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

2022
*
arXiv
*
pre-print

strength

arXiv:2105.08951v3
fatcat:xriu572dqjeqrobtr546e7j23a
*of*∙*the*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. ...##
###
The influence of a small cardinal on the product of a Lindelöf space and the irrationals

1990
*
Proceedings of the American Mathematical Society
*

We give some necessary conditions based

doi:10.1090/s0002-9939-1990-1021211-4
fatcat:6zzbipuut5hwhamziftspmznhu
*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. ...##
###
The Influence of a Small Cardinal on the Product of a Lindelof Space and the Irrationals

1990
*
Proceedings of the American Mathematical Society
*

We give some necessary conditions based

doi:10.2307/2048101
fatcat:vhm5kkdlozgwhi6udepsrtejaq
*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. ...##
###
On fixed-point theorems in synthetic computability

2017
*
Tbilisi Mathematical Journal
*

Cantor's theorem, Gödel's incompleteness theorem, and Tarski's undefinability

doi:10.1515/tmj-2017-0107
fatcat:2kmeluww3bhwxi4nva743jorka
*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. ...##
###
Horrors of topology without ${\rm AC}$: a nonnormal orderable space

1985
*
Proceedings of the American Mathematical Society
*

In

doi:10.1090/s0002-9939-1985-0796455-5
fatcat:qef53j64uvfejhigvdm544b27y
*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. ...##
###
The Fan Theorem, its strong negation, and the determinacy of games
[article]

2018
*
arXiv
*
pre-print

We prove that

arXiv:1311.6988v4
fatcat:teqxnus2drccvc4kp33kvdisyy
*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*. ...##
###
On the logical structure of choice and bar induction principles

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. ...

##
###
More reverse mathematics of the Heine-Borel Theorem

2012
*
Journal of Logic and Analysis
*

Using

doi:10.4115/jla.2012.4.6
fatcat:jn4en3nszjexjpopqgnenbqfy4
*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*...##
###
A theorem on zero cycles on surfaces

2022
*
Selecciones Matemáticas
*

In this paper we prove a result

doi:10.17268/sel.mat.2022.01.13
fatcat:lad7rmmubjbyfmc3bsu2anftzu
*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 ...##
###
First countable, countably compact, noncompact spaces
[chapter]

2007
*
Open Problems in Topology II
*

They can be destroyed without making b = c or even b = d: Section 7

doi:10.1016/b978-044452208-5/50025-9
fatcat:ybrfz4vuq5aithgilr7xkinei4
*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. ...##
###
Hybrid completeness

1998
*
Logic Journal of the IGPL
*

In

doi:10.1093/jigpal/6.4.625
fatcat:eoclnjg7vbalpflu64zay7ihey
*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 ...##
###
Brouwer's Fan Theorem as an axiom and as a contrast to Kleene's Alternative
[article]

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]. ...

##
###
Realizability Models Separating Various Fan Theorems
[chapter]

2013
*
Lecture Notes in Computer Science
*

We develop a realizability model in which

doi:10.1007/978-3-642-39053-1_35
fatcat:zzw5kii3uvapndru4s4vxyu6te
*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. ...##
###
Realizability Models Separating Various Fan Theorems
[article]

2015
*
arXiv
*
pre-print

We develop a realizability model in which

arXiv:1510.02147v1
fatcat:rht54zrb6vcwlpt3cbvb4w7nxq
*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. ...##
###
Reverse Mathematics in Bishop's Constructive Mathematics

2006
*
Philosophia Scientiæ
*

axiom

doi:10.4000/philosophiascientiae.406
fatcat:bp5dqdaysbepbfl5vjxusodtaa
*of**countable**choice*,*the*axiom*of*dependent*choice*, and*the*axiom*of*unique*choice*. ... axiom*of**choice*). ...
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