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Books on the topic 'Axiom of choice'

Author: Grafiati

Published: 4 June 2021

Last updated: 4 March 2023

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1

Kulicki, Piotr. Aksjomatyczne systemy rachunku nazw. Lublin: Wydawn. KUL, 2011.

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2

The axiom of choice. London: College Publications, 2009.

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3

The axiom of choice. Mineola, N.Y: Dover Publications, 2008.

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4

Howard, Paul. Consequences of the axiom of choice. Providence, R.I: American Mathematical Society, 1998.

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5

E, Rubin Jean, ed. Equivalents of the axiom of choice, II. Amsterdam: North-Holland, 1985.

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6

Logicheskie metody issledovanii͡a︡ diskretnykh modeleĭ vybora. Moskva: "Nauka," Glav. red. fiziko-matematicheskoĭ litry, 1989.

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7

Xuan ze gong li. Beijing: Ren min chu ban she, 2003.

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8

A, Kossov O., ed. Problema vybora i nechetkie mnozhestva. Moskva: Vses. nauchno-issl. in-t sistemnykh issledovaniĭ, 1987.

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9

1946-, Berezovskiĭ B. A., Krasnoshchekov P. S, and Institut problem upravlenii͡a︡ (Akademii͡a︡ nauk SSSR), eds. Mnogokriterialʹnai͡a︡ optimizat͡s︡ii͡a︡: Matematicheskie aspekty. Moskva: "Nauka", 1989.

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10

Blass, Andreas. Freyd's models for the independence of the axiom of choice. Providence, R.I., USA: American Mathematical Society, 1989.

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11

Volʹskiĭ, V. I. Golosovanie v malykh gruppakh: Prot͡s︡edury i metody sravnitelʹnogo analiza. Moskva: "Nauka," Glav. red. fiziko-matematicheskoĭ lit-ry, 1991.

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12

Fishburn, Peter C. The mathematics of preference, choice, and order: Essays in honor of Peter C. Fishburn. Berlin: Springer, 2008.

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13

The mathematics of preference, choice, and order: Essays in honor of Peter C. Fishburn. Berlin: Springer, 2008.

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14

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

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15

Moulin, Hervé. Axioms of cooperative decision making. Cambridge [England]: Cambridge University Press, 1988.

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16

Axiom of Choice. Springer, 2006.

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17

Jech, Thomas J. Axiom of Choice. Dover Publications, Incorporated, 2013.

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18

Axiom of Choice. Berlin/Heidelberg: Springer-Verlag, 2006. http://dx.doi.org/10.1007/11601562.

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19

Jech, Thomas J. Axiom of Choice. Dover Publications, Incorporated, 2013.

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20

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 Choice are given, including two Dutch Book paradoxes connected with the Banach–Tarski paradox. Again, all this is argued to provide evidence for causal finitism.

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21

Moore, Gregory H. Zermelo's Axiom of Choice: Its Origins, Development, and Influence (Dover Books on Mathematics). Dover Publications, 2013.

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22

Equivalents of the Axiom of Choice. Creative Media Partners, LLC, 2021.

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23

Equivalents of the Axiom of Choice. Creative Media Partners, LLC, 2021.

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24

Rubin, H., and J. E. Rubin. Equivalents of the Axiom of Choice, II. Elsevier Science & Technology Books, 1985.

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25

Equivalents of the Axiom of Choice, II. Elsevier, 1985. http://dx.doi.org/10.1016/s0049-237x(08)x7013-2.

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26

Herrlich, Horst. Axiom of Choice (Lecture Notes in Mathematics Book 1876). Springer, 2006.

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27

Tanaka, Hisao. Sentaku kori to sugaku: Hassei to ronso, soshite kakuritsu e no michi. Hatsubaijo Seiunsha, 1987.

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28

Moore, G. H. Zermelo's Axiom of Choice: Its Origins, Development, and Influence. Springer, 2012.

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29

Moore, G. H. Zermelo's Axiom of Choice: "Its Origins, Development, And Influence". Springer, 2011.

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30

Cook, Roy T. Predication, Possibility, and Choice. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198792161.003.0007.

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Bob Hale’s deflationary conception of truth equates the actual (and necessary) existence of a property or relation with the possible existence of a corresponding predicate with appropriate satisfaction conditions. After surveying recent work on developing a semi-formal framework within which to study the deflationary conception, and presenting extant results regarding the extent to which the account supports the second-order comprehension schema, this chapter examines the extent to which the deflationary account allows one to justify various second-order versions of the axiom of choice. The results are primarily negative: choice does not seem to be forthcoming on the deflationary approach, and even assuming the possibility of arbitrary countably infinite linguistic supertasks, a version of the countable axiom of choice seems to be the best that we can do.

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31

Rahman, Shahid, and Nicolas Clerbout. Linking Game-Theoretical Approaches with Constructive Type Theory: Dialogical Strategies, CTT Demonstrations and the Axiom of Choice. Springer London, Limited, 2015.

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32

Rahman, Shahid, and Nicolas Clerbout. Linking Game-Theoretical Approaches with Constructive Type Theory: Dialogical Strategies, CTT demonstrations and the Axiom of Choice. Springer, 2015.

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33

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

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34

Smullyan, Raymond M. Gödel's Incompleteness Theorems. Oxford University Press, 1992. http://dx.doi.org/10.1093/oso/9780195046724.001.0001.

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Kurt Godel, the greatest logician of our time, startled the world of mathematics in 1931 with his Theorem of Undecidability, which showed that some statements in mathematics are inherently "undecidable." His work on the completeness of logic, the incompleteness of number theory, and the consistency of the axiom of choice and the continuum theory brought him further worldwide fame. In this introductory volume, Raymond Smullyan, himself a well-known logician, guides the reader through the fascinating world of Godel's incompleteness theorems. The level of presentation is suitable for anyone with a basic acquaintance with mathematical logic. As a clear, concise introduction to a difficult but essential subject, the book will appeal to mathematicians, philosophers, and computer scientists.

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35

Pruss, Alexander R. Paradoxical Lotteries. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198810339.003.0004.

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A countably infinite fair lottery has tickets numbered 1, 2, 3, …, all of them on par. Many paradoxes arise from such lotteries. For instance, if two people get tickets from such a lottery, each will be nearly certain that the other’s ticket has a bigger number. And it will turn out that it would be possible to use an infinite lottery to manipulate a rational agent to believe empirical claims that on their evidence are extremely unlikely. It is then argued that if causal finitism is false, then such lotteries are possible, though one of the arguments has a lacuna involving the Axiom of Choice that will need to be filled in Chapter 6. The net conclusion of the chapter is that the absurdities ensuing fromsuch lotteries are further evidence that causal finitism is true, though an alternate story involving limitations of our rational faculties is also discussed.

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36

Shapiro, Stewart. Properties and Predicates, Objects and Names. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198792161.003.0006.

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Hale has articulated and defended a thesis that properties are tied to predicates in possible languages. The same goes for functions. A necessary and sufficient condition for the existence of a property or function is that it is expressed by a predicate in a possible language that can be understood by finite beings, like us humans. The purpose of this chapter is to assess this view and determine how it fares against the output of standard mathematics. In order to interpret mathematics, Hale must defend some prima facie implausible claims about what sorts of languages are possible, for us finite beings. The only alternative is for Hale to reject, on purely philosophical grounds, large chunks of contemporary mathematics. The relevant mathematics is whatever relies on impredicative definitions and the axiom of choice. This includes his, and Crispin Wright’s, abstraction-based neologicism, which has been shown to rely indispensably on impredicative definitions.

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37

Lukiv, Dan. Ten Axioms of Choice Theory, and Seven Caring Versus Deadly Habits, Based on William Glasser's Choice Theory: Socio-Emotional Discussion Groups. Independently Published, 2016.

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38

Ten Axioms, and Seven Caring Versus Deadly Habits--Based on William Glasser's Choice Theory: Socio-Emotional Discussion Groups. Independently Published, 2017.

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39

Sugden, Robert. The Community of Advantage. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198825142.001.0001.

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Normative analysis in economics has usually aimed at satisfying individuals’ preferences. Its conclusions have supported a long-standing liberal tradition of economics that values economic freedom and views markets favourably. However, behavioural research shows that individuals’ preferences, as revealed in choices, are often unstable, and vary according to contextual factors that seem irrelevant for welfare. The Community of Advantage proposes a reformulation of normative economics that is compatible with what is now known about the psychology of choice. Other such reformulations have assumed that people have well-defined ‘latent’ preferences which, because of psychologically-induced errors, are not always revealed in actual choices. According to these reformulations, the economist’s job is to reconstruct latent preferences and to design policies to satisfy them. The argument of this book is that latent preference and error are psychologically ungrounded concepts, and that economics needs to be more radical in giving up rationality assumptions. The book advocates a kind of normative economics that does not use the concept of preference. Its recommendations are addressed, not to an imagined ‘social planner’, but to citizens, viewed as potential parties to mutually beneficial agreements. Its normative criterion is the provision of opportunities for individuals to participate in voluntary transactions. Using this approach, many of the normative conclusions of the liberal tradition are reconstructed. It is argued that a well-functioning market economy is an institution that individuals have reason to value, whether or not their preferences satisfy conventional axioms of rationality, and that individuals’ motivations in such an economy can be cooperative rather than self-interested.

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40

Button, Tim, and Sean Walsh. Internal categoricity and the natural numbers. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198790396.003.0010.

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The simple conclusion of the preceding chapters is that moderate modelism fails. But this leaves us with a choice between abandoning moderation and abandoning modelism. The aim of this chapter, and the next couple of chapters, is to outline a speculative way to save moderation by abandoning modelism. The idea is to do metamathematics without semantics, by working deductively in a higher-order logic. In this chapter, the focus is on the internal categoricity of arithmetic. After formalising an internal notion of a model of the Peano axioms, we show how to internalise Dedekind’s Categority Theorem. The resulting “intolerance” of Peano arithmetic provides internalists with a way to draw the distinction between algebraic and univocal theories. In the appendices, we discuss how this relates to Parsons’ important work, and establish a certain dependence of the internal categoricity theorem on higher-order logic.

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41

Wagner, R. Harrison. Rationalism and Security. Oxford University Press, 2017. http://dx.doi.org/10.1093/acrefore/9780190846626.013.285.

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In 1969, the game theorist John Harsanyi wrote an article criticizing the two main postulates of the general theory of social behavior prevalent at the time: the functionalist approach to the explanation of social institutions and the conformist approach to the explanation of individual behavior. According to Harsanyi, functionalist and conformist theories overstated the degree of consensus in societies, could not account for change, and described observed behavior without explaining it. Harsanyi proposed an alternative approach provided by theories based on the concept of rational choice (rational behavior, or rational decision-making). His goal was to develop a hypothetico-deductive theory explaining (and possibly predicting) a large number of empirical facts from a few relatively simple theoretical assumptions or axioms. Among students of international politics, Harsanyi’s approach sparked a controversy about rationalism. However, some critics of rationalism do not distinguish clearly between the interest-based theories Harsanyi criticized and the rational choice methods he advocated, and some even confuse both with neoclassical economics. In order to understand the issues raised in the controversy about rationalism, it is helpful to look at interest-based theories of politics and their relation to neoclassical economics. Game theory has provided a useful framework for the intellectual agenda outlined by Harsanyi, especially in the area of international security.

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42

Geometric Set Theory. American Mathematical Society, 2020.

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