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

Frank Waaldijk
2005 Foundations of Science  
We discuss 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 .  ... 
doi:10.1007/s10699-004-3065-z fatcat:ybgsxvnixrbj5cswzpyzkb2bcq

Dependent choice as a termination principle [article]

Thomas Powell
2019 arXiv   pre-print
We introduce a new formulation 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 [✄] .  ... 
arXiv:1902.09539v1 fatcat:y653pjknu5ecbeaplffmrvcbuu

Parametrised bar recursion: A unifying framework for realizability interpretations of classical dependent choice [article]

Thomas Powell
2015 arXiv   pre-print
From this proof, 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.  ... 
arXiv:1411.0457v2 fatcat:qheau2jvtjebpofgrk4lojwiom

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]

Joan Rand Moschovakis
2021 arXiv   pre-print
In 2002 Robert Solovay proved that a subsystem BI 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  ... 
arXiv:2101.05878v1 fatcat:tbn3xgu6rnemhbsxivxtr74jcu

Some axioms for constructive analysis

Joan Rand Moschovakis, Garyfallia Vafeiadou
2012 Archive for Mathematical Logic  
This note explores 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."  ... 
doi:10.1007/s00153-012-0273-z fatcat:wkvvt7hifrdqnasn52waffaa34

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

MARTÍN ESCARDÓ, PAULO OLIVA
2015 Journal of Symbolic Logic (JSL)  
We also show that one iterated product is 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).  ... 
doi:10.1017/jsl.2014.82 fatcat:na62li7nwfbkpiw2katou4yhiy

APPLICATIONS OF INDUCTIVE DEFINITIONS AND CHOICE PRINCIPLES TO PROGRAM SYNTHESIS [chapter]

Ulrich Berger, Monika Seisenberger
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  ... 
doi:10.1093/acprof:oso/9780198566519.003.0008 fatcat:n4z453pujrenfgte7abt4tkjhi

Bar Recursion and Products of Selection Functions [article]

Martin Escardo, Paulo Oliva
2014 arXiv   pre-print
We also show that one iterated product is 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.  ... 
arXiv:1407.7046v3 fatcat:e6ruek3vizhyxhqsnawuoda35e

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

William Julian, Fred Richman
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  ... 
doi:10.2140/pjm.1984.111.333 fatcat:jpk27kwjjbb4bd6j2747owcjue

A computational interpretation of open induction

U. Berger
2004 Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 2004.  
We study 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.  ... 
doi:10.1109/lics.2004.1319627 dblp:conf/lics/Berger04 fatcat:aihxty33a5evzetc57jlzwbiz4

Reconsidering recursion in syntactic theory

Marcus Tomalin
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 .  ... 
doi:10.1016/j.lingua.2006.11.001 fatcat:grppvowbmne4hiil6wbxmg56wa

Spector bar recursion over finite partial functions [article]

Paulo Oliva, Thomas Powell
2015 arXiv   pre-print
We conclude by formally establishing that our new 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.  ... 
arXiv:1410.6361v3 fatcat:t7l3pg6vg5hstksp6iv6y2rnzi
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