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A computational interpretation of Zorn's lemma [article]

Thomas Powell
2020 arXiv   pre-print
We give a computational interpretation to an abstract instance of Zorn's lemma formulated as a wellfoundedness principle in the language of arithmetic in all finite types.  ...  We show that a realizer for the functional interpretation of open induction over the lexicographic ordering on sequences follows as a simple application of our main results.  ...  What other applications of our abstract computational interpretation of Zorn's lemma are possible?  ... 
arXiv:2001.03540v1 fatcat:bq3i543pdfcp3mirjtgm3eq6vu

Well quasi-orders and the functional interpretation [article]

Thomas Powell
2017 arXiv   pre-print
question of the constructive meaning of Zorn's lemma and the notion of recursion over the non-wellfounded lexicographic ordering on infinite sequences.  ...  The main focus is on the interpretation of Nash-Williams' famous minimal bad sequence construction, and the exploration of a number of much broader problems which are related to this, particularly the  ...  In developing the ideas of this paper I have benefited greatly from numerous illuminating conversations with Ulrich Berger, Paulo Oliva and Monika Seisenberger.  ... 
arXiv:1706.02881v1 fatcat:27jdif64vbafdfses6zsproyq4

Flabby and injective objects in toposes [article]

Ingo Blechschmidt
2018 arXiv   pre-print
In the special case of localic toposes, this notion reduces to the common notion of flabby sheaves, yielding a site-independent characterization of flabby sheaves.  ...  We introduce a general notion of flabby objects in elementary toposes and study their basic properties.  ...  Most of the work for this note was carried out at the University of Augsburg and the Max Planck Institute for Mathematics in the Sciences in Leipzig.  ... 
arXiv:1810.12708v1 fatcat:leeb2dm2rfegnmfzkqjg3uhj4q

Completeness of resolution revisited

Marc Bezem
1990 Theoretical Computer Science  
Moreover, by using Zorn's Lemma in dealing with infinite sets of ground clauses, we obtain cor~pletz zss proofs which are completely independent of the cardinalitj of both the language and the set of clauses  ...  By a novel argument we prove the completeness of (ground) resolution.  ...  Zorn's Lemma Let c be a partial ordering on a set S. A chain in S is a totally ordered subset of S (i.e. satisfying the trichotomy axiom).  ... 
doi:10.1016/0304-3975(90)90139-9 fatcat:eo24tcnsvndbdanrveqdeuylo4

Prioritising Preference Relations [chapter]

Mark Ryan
1993 Theory and Formal Methods 1993  
and the definition of best elements uses a generalisation of lexicographic ordering.  ...  We describe some ideas and results about the following problem: Given a set, a family of \preference relations" on the set, and a \priority" among those preference relations, which elements of the set  ...  Thanks to the referees Reinhold Heckman, Roy Crole and Abbas Edalat for useful comments on a draft of this paper.  ... 
doi:10.1007/978-1-4471-3503-6_21 fatcat:tlbr7c6p55hyhj4moc2727l2b4

Existence and uniqueness of the real closure of an ordered field without Zorn's Lemma

Tomas Sander
1991 Journal of Pure and Applied Algebra  
., Existence and uniqueness of the real closure of an ordered field without Zorn's Lemma, Journal of Pure and Applied Algebra 73 (1991) 165-180.  ...  In this paper the existence of a real closure of an ordered field is given without &he use of the axiom of choice.  ...  Friedrichsdorf and A. Prestel.  ... 
doi:10.1016/0022-4049(91)90110-n fatcat:xtlm4vw2ojffzmvix34aqcrkwu

Page 8474 of Mathematical Reviews Vol. , Issue 2004k [page]

2004 Mathematical Reviews  
From the text: “In classical set theory, Zorn’s lemma is equivalent to the axiom of choice and a host of other principles and theo- rems.  ...  extension has a computable subchain of type 7. | M.  ... 

Closures on CPOs Form Complete Lattices

Francesco Ranzato
1999 Information and Computation  
Since the concrete domain D is generally assumed to be a complete lattice, by Ward's theorem, uco(D) turns out to be the so-called (complete) lattice of abstract interpretations of D (cf.  ...  An example of application in abstract interpretation theory is given. ] is not hard to derive Ward's (1942, Theorem 4.2) basic theorem stating that if C is a complete lattice then uco(C) is a complete  ...  This work has been partly supported by an individual grant 202.12199 from Comitato 12``Scienza e Tecnologie dell'Informazione'' of Italian CNR and by a postdoctoral fellowship from UniversitaÁ di Padova  ... 
doi:10.1006/inco.1999.2801 fatcat:mgrdpthuu5cjjbsrodyghqn77e

A generalization of Naundorf's fixpoint theorem

Wim H. Hesselink
2000 Theoretical Computer Science  
A condition is presented that implies that F has a unique ÿxpoint. This is a generalization of a theorem of Naundorf.  ...  Given is an ordered set in which every chain has an upper bound and every pair of elements has a greatest lower bound. Let Z be its set of maximal elements and let F be a function from Z to Z.  ...  According to [1] , Zorn's Lemma asserts that every inductive poset has a maximal element. We need the following stronger version: Lemma. Let L be a chain in an inductive poset X .  ... 
doi:10.1016/s0304-3975(00)00202-4 fatcat:4f3fijurdbdv3gdywbmqwcjsfy

Six impossible rings

Wilfrid Hodges
1974 Journal of Algebra  
M will satisfy all of the axioms of Zermelo-Fraenkel set theory except Zorn's Lemma (= Axiom of Choice).  ...  Hence each ring with its accompanying universe of sets will constitute a proof that some well known theorem about rings cannot be proved without using Zorn's Lemma.  ...  ZF does not include Zorn's Lemma or any form of the Axiom of Choice; we write ZFC for ZF with Zorn's Lemma added. (See Cohen [2] for a fuller account.)  ... 
doi:10.1016/0021-8693(74)90065-9 fatcat:3eomq2r66jfddfyxyr5tmejgv4

Page 1022 of Mathematical Reviews Vol. 16, Issue 4 [page]

1955 Mathematical Reviews  
IBM Type 701 Computer. Biichi, J. R. Axiom of choice and Zorn’s lemma. Buckel, W. Kreise der euklidischen Ebene. Kegelschnitte der projektiven Ebene. Biickner, H., Weyl, F.  ...  A. C., Houthakker, H. S., and Prais, S. J. Electronic computation in economic statistics. Brown, W. F., and Thomas, T. Y. Pressure derivatives behind shocks.  ... 

Constructive theory of Banach algebras

Coquand, Spitters
2010 Journal of Logic and Analysis  
We present a way to organize a constructive development of the theory of Banach algebras, inspired by works of Cohen, de Bruijn and Bishop.  ...  point-free, description of the spectrum of a Banach algebra.  ...  Acknowledgements We would like to thank the referees for their suggestions which helped to improve the presentation of the paper.  ... 
doi:10.4115/jla.2010.2.11 fatcat:o532l7jfbjeh3oaxqpkuqj3roy

An interpretation of system F through bar recursion

Valentin Blot
2017 2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)  
There are two possible computational interpretations of second-order arithmetic: Girard's system F or Spector's bar recursion and its variants.  ...  This translation relies on a realizability interpretation of second-order arithmetic.  ...  ) normrc can be shown to be the computational interpretation of the proof of lemma 2: Lemma 12.  ... 
doi:10.1109/lics.2017.8005066 dblp:conf/lics/Blot17 fatcat:b7sqz3fsqrbwhitpcztz7vxi2a

The Computational Significance of Hausdorff's Maximal Chain Principle [chapter]

Peter Schuster, Daniel Wessel
2020 Lecture Notes in Computer Science  
As a fairly frequent form of the Axiom of Choice about relatively simple structures (posets), Hausdorff's Maximal Chain Principle appears to be little amenable to computational interpretation.  ...  This achievement enables us to present a generalisation of Hausdorff's principle first as a semantic and then as a syntactical conservation theorem.  ...  The present study was carried out within the projects "A New Dawn of Intuitionism: Mathematical and Philosophical Advances" (ID 60842) funded by the John Templeton Foundation, and "Reducing complexity  ... 
doi:10.1007/978-3-030-51466-2_21 fatcat:m7ios446cfeahni63eorjtsaem

The Uniform Distributions Puzzle

Luc Lauwers
2009 Social Science Research Network  
Kadane and O'Hagan discuss the existence of a uniform probability on the set of natural numbers (they provide a sufficient and necessary condition for the existence of such a uniform probability).  ...  I show that a purely finitely additive measure on the set of natural numbers cannot be constructed (its existence needs some form of the Axiom of Choice, the prototype of a nonconstructive axiom).  ...  Among the applications of AC, we mention Zorn's Lemma, the theorem of Hahn-Banach, and the existence of free ultrafilters.  ... 
doi:10.2139/ssrn.1393712 fatcat:dhen7i27angslcglkmeawfadja
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