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Page 2 of Mathematical Reviews Vol. 15, Issue 1 [page]

1954 Mathematical Reviews  
Investigation of the equivalence of the axiom of choice and Zorn’s lemma from the viewpoint of the hierarchy of types. J. Symbolic Logic 18, 125-135 (1953).  ...  Zorn’s lemma for sets of type a: (Frocea)) Pra Wr]> (Axa) (ta) (rxu > rux) or (rocaa))ZL°r, where Pr is the (natural) formalization of ‘y quasi-orders the type a’ and Wr of ‘every r-chain has an r-upper  ... 

JSL volume 18 issue 2 Cover and Front matter

1953 Journal of Symbolic Logic (JSL)  
QUINE . 119 Investigation of the equivalence of t h e axiom of choice a n d Zorn's lemma from the viewpoint of t h e hierarchy of types. By J.  ...  All manuscripts should be typewritten with wide margin and wide spacing between the lines, and the author should keep a complete copy; footnotes should be numbered consecutively and should ako be typed  ... 
doi:10.1017/s0022481200097747 fatcat:ijsiq3owofawhpfnrriy5qezle

JSL volume 18 issue 4 Cover and Front matter and Errata

1953 Journal of Symbolic Logic (JSL)  
All manuscripts should be typewritten with wide margin and wide spacing between the lines, and the author should keep a complete copy; footnotes should be numbered consecutively and should also be typed  ...  ., published quarterly, in the months of March, June, September, and December. The JOURNAL is published for the Association by N.V. Erven P. Noordhoff, Publishers, Groningen, The Netherlands.  ...  QUINE 119 Investigation of the equivalence of the axiom of choice and Zorn's lemma from the viewpoint of the hierarchy of types. By J. RICHARD BUCHI 125Some formal relative consistency proofs.  ... 
doi:10.1017/s0022481200088757 fatcat:fdf4snqq2rd7tkqjmtd4begcxi

Game logic and its applications I

Mamoru Kaneko, Takashi Nagashima
1996 Studia Logica: An International Journal for Symbolic Logic  
The reason for an infinitary extension is to express the common knowledge concept explicitly. Depending upon the choice of axioms on the knowledge operators, there is a hierarchy of logics.  ...  This paper provides a logic framework for investigations of game theoretical problems. We adopt an infinitary extension of classical predicate logic as the base logic of the framework.  ...  There still remain important problems in this direction from the viewpoint of both logic and game theory. These will be discussed in Kaneko [10] .  ... 
doi:10.1007/bf00370838 fatcat:rdsaefbgk5anje24ktqkbjzttq

Set Theory and the Analyst [article]

N. H. Bingham, A. J. Ostaszewski
2018 arXiv   pre-print
The bulk of the text is devoted to a summary, intended for the working analyst, of the extensive background in set theory and logic needed to discuss such matters: to quote from the Preface of Kelley [  ...  This survey is motivated by specific questions arising in the similarities and contrasts between (Baire) category and (Lebesgue) measure -- category-measure duality and non-duality, as it were.  ...  AC is most often used in the (equivalent) form of Zorn's Lemma of 1935 (a misnomer, as the result is due to Kuratowski in 1922 , but the usage is now established).  ... 
arXiv:1801.09149v1 fatcat:oppiwcgt4jcsdbdg2czoepnopu

Universes of Fuzzy Sets and Axiomatizations of Fuzzy Set Theory. Part I: Model-Based and Axiomatic Approaches

Siegfried Gottwald
2006 Studia Logica: An International Journal for Symbolic Logic  
For classical sets one has with the cumulative hierarchy of sets, with axiomatizations like the system ZF, and with the category SET of all sets and mappings standard approaches toward global universes  ...  Rasiowa and C. Rauszer, and held in Warsaw and Ruciane-Nida from 23rd to  ...  pairing union powerset ∈-induction (i.e. foundation) separation comprehension infinity and together with two further axioms giving a version of Zorn's lemma and claiming the existence of the support of  ... 
doi:10.1007/s11225-006-7197-8 fatcat:vxo5u3lvy5g53ewkj5bdddfa6y

A Semi-Constructive Approach to the Hyperreal Line [article]

Guillaume Massas
2022 arXiv   pre-print
mathematics, i.e., does not assume any fragment of the Axiom of Choice beyond the Axiom of Dependent Choices.  ...  I discuss the technical advantages of ^†ℛ over some other alternative approaches to nonstandard analysis and argue that it is well-suited to address some of the philosophical and methodological concerns  ...  References Acknowledgments I would like to thank Wes Holliday and Anna Bellomo for their helpful comments on earlier versions of this manuscript.  ... 
arXiv:2201.10818v1 fatcat:cik42jnxazgvdizeblkdxnvqc4

Set theory and the analyst

Nicholas H. Bingham, Adam J. Ostaszewski
2018 European Journal of Mathematics  
Acknowledgements We are grateful to a number of friends and colleagues for their comments. We thank in particular Wilfrid Hodges, Akihiro Kanamori, Adrian Mathias, the Referees, and the Editor.  ...  , and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.  ...  AC is most often used in the (equivalent) form of Zorn's Lemma of 1935 (a misnomer, as the result is due to Kuratowski in 1922, but the usage is now established).  ... 
doi:10.1007/s40879-018-0278-1 fatcat:d4pspttcwjb45hmppqwugcljdy

On the Axioms of Causal Set Theory [article]

Benjamin F. Dribus
2013 arXiv   pre-print
arising from the asymptotic enumeration of Kleitman and Rothschild.  ...  The paper culminates in the theory of co-relative histories and kinematic schemes, combining the causal metric hypothesis, the histories approach to quantum theory, and Grothendieck's relative viewpoint  ...  In Zermelo-Fraenkel set theory, the order extension principle may be proven using the power set axiom and Zorn's lemma.  ... 
arXiv:1311.2148v3 fatcat:nw4itctovnhrfcgiq7zxj4jg7a

Swinging types=functions+relations+transition systems

Peter Padawitz
2000 Theoretical Computer Science  
Swinging types combine equational, Horn and modal logic for the purpose of using evaluation and proof rules from all three logics for rapid prototyping and veriÿcation.  ...  Swinging types admit the integrated design and analysis of these components and aspects.  ...  It is easy to show that the union of each totally ordered subset of F is again in F. Hence by Zorn's Lemma, F has maximal element.  ... 
doi:10.1016/s0304-3975(00)00171-7 fatcat:rgolg2rgwvglvgruklxmzywg4u

Self-similarity in the Foundations [article]

Paul K. Gorbow
2018 arXiv   pre-print
A key axiom of this theory expresses that its structures have an endofunctor with natural properties.  ...  ZF, and an analogue of a theorem of Gaifman to the effect that certain countable models of set theory can be elementarily end-extended to a model with many automorphisms whose sets of fixed points equal  ...  By Zorn's lemma it suffices to show that for any ordinal α and for any ⊆-increasing sequence F = (F ξ ) ξ<α of filters on P, the union G = ξ<α F ξ is a filter. But this follows from that ∀x, y ∈ G.  ... 
arXiv:1806.11310v1 fatcat:iwnf2v6qa5ctld3rkevnqkywvq

Roitman Judith. Introduction to modern set theory. Pure and applied mathematics. John Wiley & Sons, New York etc. 1990, xiii + 156 pp

J. R. Shoenfield
1991 Journal of Symbolic Logic (JSL)  
for my father, who loved mathematics The third chapter introduces all of the axioms except regularity and choice, formally defines the natural numbers, and gives examples of models of the axioms, with  ...  Martin's axiom, stationary sets and ♦, and cardinal invariants of the reals. A brief word about formality.  ...  And, like the proof of theorem 1.43, we need the axiom of dependent choice, a weak form of the axiom of choice. There are more global consequences of the axiom of regularity.  ... 
doi:10.2307/2274720 fatcat:l54c2t7iabdrvltayrlyy23u3m

Constructive Reverse Mathematics [article]

Hannes Diener
2020 arXiv   pre-print
An introduction and overview of constructive reverse mathematics.  ...  Here the last axiom is, as usual, actually an axiom schema for all statements ϕ. And, of course, all variables are of a fixed natural number type.  ...  That 3 implies 1 follows from Lemma 3.1.3. The proof of the equivalence of 4 and WKL can be found in [66] .  ... 
arXiv:1804.05495v3 fatcat:qrdantvyinettnusj4im7wx3bq

Causal structures and causal boundaries

Alfonso García-Parrado, José M M Senovilla
2005 Classical and quantum gravity  
We give an up-to-date perspective with a general overview of the theory of causal properties, the derived causal structures, their classification and applications, and the definition and construction of  ...  causal boundaries and of causal symmetries, mostly for Lorentzian manifolds but also in more abstract settings.  ...  Financial support under grants BFM2000-0018 and FIS2004-01626 of the Spanish CICyT and no. 9/UPV 00172.310-14456/2002 of the University of the Basque Country is also acknowledged.  ... 
doi:10.1088/0264-9381/22/9/r01 fatcat:yj6eysurqvfojmnkp2co4qtleq

Iterated Belief Revision Under Resource Constraints: Logic as Geometry [article]

Dan P. Guralnik, Daniel E. Koditschek
2018 arXiv   pre-print
a two-way dictionary from inference to geometry, leading to immense savings in computation, at a cost in the quality of representation that can be quantified in terms of topological invariants.  ...  Finally, we present simulation results to illustrate the viability of the approach, and close with a discussion of the strengths, weaknesses, and potential development of UMA-based learners.  ...  A standard application of Zorn's lemma shows that any G-coherent subset of Σ is contained in an element of G • . Note also that G • ⊆ C(G). Example 2.13 (the orthogonal PCR and the Hamming cube).  ... 
arXiv:1812.08313v1 fatcat:c62dcxwryrcyvhrgbdwmfjetxq
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