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On the Foundations of Constructive Mathematics – Especially in Relation to the Theory of Continuous Functions

2005
*
Foundations of Science
*

We discuss

doi:10.1007/s10699-004-3065-z
fatcat:ybgsxvnixrbj5cswzpyzkb2bcq
*the*foundations*of*constructive mathematics, including*recursive*mathematics*and*intuitionism, in relation to classical mathematics. ... Finally, in*the*appendix we offer bish an elegant topological definition*of*'locally compact', which unlike*the*current definition is*equivalent*to*the*usual classical*and*/or intuitionistic definition ... At*the*same time we start constructing*the**recursive**bar*R 40 , which can be enumerated as a series*of**open*intervals (I m ) m∈N . ...##
###
Dependent choice as a termination principle
[article]

2019
*
arXiv
*
pre-print

We introduce a new formulation

arXiv:1902.09539v1
fatcat:y653pjknu5ecbeaplffmrvcbuu
*of**the*axiom*of*dependent choice that can be viewed as an abstract termination principle, which generalises*the**recursive*path orderings used to establish termination*of*... We consider several variants*of*our termination principle,*and*relate them to general termination theorems in*the*literature. ... For*the*axiom*of**open*induction, a corresponding recursor called*open**recursion*has been considered by Berger [1]*and*shown to give a direct realizability interpretation to OI [✄] . ...##
###
Parametrised bar recursion: A unifying framework for realizability interpretations of classical dependent choice
[article]

2015
*
arXiv
*
pre-print

From this proof,

arXiv:1411.0457v2
fatcat:qheau2jvtjebpofgrk4lojwiom
*the*soundness*of*most*of**the*existing*bar**recursive*realizability interpretations*of*choice, including those based on*the*Berardi-Bezem-Coquand functional, modified realizability*and**the*... In many cases, these are achieved by extending*the*base interpreting system*of*primitive*recursive*functionals with some form*of**bar**recursion*, which realizes*the*negative translation*of*either countable ... However, any formula*equivalent*to one*of**the*form B(u) :≡ ∃nP ([u](n)) is*open*in both senses. ...##
###
Page 607 of Mathematical Reviews Vol. , Issue 87b
[page]

1987
*
Mathematical Reviews
*

*The*answer is negative; there will be a

*recursive*functional

*of*type 3 that cannot be generated from S1-S9

*and*

*bar*-

*recursion*. ... Secondly, he answers a problem formulated by Hyland: Is a scheme

*of*

*bar*-

*recursion*sufficient, together with S1-S9, to generate

*the*countably

*recursive*functionals? ...

##
###
Solovay's Relative Consistency Proof for FIM and BI
[article]

2021
*
arXiv
*
pre-print

In 2002 Robert Solovay proved that a subsystem BI

arXiv:2101.05878v1
fatcat:tbn3xgu6rnemhbsxivxtr74jcu
*of*classical second order arithmetic, with*bar*induction*and*arithmetical countable choice, can be negatively interpreted in*the*neutral subsystem BSK ... Combining this result with Kleene's formalized*recursive*realizability, he established (in primitive*recursive*arithmetic PRA) that FIM + MP*and*BI have*the*same consistency strength. ... ) Axioms that insure that*the*type 1 variables are closed under Turing jump*and*"*recursive*in"*and*contain all*recursive*functions. (4)*The*key axiom*of**Bar*-Induction: If R is a linear ordering on omega ...##
###
Some axioms for constructive analysis

2012
*
Archive for Mathematical Logic
*

This note explores

doi:10.1007/s00153-012-0273-z
fatcat:wkvvt7hifrdqnasn52waffaa34
*the*common core*of*constructive, intuitionistic,*recursive**and*classical analysis from an axiomatic standpoint. ... In addition to clarifying*the*relation between Kleene's*and*Troelstra's minimal formal theories*of*numbers*and*number-theoretic sequences, we propose some modified choice principles*and*other function ...*The*first author also thanks*the*organizers*and*participants*of**the*2010 Chiemsee conference "Constructive Mathematics: Proofs*and*Computations." ...##
###
Page 11 of Mathematical Reviews Vol. 41, Issue 1
[page]

1971
*
Mathematical Reviews
*

{

*The*author credits*the*idea for*the*use*of**bar*-induction to K. Schiitte [J. ... Finally, an*open*problem that should have a negative answer is posed: Given a fixed numbered field F, does there exist a*recursive*enumeration*of**the*class*of*all constructive extensions*of*F, to which ...##
###
BAR RECURSION AND PRODUCTS OF SELECTION FUNCTIONS

2015
*
Journal of Symbolic Logic (JSL)
*

We also show that one iterated product is

doi:10.1017/jsl.2014.82
fatcat:na62li7nwfbkpiw2katou4yhiy
*equivalent*over system T to Spector's*bar**recursion*, whereas*the*other is T-*equivalent*to modified*bar**recursion*. ... Modified*bar**recursion*itself is shown to arise directly from*the*iteration*of*a different binary product*of*'skewed' selection functions. ...*The*authors would like to thank Ulrich Berger for suggesting some improvements on an earlier version*of**the*paper,*and*Radu Grigore for drawing*the*diagram*of*Figure 1 in LaTeX (tikz). ...##
###
APPLICATIONS OF INDUCTIVE DEFINITIONS AND CHOICE PRINCIPLES TO PROGRAM SYNTHESIS
[chapter]

2005
*
From Sets and Types to Topology and Analysis
*

*The*first method removes any reference to infinite sequences

*and*transforms

*the*theorem into a system

*of*inductive definitions,

*the*other applies a combination

*of*Gödel's negativeand Friedman's A-translation ... We also discuss some proof-theoretic optimizations that were crucial for

*the*formalization

*and*implementation

*of*this work in

*the*interactive proof system Minlog. 1 Higman's lemma is used, for example, ...

*The*statement ∀ws.Bars (Folder ws) →

*Bar*A * ws, which is

*equivalent*to (+), can be proven by main induction on

*the*number

*of*letters that do not occur as an end letter in ws

*and*side induction on

*Bars*...

##
###
Bar Recursion and Products of Selection Functions
[article]

2014
*
arXiv
*
pre-print

We also show that one iterated product is

arXiv:1407.7046v3
fatcat:e6ruek3vizhyxhqsnawuoda35e
*equivalent*over system T to Spector's*bar**recursion*, whereas*the*other is T-*equivalent*to modified*bar**recursion*. ... Modified*bar**recursion*itself is shown to arise directly from*the*iteration*of*a different binary product*of*"skewed" selection functions. ...*The*authors would like to thank Ulrich Berger, Thomas Powell*and*in particular*the*anonymous referee for suggesting numerous improvements*and*spotting inaccuracies in earlier versions*of**the*paper. ...##
###
Page 583 of Mathematical Reviews Vol. , Issue 93b
[page]

1993
*
Mathematical Reviews
*

*The*morphisms

*of*S7 [resp., S7] are

*equivalence*classes

*of*indices

*of*partial

*recursive*functions, indices a

*and*b being

*equivalent*if it is provable in T that for all n [resp., for each n it is provable ... In addition, we demonstrate that ACA}, ACAo+(

*bar*rule)

*and*ACAo+(

*bar*induction)” prove

*the*same I!-sentences.” 93b:03105 03F30 03D15 03F35 Takeuti, Gaisi (1-IL) A second order version

*of*S',

*and*U}. ...

##
###
A uniformly continuous function on [0,1] that is everywhere different from its infimum

1984
*
Pacific Journal of Mathematics
*

*The*existence

*of*such a function is shown to be

*equivalent*to

*the*constructive denial

*of*Kδnig's lemma. ... An example

*of*a uniformly continuous function on [0,1] that is everywhere different from its infimum is constructed in

*the*context

*of*Bishop's constructive mathematics using a consequence

*of*Chruch's thesis ... When he raises

*the*problem

*of*whether every positive, uniformly continuous function on [0,1] has a positive infimum he points out that this is

*equivalent*to every compact subset

*of*

*the*

*open*unit disc U ...

##
###
A computational interpretation of open induction

2004
*
Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 2004.
*

We study

doi:10.1109/lics.2004.1319627
dblp:conf/lics/Berger04
fatcat:aihxty33a5evzetc57jlzwbiz4
*the*proof-theoretic*and*computational properties*of**open*induction, a principle which is classically*equivalent*to Nash-Williams' minimal-bad-sequence argument*and*also to (countable) dependent ... P is*the*computational content*of**the*(negative-andtranslated) classical proof*of*¥ £ ¦ using*the*principle*of**open*induction (*and*a realizer thereof). ... Proposition 3.4*The*principles*of*dependent choice,*open*induction*and*update induction are classically (i.e. provably in q Y n h )*equivalent*. ...##
###
Reconsidering recursion in syntactic theory

2007
*
Lingua
*

*The*use

*of*

*recursive*devices in formal grammars in

*the*1950s is summarised,

*and*

*the*work

*of*

*Bar*-Hillel

*and*Chomsky from this period is analysed in some detail. ... (24/03/05) focused primarily upon

*the*role

*of*

*recursion*

*and*

*the*interfaces in syntactic theory. ...

*The*

*equivalence*

*of*

*the*computable to

*the*l-definable functions (

*and*hence to

*the*general

*recursive*functions) was proved by Turing 1937 . ...

##
###
Spector bar recursion over finite partial functions
[article]

2015
*
arXiv
*
pre-print

We conclude by formally establishing that our new

arXiv:1410.6361v3
fatcat:t7l3pg6vg5hstksp6iv6y2rnzi
*bar*recursor is primitive*recursively**equivalent*to*the*original Spector*bar**recursion*,*and*thus defines*the*same class*of*functionals when added to Gödel's ... We introduce a new, demand-driven variant*of*Spector's*bar**recursion*in*the*spirit*of**the*Berardi-Bezem-Coquand functional. ...*Equivalence**of*BR*and*sBR In this section we prove that*the**recursion*schemata BR*and*sBR are actually primitive*recursively**equivalent*. ...
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