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On the logical structure of choice and bar induction principles [article]

Nuria Brede, Hugo Herbelin
2022 arXiv   pre-print
strength ofthe general axiom of choice expressed as ∀ a∃β R(a, b) ⇒∃α∀ a R(α,(a α (a))) when T is a filter that derives point-wise from a relation R on A x B without introducing further constraints,∙  ...  We develop an approach to choice principles and their contrapositive bar-induction principles as extensionality schemes connecting an "intensional" or "effective" view of respectively ill-and well-foundedness  ...  The second author thanks Valentin Blot and Étienne Miquey for numerous fruitful discussions on the axiom of dependent choice and bar induction.  ... 
arXiv:2105.08951v3 fatcat:xriu572dqjeqrobtr546e7j23a

The influence of a small cardinal on the product of a Lindelöf space and the irrationals

L. Brian Lawrence
1990 Proceedings of the American Mathematical Society  
We give some necessary conditions based on the minimum cardinality of a <* unbounded family in ma>.  ...  It is unknown whether there is in ZFC a Lindelöf space whose product with the irrationals is nonnormal.  ...  Acknowledgment The author thanks the referees for a number of very helpful suggestions.  ... 
doi:10.1090/s0002-9939-1990-1021211-4 fatcat:6zzbipuut5hwhamziftspmznhu

The Influence of a Small Cardinal on the Product of a Lindelof Space and the Irrationals

L. Brian Lawrence
1990 Proceedings of the American Mathematical Society  
We give some necessary conditions based on the minimum cardinality of a <* unbounded family in ma>.  ...  It is unknown whether there is in ZFC a Lindelöf space whose product with the irrationals is nonnormal.  ...  Acknowledgment The author thanks the referees for a number of very helpful suggestions.  ... 
doi:10.2307/2048101 fatcat:vhm5kkdlozgwhi6udepsrtejaq

On fixed-point theorems in synthetic computability

Andrej Bauer
2017 Tbilisi Mathematical Journal  
Cantor's theorem, Gödel's incompleteness theorem, and Tarski's undefinability of truth are all instances of the contrapositive form of the theorem.  ...  We study Lawvere's fixed-point theorem in synthetic computability, which is higher-order intuitionistic logic augmented with the Axiom of Countable Choice, Markov's principle, and the Enumeration axiom  ...  The first one is the Axiom of Countable Choice: Axiom 2.1. A total relation whose domain is N contains the graph of a function.  ... 
doi:10.1515/tmj-2017-0107 fatcat:2kmeluww3bhwxi4nva743jorka

Horrors of topology without ${\rm AC}$: a nonnormal orderable space

Eric K. van Douwen
1985 Proceedings of the American Mathematical Society  
In the absence of AC there can be a space which is not normal, yet which is orderable and is the topological sum of countably many compact countable metrizable spaces.  ...  D As pointed out by Jech, this can be proved the same way one proves that a countable collection of doubletons need not have a choice function [J2, pp. 198-201, 208] (y) VA < / g Z[Z¿ < Z" i.e.  ...  Let «f AC denote the following weak form of AC: A countable collection S has a choice function if there is wfAC: ( <z: Z g .S") such that (Z, <z) has order type f for each ZeJ. 2.1. Theorem.  ... 
doi:10.1090/s0002-9939-1985-0796455-5 fatcat:qef53j64uvfejhigvdm544b27y

The Fan Theorem, its strong negation, and the determinacy of games [article]

Wim Veldman
2018 arXiv   pre-print
We prove that the Fan Theorem is equivalent to contrapositions of a number of intuitionistically accepted axioms of countable choice and that Kleene's Alternative is equivalent to strong negations of these  ...  IIn the context of a weak formal theory called Basic Intuitionistic Mathematics BIM, we study Brouwer's Fan Theorem and a strong negation of the Fan Theorem, Kleene's Alternative (to the Fan Theorem).  ...  In Section 6 we formulate some special cases of the Second Axiom Scheme of Countable Choice AC ω,N , among them Π 0 1 -AC ω,C , the Π 0 1 -Axiom of Countable Compact Choice.  ... 
arXiv:1311.6988v4 fatcat:teqxnus2drccvc4kp33kvdisyy

On the logical structure of choice and bar induction principles

Nuria Brede, Hugo Herbelin
2021 2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)  
of the general axiom of choice expressed as ∀a ∃b R(a, b) ⇒ ∃α ∀a R(a, α(a))) when T is a filter that derives point-wise from a relation R on A × B without introducing further constraints, • the strength  ...  of the Boolean Prime Filter Theorem / Ultrafilter Theorem if B is the two-element set B, • the strength of the axiom of dependent choice if A = N, and up to weak classical reasoning GBIABT intuitionistically  ...  The second author thanks Valentin Blot and Étienne Miquey for numerous fruitful discussions on the axiom of dependent choice and bar induction.  ... 
doi:10.1109/lics52264.2021.9470523 fatcat:bfwohvtglfb6jlijnuns653w3u

More reverse mathematics of the Heine-Borel Theorem

Jeffry Hirst, Jessica Miller
2012 Journal of Logic and Analysis  
Using the techniques of reverse mathematics, we characterize subsets X ⊆ [0, 1] in terms of the strength of HB(X), the Heine-Borel Theorem for the subset.  ...  We introduce W(X), formalizing the notion that the Heine-Borel Theorem for X is weak, and S(X), formalizing the notion that the theorem is strong.  ...  The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation. Portions of the  ... 
doi:10.4115/jla.2012.4.6 fatcat:jn4en3nszjexjpopqgnenbqfy4

A theorem on zero cycles on surfaces

Rina Paucar R., Instituto de Matemáticas y Ciencias Afines, Universidad Nacional de Ingeniería, Lima, Perú., Claudia Schoemann, Laboratoire Gaati, Universite de la Polynesie Francaise, Papeete, French Polynesia.
2022 Selecciones Matemáticas  
In this paper we prove a result on 0-cycles on surfaces as an application of the theorem on the kernel of the Gysin homomorphism of Chow groups of 0-cycles of degree zero induced by the embedding of a  ...  curve into a surface, and we study the connection of this result with Bloch's conjecture and constant cycles curves.  ...  We are grateful to Qixiao Ma, Matthias Paulsen and Joe Palacios for many detailed comments and for patiently answered the emails concerning the main result of this paper. ORCID and License  ... 
doi:10.17268/sel.mat.2022.01.13 fatcat:lad7rmmubjbyfmc3bsu2anftzu

First countable, countably compact, noncompact spaces [chapter]

Peter J. Nyikos
2007 Open Problems in Topology II  
They can be destroyed without making b = c or even b = d: Section 7 of [11] gives one simple way, but with some choices of ground model the final model still has good spaces that are not ω-bounded, and  ...  Making the whole space countably compact relies on the existence of numerous tight (ω 1 , c * )-gaps and (c, ω * 1 )-gaps.  ... 
doi:10.1016/b978-044452208-5/50025-9 fatcat:ybrfz4vuq5aithgilr7xkinei4

Hybrid completeness

P Blackburn
1998 Logic Journal of the IGPL  
In the case of L(#), on the other hand, although we can still formulate a theory of labeling, the Barcan analogs are not valid.  ...  In the case of L(8), we can combine these ideas relatively straightforwardly with the aid of analogs of the Barcan axioms coupled with a modal theory of labeling.  ...  So, by the contrapositive of the Q2 axiom, #x is in as required. a Theorem 38 (Completeness) Every consistent set of formulae in a countable language L(#) is satis able in a rooted and countable standard  ... 
doi:10.1093/jigpal/6.4.625 fatcat:eoclnjg7vbalpflu64zay7ihey

Brouwer's Fan Theorem as an axiom and as a contrast to Kleene's Alternative [article]

Wim Veldman
2022 arXiv   pre-print
The task of finding equivalents of Kleene's Alternative is, intuitionistically, a nontrivial extension of finding equivalents of the Fan Theorem, although there is a certain symmetry in the arguments that  ...  The class of the compact closed-and-separable sets and also the class of the closed-and-separable sets that are positively noncompact are characterized in many different ways and a host of equivalents  ...  Notions of finiteness. Let α be given. We consider the set D α := {n | α(n) = 0}, the subset of ω decided by α. We define: D α is finite if and only if ∃n∀m ≥ n[α(m) = 0].  ... 
arXiv:1106.2738v5 fatcat:thvifaiklbgehbqsljxua5dkpu

Realizability Models Separating Various Fan Theorems [chapter]

Robert S. Lubarsky, Michael Rathjen
2013 Lecture Notes in Computer Science  
We develop a realizability model in which the realizers are the reals not just Turing computable in a fixed real but rather the reals in a countable ideal of Turing degrees.  ...  This is then applied to prove several separation results involving variants of the Fan Theorem.  ...  Within the realizability model, the same example shows IZF does not prove FAN. Perhaps a word should be said on the choice of the ground theory.  ... 
doi:10.1007/978-3-642-39053-1_35 fatcat:zzw5kii3uvapndru4s4vxyu6te

Realizability Models Separating Various Fan Theorems [article]

Robert S. Lubarsky, Michael Rathjen
2015 arXiv   pre-print
We develop a realizability model in which the realizers are the reals not just Turing computable in a fixed real but rather the reals in a countable ideal of Turing degrees.  ...  This is then applied to prove several separation results involving variants of the Fan Theorem.  ...  Within the realizability model, the same example shows IZF does not prove FAN. Perhaps a word should be said on the choice of the ground theory.  ... 
arXiv:1510.02147v1 fatcat:rht54zrb6vcwlpt3cbvb4w7nxq

Reverse Mathematics in Bishop's Constructive Mathematics

Hajime Ishihara
2006 Philosophia Scientiæ  
axiom of countable choice, the axiom of dependent choice, and the axiom of unique choice.  ...  axiom of choice).  ... 
doi:10.4000/philosophiascientiae.406 fatcat:bp5dqdaysbepbfl5vjxusodtaa
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