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Relevant bibliographies by topics / Axiom of choice. Set theory

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Academic literature on the topic 'Axiom of choice. Set theory'

Author: Grafiati

Published: 4 June 2021

Last updated: 6 February 2022

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Journal articles on the topic "Axiom of choice. Set theory"

1

MORILLON, MARIANNE. "MULTIPLE CHOICES IMPLY THE INGLETON AND KREIN–MILMAN AXIOMS." Journal of Symbolic Logic 85, no. 1 (2019): 439–55. http://dx.doi.org/10.1017/jsl.2019.48.

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AbstractIn set theory without the Axiom of Choice, we consider Ingleton’s axiom which is the ultrametric counterpart of the Hahn–Banach axiom. We show that in ZFA, i.e., in the set theory without the Axiom of Choice weakened to allow “atoms,” Ingleton’s axiom does not imply the Axiom of Choice (this solves in ZFA a question raised by van Rooij, [27]). We also prove that in ZFA, the “multiple choice” axiom implies the Krein–Milman axiom. We deduce that, in ZFA, the conjunction of the Hahn–Banach, Ingleton and Krein–Milman axioms does not imply the Axiom of Choice.

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van Lambalgen, Michiel. "Independence, randomness and the axiom of choice." Journal of Symbolic Logic 57, no. 4 (1992): 1274–304. http://dx.doi.org/10.2307/2275368.

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AbstractWe investigate various ways of introducing axioms for randomness in set theory. The results show that these axioms, when added to ZF, imply the failure of AC. But the axiom of extensionality plays an essential role in the derivation, and a deeper analysis may ultimately show that randomness is incompatible with extensionality.

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Halbeisen, Lorenz, and Saharon Shelah. "Relations Between Some Cardinals in the Absence of the Axiom of Choice." Bulletin of Symbolic Logic 7, no. 2 (2001): 237–61. http://dx.doi.org/10.2307/2687776.

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AbstractIf we assume the axiom of choice, then every two cardinal numbers are comparable. In the absence of the axiom of choice, this is no longer so. For a few cardinalities related to an arbitrary infinite set, we will give all the possible relationships between them, where possible means that the relationship is consistent with the axioms of set theory. Further we investigate the relationships between some other cardinal numbers in specific permutation models and give some results provable without using the axiom of choice.

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Dihoum, Eman, and Michael Rathjen. "Preservation of choice principles under realizability." Logic Journal of the IGPL 27, no. 5 (2019): 746–65. http://dx.doi.org/10.1093/jigpal/jzz002.

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AbstractEspecially nice models of intuitionistic set theories are realizability models $V({\mathcal A})$, where $\mathcal A$ is an applicative structure or partial combinatory algebra. This paper is concerned with the preservation of various choice principles in $V({\mathcal A})$ if assumed in the underlying universe $V$, adopting Constructive Zermelo–Fraenkel as background theory for all of these investigations. Examples of choice principles are the axiom schemes of countable choice, dependent choice, relativized dependent choice and the presentation axiom. It is shown that any of these axiom

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Spector, Mitchell. "Ultrapowers without the axiom of choice." Journal of Symbolic Logic 53, no. 4 (1988): 1208–19. http://dx.doi.org/10.1017/s0022481200028024.

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AbstractA new method is presented for constructing models of set theory, using a technique of forming pseudo-ultrapowers. In the presence of the axiom of choice, the traditional ultrapower construction has proven to be extremely powerful in set theory and model theory; if the axiom of choice is not assumed, the fundamental theorem of ultrapowers may fail, causing the ultrapower to lose almost all of its utility. The pseudo-ultrapower is designed so that the fundamental theorem holds even if choice fails; this is arranged by means of an application of the omitting types theorem. The general the

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Rathjen, Michael. "Power Kripke–Platek set theory and the axiom of choice." Journal of Logic and Computation 30, no. 1 (2020): 447–57. http://dx.doi.org/10.1093/logcom/exaa020.

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Abstract While power Kripke–Platek set theory, ${\textbf{KP}}({\mathcal{P}})$, shares many properties with ordinary Kripke–Platek set theory, ${\textbf{KP}}$, in several ways it behaves quite differently from ${\textbf{KP}}$. This is perhaps most strikingly demonstrated by a result, due to Mathias, to the effect that adding the axiom of constructibility to ${\textbf{KP}}({\mathcal{P}})$ gives rise to a much stronger theory, whereas in the case of ${\textbf{KP}}$, the constructible hierarchy provides an inner model, so that ${\textbf{KP}}$ and ${\textbf{KP}}+V=L$ have the same strength. This pa

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DEISER, OLIVER. "AN AXIOMATIC THEORY OF WELL-ORDERINGS." Review of Symbolic Logic 4, no. 2 (2011): 186–204. http://dx.doi.org/10.1017/s1755020310000390.

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We introduce a new simple first-order framework for theories whose objects are well-orderings (lists). A system ALT (axiomatic list theory) is presented and shown to be equiconsistent with ZFC (Zermelo Fraenkel Set Theory with the Axiom of Choice). The theory sheds new light on the power set axiom and on Gödel’s axiom of constructibility. In list theory there are strong arguments favoring Gödel’s axiom, while a bare analogon of the set theoretic power set axiom looks artificial. In fact, there is a natural and attractive modification of ALT where every object is constructible and countable. In

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Akihiro Kanamori. "Zermelo and Set Theory." Bulletin of Symbolic Logic 10, no. 4 (2004): 487–553. http://dx.doi.org/10.2178/bsl/1102083759.

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Ernst Friedrich Ferdinand Zermelo (1871–1953) transformed the set theory of Cantor and Dedekind in the first decade of the 20th century by incorporating the Axiom of Choice and providing a simple and workable axiomatization setting out generative set-existence principles. Zermelo thereby tempered the ontological thrust of early set theory, initiated the delineation of what is to be regarded as set-theoretic, drawing out the combinatorial aspects from the logical, and established the basic conceptual framework for the development of modern set theory. Two decades later Zermelo promoted a distin

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WEBER, ZACH. "TRANSFINITE CARDINALS IN PARACONSISTENT SET THEORY." Review of Symbolic Logic 5, no. 2 (2012): 269–93. http://dx.doi.org/10.1017/s1755020312000019.

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This paper develops a (nontrivial) theory of cardinal numbers from a naive set comprehension principle, in a suitable paraconsistent logic. To underwrite cardinal arithmetic, the axiom of choice is proved. A new proof of Cantor’s theorem is provided, as well as a method for demonstrating the existence of large cardinals by way of a reflection theorem.

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Andréka, H., Á. Kurucz, and I. Németi. "Connections between axioms of set theory and basic theorems of universal algebra." Journal of Symbolic Logic 59, no. 3 (1994): 912–23. http://dx.doi.org/10.2307/2275917.

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Abstract.One of the basic theorems in universal algebra is Birkhoff's variety theorem: the smallest equationally axiomatizable class containing a class K of algebras coincides with the class obtained by taking homomorphic images of subalgebras of direct products of elements of K. G. Grätzer asked whether the variety theorem is equivalent to the Axiom of Choice. In 1980, two of the present authors proved that Birkhoff's theorem can already be derived in ZF. Surprisingly, the Axiom of Foundation plays a crucial role here: we show that Birkhoff's theorem cannot be derived in ZF + AC \{Foundation}

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Dissertations / Theses on the topic "Axiom of choice. Set theory"

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Pace, Dennis. "Axiom of Choice: Equivalences and Applications." Youngstown State University / OhioLINK, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=ysu1340993084.

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Chad, Ben. "Two-point sets." Thesis, University of Oxford, 2010. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.589611.

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This thesis concerns two-point sets, which are subsets of the real plane which intersect every line in exactly two points. The existence of two-point sets was first shown in 1914 by Mazukiewicz, and since this time, the properties of these objects have been of great intrigue to mathematicians working in both topology and set theory. Arguably, the most famous problem about two-point sets is concerned with their so-called "descriptive complexity"; it remains open, and it appears to be deep. An informal interpretation of the problem, which traces back at least to Erdos, is: The term "two-point" s

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Törnkvist, Robin. "Tychonoff's theorem and its equivalence with the axiom of choice." Thesis, Umeå universitet, Institutionen för matematik och matematisk statistik, 2015. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-107423.

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In this essay we give an elementary introduction to topology so that we can prove Tychonoff’s theorem, and also its equivalence with the axiom of choice.<br>Denna uppsats tillhandahåller en grundläggande introduktion till topologi för att sedan bevisa Tychonoff’s theorem, samt dess ekvivalens med urvalsaxiomet.

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Bartley, Michael George. "The axiom : 'all free groups are projective in Ab(E)'." Thesis, University of Bristol, 1988. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.330322.

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May, Russell J. "A Collapsing Result Using the Axiom of Determinancy and the Theory of Possible Cofinalities." Thesis, University of North Texas, 2001. https://digital.library.unt.edu/ark:/67531/metadc2789/.

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Assuming the axiom of determinacy, we give a new proof of the strong partition relation on ω1. Further, we present a streamlined proof that J<λ+(a) (the ideal of sets which force cof Π α < λ) is generated from J<λ+(a) by adding a singleton. Combining these results with a polarized partition relation on ω1

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Iaria, Alessandro. "Essays on choice set heterogeneity in demand estimation." Thesis, University of Warwick, 2014. http://wrap.warwick.ac.uk/62609/.

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The estimation of demand systems is a key activity in empirical economics. The most prominent use of demand estimates relates to the computation of social welfare changes due to modifications in the economic environment. Examples include the introduction of a new tax (Griffith et al. 2010), a change in the regulation of an industry (Crawford & Yurukoglu, 2011), the introduction of a new product (Petrin, 2002), mergers between companies (Nevo, 2000, 2001), the legalization of soft drugs (Jacobi & Sovinsky, 2012). Any sort of welfare exercise, essential for policy making, would require—ideally—p

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Holshouser, Jared Kenneth. "Partition Properties for Non-Ordinal Sets under the Axiom of Determinacy." Thesis, University of North Texas, 2017. https://digital.library.unt.edu/ark:/67531/metadc984121/.

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In this paper we explore coloring theorems for the reals, its quotients, cardinals, and their combinations. This work is done under the scope of the axiom of determinacy. We also explore generalizations of Mycielski's theorem and show how these can be used to establish coloring theorems. To finish, we discuss the strange realm of long unions.

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Malmberg, Hannes. "Random Choice over a Continuous Set of Options." Licentiate thesis, Stockholms universitet, Matematiska institutionen, 2013. http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-89917.

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Random choice theory has traditionally modeled choices over a -nite number of options. This thesis generalizes the literature by studyingthe limiting behavior of choice models as the number of optionsapproach a continuum.The thesis uses the theory of random elds, extreme value theoryand point processes to calculate this limiting behavior. For a numberof distributional assumptions, we can give analytic expressions forthe limiting probability distribution of the characteristics of the bestchoice. In addition, we also outline a straightforward extension to ourtheory which would signicantly relax

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Enflo, Karin. "Measures of Freedom of Choice." Doctoral thesis, Uppsala universitet, Avdelningen för praktisk filosofi, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-179078.

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This thesis studies the problem of measuring freedom of choice. It analyzes the concept of freedom of choice, discusses conditions that a measure should satisfy, and introduces a new class of measures that uniquely satisfy ten proposed conditions. The study uses a decision-theoretical model to represent situations of choice and a metric space model to represent differences between options. The first part of the thesis analyzes the concept of freedom of choice. Different conceptions of freedom of choice are categorized into evaluative and non-evaluative, as well as preference-dependent and pref

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Lotan, Tsippy. "Modeling route choice behavior in the presence of information using concepts from fuzzy set theory and approximate reasoning." Thesis, Massachusetts Institute of Technology, 1992. http://hdl.handle.net/1721.1/12901.

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Books on the topic "Axiom of choice. Set theory"

1

The axiom of choice. Dover Publications, 2008.

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E, Shaposhnikov D., ред. Mnogokriterialʹnyĭ vybor s uchetom individualʹnykh predpochteniĭ. Rossiĭskai͡a︡ akademii͡a︡ nauk, In-t prikladnoĭ fiziki, 1994.

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3

Blass, Andreas. Freyd's models for the independence of the axiom of choice. American Mathematical Society, 1989.

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Segal, Uzi. Two-stage lotteries without the reduction axiom. University of Toronto, 1987.

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author, Malik D. S., and Clark Terry D. author, eds. Application of fuzzy logic to social choice theory. CRC Press, Taylor & Francis Group, 2015.

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Ferro, Alfredo. Decision procedures for elementary sublanguages of set theory. VII. Validity in set theory when a choice operator is present. Courant Institute of Mathematical Sciences, New York University, 1986.

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Pruss, Alexander R. The Axiom of Choice Machine. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198810339.003.0006.

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This is a mainly technical chapter concerning the causal embodiment of the Axiom of Choice from set theory. The Axiom of Choice powered a construction of an infinite fair lottery in Chapter 4 and a die-rolling strategy in Chapter 5. For those applications to work, there has to be a causally implementable (though perhaps not compatible with our laws of nature) way to implement the Axiom of Choice—and, for our purposes, it is ideal if that involves infinite causal histories, so the causal finitist can reject it. Such a construction is offered. Moreover, other paradoxes involving the Axiom of Cho

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Weaver, Nik. Forcing for Mathematicians. World Scientific Publishing Co Pte Ltd, 2014.

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The Covering Property Axiom, CPA: A Combinatorial Core of the Iterated Perfect Set Model (Cambridge Tracts in Mathematics). Cambridge University Press, 2004.

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Injective Choice Functions (Lecture Notes in Mathematics, Vol 1238). Springer, 1987.

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Book chapters on the topic "Axiom of choice. Set theory"

1

Dasgupta, Abhijit. "Alephs, Cofinality, and the Axiom of Choice." In Set Theory. Springer New York, 2013. http://dx.doi.org/10.1007/978-1-4614-8854-5_10.

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Adamson, Iain T. "Equivalents of the Axiom of Choice." In A Set Theory Workbook. Birkhäuser Boston, 1998. http://dx.doi.org/10.1007/978-0-8176-8138-8_11.

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De La Cruz, Omar, and Carlos Augusto Di Prisco. "Weak Forms of the Axiom of Choice and Partitions of Infinite Sets." In Set Theory. Springer Netherlands, 1998. http://dx.doi.org/10.1007/978-94-015-8988-8_4.

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Vaught, Robert L. "The Axiom of Regularity." In Set Theory. Birkhäuser Boston, 2001. http://dx.doi.org/10.1007/978-1-4612-0835-8_9.

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Miller, Arnold W. "Martin’s Axiom." In Descriptive Set Theory and Forcing. Springer Berlin Heidelberg, 1995. http://dx.doi.org/10.1007/978-3-662-21773-3_5.

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Miller, Arnold W. "Martin’s axiom and Constructibility." In Descriptive Set Theory and Forcing. Springer Berlin Heidelberg, 1995. http://dx.doi.org/10.1007/978-3-662-21773-3_23.

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Fontanella, Laura. "How to Choose New Axioms for Set Theory?" In Reflections on the Foundations of Mathematics. Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-15655-8_2.

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Henle, James M. "Choice and Infinitesimals." In An Outline of Set Theory. Springer New York, 1986. http://dx.doi.org/10.1007/978-1-4613-8680-3_10.

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Henle, James M. "Choice and Infinitesimals." In An Outline of Set Theory. Springer New York, 1986. http://dx.doi.org/10.1007/978-1-4613-8680-3_20.

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Henle, James M. "Choice and Infinitesimals." In An Outline of Set Theory. Springer New York, 1986. http://dx.doi.org/10.1007/978-1-4613-8680-3_30.

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Conference papers on the topic "Axiom of choice. Set theory"

1

Zhou, Chunlai, Biao Qin, and Xiaoyong Du. "A Savage-style Utility Theory for Belief Functions." In Twenty-Seventh International Joint Conference on Artificial Intelligence {IJCAI-18}. International Joint Conferences on Artificial Intelligence Organization, 2018. http://dx.doi.org/10.24963/ijcai.2018/712.

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In this paper, we provide an axiomatic justification for decision making with belief functions by studying the belief-function counterpart of Savage's Theorem where the state space is finite and the consequence set is a continuum [l, M] (l<M). We propose six axioms for a preference relation over acts, and then show that this axiomatization admits a definition of qualitative belief functions comparing preferences over events that guarantees the existence of a belief function on the state space. The key axioms are uniformity and an analogue of the independence axiom. The uniformity axiom is u

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Fu, Hui, Na Liu, Jun Liang, Adam J. Pel, and Serge P. Hoogendoorn. "Modeling and Simulation of Evacuation Route Choice Behavior Using Fuzzy Set Theory." In 2015 IEEE 18th International Conference on Intelligent Transportation Systems - (ITSC 2015). IEEE, 2015. http://dx.doi.org/10.1109/itsc.2015.218.

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Subbarayan, G., D. L. Bartel, and D. L. Taylor. "A Method for Comparative Performance Evaluation of Structural Optimization Codes: Part I — Theory." In ASME 1988 Design Technology Conferences. American Society of Mechanical Engineers, 1988. http://dx.doi.org/10.1115/detc1988-0028.

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Abstract This paper presents a systematic procedure for the comparative performance evaluation of nonlinear programming codes intended for applications in structural optimization. Part I discusses the issues in the evaluation of nonlinear programming codes and proposes a performance evaluation scheme for structural optimization codes. Aspects of performance evaluation such as the choice of test problems and appropriate set of performance criteria for structural optimization codes are described. A procedure to analyze codes based on their observed geometric behavior for test problems is also pr

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Novaro, Arianna, Umberto Grandi, Dominique Longin, and Emiliano Lorini. "Goal-Based Collective Decisions: Axiomatics and Computational Complexity." In Twenty-Seventh International Joint Conference on Artificial Intelligence {IJCAI-18}. International Joint Conferences on Artificial Intelligence Organization, 2018. http://dx.doi.org/10.24963/ijcai.2018/65.

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We study agents expressing propositional goals over a set of binary issues to reach a collective decision. We adapt properties and rules from the literature on Social Choice Theory to our setting, providing an axiomatic characterisation of a majority rule for goal-based voting. We study the computational complexity of finding the outcome of our rules (i.e., winner determination), showing that it ranges from Nondeterministic Polynomial Time (NP) to Probabilistic Polynomial Time (PP).

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Subbarayan, G., D. L. Bartel, and D. L. Taylor. "A Method for Comparative Performance Evaluation of Structural Optimization Codes: Part II — Applications." In ASME 1988 Design Technology Conferences. American Society of Mechanical Engineers, 1988. http://dx.doi.org/10.1115/detc1988-0029.

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Abstract This paper presents a systematic procedure for the comparative performance evaluation of nonlinear programming codes intended for applications in structural optimization. Part I discusses the issues in the evaluation of nonlinear programming codes and proposes a performance evaluation scheme for structural optimization codes. Aspects of performance evaluation such as the choice of test problems and appropriate set of performance criteria for structural optimization codes are described. A procedure to analyze codes based on their observed geometric behavior for test problems is also pr

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Mata Diaz, Amilcar, and Ramon Pino Perez. "Impossibility in Belief Merging (Extended Abstract)." In Twenty-Seventh International Joint Conference on Artificial Intelligence {IJCAI-18}. International Joint Conferences on Artificial Intelligence Organization, 2018. http://dx.doi.org/10.24963/ijcai.2018/799.

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With the aim of studying social properties of belief merging and having a better understanding of impossibility, we extend in three ways the framework of logic-based merging introduced by Konieczny and Pino Perez. First, at the level of representation of the information, we pass from belief bases to complex epistemic states. Second, the profiles are represented as functions of finite societies to the set of epistemic states (a sort of vectors) and not as multisets of epistemic states. Third, we extend the set of rational postulates in order to consider the epistemic versions of the classical p

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Harrenstein, Paul, Paolo Turrini, and Michael Wooldridge. "Characterising the Manipulability of Boolean Games." In Twenty-Sixth International Joint Conference on Artificial Intelligence. International Joint Conferences on Artificial Intelligence Organization, 2017. http://dx.doi.org/10.24963/ijcai.2017/150.

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The existence of (Nash) equilibria with undesirable properties is a well-known problem in game theory, which has motivated much research directed at the possibility of mechanisms for modifying games in order to eliminate undesirable equilibria, or induce desirable ones. Taxation schemes are a well-known mechanism for modifying games in this way. In the multi-agent systems community, taxation mechanisms for incentive engineering have been studied in the context of Boolean games with costs. These are games in which each player assigns truth-values to a set of propositional variables she uniquely

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Pumhoessel, Thomas, Peter Hehenberger, and Klaus Zeman. "Preserving Stability Properties in Reduced Models of Time-Periodic Systems Using Proper Orthogonal Decomposition." In ASME 2011 International Mechanical Engineering Congress and Exposition. ASMEDC, 2011. http://dx.doi.org/10.1115/imece2011-63435.

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The necessity of providing reduced models of dynamical systems is growing continuously. Model-based control and model-based design are exemplary fields of applications. In this contribution, the reduction of a controlled drivetrain of a rolling mill using the method of Proper Orthogonal Decomposition is investigated, where the specific choice of the control law leads to equations of motion with time-periodic coefficients. Depending on amplitudes and frequency parameters of the time-periodic coefficients, artificial damping is introduced, referred to as parametric control. The maximum damping e

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Petrolo, Marco, and Erasmo Carrera. "Best Structural Theories for Free Vibrations of Sandwich Composites via Machine Learning." In ASME 2019 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2019. http://dx.doi.org/10.1115/imece2019-10296.

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Abstract This work presents a novel methodology for the development of refined structural theories for the modal analysis of sandwich composites. Such a methodology combines three well-established techniques, namely, the Carrera Unified Formulation (CUF), the Axiomatic/Asymptotic Method (AAM), and Artificial Neural Networks (NN). CUF generates structural theories and finite element arrays hierarchically. CUF provides the training set for the NN in which the structural theories are inputs and the natural frequencies targets. AAM evaluates the influence of each generalized displacement variable,

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Ghosh, Dipanjan D., and Andrew Olewnik. "Computationally Efficient Imprecise Uncertainty Propagation in Engineering Design and Decision Making." In ASME 2012 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/detc2012-70419.

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Modeling uncertainty through probabilistic representation in engineering design is common and important to decision making that considers risk. However, representations of uncertainty often ignore elements of “imprecision” that may limit the robustness of decisions. Further, current approaches that incorporate imprecision suffer from computational expense and relatively high solution error. This work presents the Computationally Efficient Imprecise Uncertainty Propagation (CEIUP) method which draws on existing approaches for propagation of imprecision and integrates sparse grid numerical integ

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