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A computational interpretation of Zorn's lemma
[article]

2020
*
arXiv
*
pre-print

We give

arXiv:2001.03540v1
fatcat:bq3i543pdfcp3mirjtgm3eq6vu
*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? ...##
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Well quasi-orders and the functional interpretation
[article]

2017
*
arXiv
*
pre-print

question

arXiv:1706.02881v1
fatcat:27jdif64vbafdfses6zsproyq4
*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. ...##
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Flabby and injective objects in toposes
[article]

2018
*
arXiv
*
pre-print

In the special case

arXiv:1810.12708v1
fatcat:leeb2dm2rfegnmfzkqjg3uhj4q
*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. ...##
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Completeness of resolution revisited

1990
*
Theoretical Computer Science
*

Moreover, by using

doi:10.1016/0304-3975(90)90139-9
fatcat:eo24tcnsvndbdanrveqdeuylo4
*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). ...##
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Prioritising Preference Relations
[chapter]

1993
*
Theory and Formal Methods 1993
*

and the definition

doi:10.1007/978-1-4471-3503-6_21
fatcat:tlbr7c6p55hyhj4moc2727l2b4
*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. ...##
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Existence and uniqueness of the real closure of an ordered field without Zorn's Lemma

1991
*
Journal of Pure and Applied Algebra
*

., Existence and uniqueness

doi:10.1016/0022-4049(91)90110-n
fatcat:xtlm4vw2ojffzmvix34aqcrkwu
*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. ...##
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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

1999
*
Information and Computation
*

Since the concrete domain D is generally assumed to be

doi:10.1006/inco.1999.2801
fatcat:mgrdpthuu5cjjbsrodyghqn77e
*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 ...##
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A generalization of Naundorf's fixpoint theorem

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 . ...

##
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Six impossible rings

1974
*
Journal of Algebra
*

M will satisfy all

doi:10.1016/0021-8693(74)90065-9
fatcat:3eomq2r66jfddfyxyr5tmejgv4
*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.) ...##
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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. ...##
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Constructive theory of Banach algebras

2010
*
Journal of Logic and Analysis
*

We present

doi:10.4115/jla.2010.2.11
fatcat:o532l7jfbjeh3oaxqpkuqj3roy
*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. ...##
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An interpretation of system F through bar recursion

2017
*
2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
*

There are two possible

doi:10.1109/lics.2017.8005066
dblp:conf/lics/Blot17
fatcat:b7sqz3fsqrbwhitpcztz7vxi2a
*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. ...##
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The Computational Significance of Hausdorff's Maximal Chain Principle
[chapter]

2020
*
Lecture Notes in Computer Science
*

As

doi:10.1007/978-3-030-51466-2_21
fatcat:m7ios446cfeahni63eorjtsaem
*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 ...##
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The Uniform Distributions Puzzle

2009
*
Social Science Research Network
*

Kadane and O'Hagan discuss the existence

doi:10.2139/ssrn.1393712
fatcat:dhen7i27angslcglkmeawfadja
*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. ...
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