of which introduces some form of "continuous induction", in many cases with no reference to past precedent: [Kh23], [Pe26], [Kh49], [Du57], [Fo57], [MR68], [Sh72], [Be82], [Le82], [Sa84], [Do03], [Ka07], [Ha11].…”

Characterizing Positively Invariant Sets: Inductive and Topological Methods

“…Lemma 7 (Continuity induction Hathaway [2011]). A predicate P (t) holds for all t ∈ [0, T ], where T > 0, if and only if:…”