CANTORIAN SET THEORY

2018 ◽  
Vol 24 (4) ◽  
pp. 393-451
Author(s):  
ALEX OLIVER ◽  
TIMOTHY SMILEY
Keyword(s):  
Set Theory ◽  
Definition Of ◽  
Almost All ◽  
Ordered Pairs

AbstractAlmost all set theorists pay at least lip service to Cantor’s definition of a set as a collection of many things into one whole; but empty and singleton sets do not fit with it. Adapting Dana Scott’s axiomatization of the cumulative theory of types, we present a ‘Cantorian’ system which excludes these anomalous sets. We investigate the consequences of their omission, examining their claim to a place on grounds of convenience, and asking whether their absence is an obstacle to the theory’s ability to represent ordered pairs or to support the arithmetization of analysis or the development of the theory of cardinals and ordinals.

scholarly journals Final coalgebras as greatest fixed points in ZF set theory

1999 ◽  
Vol 9 (5) ◽  
pp. 545-567 ◽  
Author(s):  
LAWRENCE C. PAULSON
Keyword(s):  
Fixed Points ◽  
Set Theory ◽  
Theorem Prover ◽  
Infinite Length ◽  
Special Cases ◽  
Definition Of ◽  
Recursive Definitions ◽  
Ordered Pairs

A special final coalgebra theorem, in the style of Aczel (1988), is proved within standard Zermelo–Fraenkel set theory. Aczel's Anti-Foundation Axiom is replaced by a variant definition of function that admits non-well-founded constructions. Variant ordered pairs and tuples, of possibly infinite length, are special cases of variant functions. Analogues of Aczel's solution and substitution lemmas are proved in the style of Rutten and Turi (1993). The approach is less general than Aczel's, but the treatment of non-well-founded objects is simple and concrete. The final coalgebra of a functor is its greatest fixedpoint.Compared with previous work (Paulson, 1995a), iterated substitutions and solutions are considered, as well as final coalgebras defined with respect to parameters. The disjoint sum construction is replaced by a smoother treatment of urelements that simplifies many of the derivations.The theory facilitates machine implementation of recursive definitions by letting both inductive and coinductive definitions be represented as fixed points. It has already been applied to the theorem prover Isabelle (Paulson, 1994).

Set—Theoretical Representations of Ordered Pairs and Their Adequacy for the Logic of Relations

1982 ◽  
Vol 12 (2) ◽  
pp. 353-374 ◽  
Author(s):  
Randall R. Dipert
Keyword(s):  
Twentieth Century ◽  
Mathematical Logic ◽  
Set Theory ◽  
Foundations Of Mathematics ◽  
Norbert Wiener ◽  
Precise Formulation ◽  
Definition Of ◽  
Ontological Economy ◽  
And Function ◽  
Ordered Pairs

One of the most significant discoveries of early twentieth century mathematical logic was a workable definition of ‘ordered pair’ totally within set theory. Norbert Wiener, and independently Casimir Kuratowski, are usually credited with this discovery. A definition of ‘ordered pair’ held the key to the precise formulation of the notions of ‘relation’ and ‘function’ — both of which are probably indispensable for an understanding of the foundations of mathematics. The set-theoretic definition of ‘ordered pair’ thus turned out to be a key victory for logicism, providing one admits set theory is logic. The definition also was instrumental in achieving the appearance of ontological economy — since it seemed only sets were needed — although this feature was emphasized only later.

Set theory with a filter quantifier

1983 ◽  
Vol 48 (2) ◽  
pp. 263-287 ◽  
Author(s):  
Matt Kaufmann
Keyword(s):  
Set Theory ◽  
Regular Cardinal ◽  
Completeness Theorem ◽  
Large Cardinal ◽  
Second Order ◽  
Class Theory ◽  
Natural Extensions ◽  
Definition Of ◽  
Almost All

The incompleteness of ZF set theory leads one to look for natural extensions of ZF in which one can prove statements independent of ZF which appear to be “true”. One approach has been to add large cardinal axioms. Or, one can investigate second-order expansions like Kelley-Morse class theory, KM. In this paper we look at a set theory ZF(aa), with an added quantifier aa which ranges over ordinals. The “aa” stands for “almost all”, and although we will consider interpretations in terms of the closed unbounded filter on a regular cardinal κ, we will consider other interpretations also.We start in §1 by giving the axioms for the theory ZF(aa) and presenting a completeness theorem which gives a model-theoretic definition of ZF(aa). In §2 we investigate set theory with a satisfaction predicate and interpret it in a fragment of ZF(aa). In §3 we generalize the methods of §2 to obtain a hierarchy of satisfaction predicates. We use these predicates to prove reflection theorems, as well as to prove the consistency of certain fragments of ZF(aa). Next, in §4 we discuss expandability of models of ZF to models of fragments of ZF(aa) and of Kelley-Morse. We conclude in §5 with a discussion of an extension ZF(aa) + DET of ZF(aa) in which the quantifier aa is self-dual.

scholarly journals Frege’s puzzle and the ex ante Pareto principle

2020 ◽  
Author(s):  
Anna Mahtani
Keyword(s):  
Policy Choice ◽  
Choice Situation ◽  
Pareto Principle ◽  
Social Situations ◽  
Central Importance ◽  
Ex Ante ◽  
Definition Of ◽  
Almost All ◽  
Rational Behaviour ◽  
Fixed Population

Abstract The ex ante Pareto principle has an intuitive pull, and it has been a principle of central importance since Harsanyi’s defence of utilitarianism (to be found in e.g. Harsanyi, Rational behaviour and bargaining equilibrium in games and social situations. CUP, Cambridge, 1977). The principle has been used to criticize and refine a range of positions in welfare economics, including egalitarianism and prioritarianism. But this principle faces a serious problem. I have argued elsewhere (Mahtani, J Philos 114(6):303-323 2017) that the concept of ex ante Pareto superiority is not well defined, because its application in a choice situation concerning a fixed population can depend on how the members of that population are designated. I show in this paper that in almost all cases of policy choice, there will be numerous sets of rival designators for the same fixed population. I explore two ways that we might complete the definition of ex ante Pareto superiority. I call these the ‘supervaluationist’ reading and the ‘subvaluationist’ reading. I reject the subvaluationist reading as uncharitable, and argue that the supervaluationist reading is the most promising interpretation of the ex ante Pareto principle. I end by exploring some of the implications of this principle for prioritarianism and egalitarianism.

scholarly journals Legal status of cryptocurrency in Asian countries

2021 ◽  
Vol 66 ◽  
pp. 96-102
Author(s):  
V.M. Logoida
Keyword(s):  
Civil Rights ◽  
Government Regulation ◽  
Legal Status ◽  
Legal Regulation ◽  
Intangible Asset ◽  
Asian Countries ◽  
Legal Nature ◽  
Liberal Approach ◽  
Definition Of ◽  
Almost All

The article is devoted to the study of the experience of legal regulation of the legal status of cryptocurrencies and transactions with them in Asian countries (except for the People's Republic of China and Asian countries - members of the Commonwealth of Independent States, as the author examined them in separate publications). In the article the author, based on the study of regulations, administrative and judicial practice of all major countries in this part of the world, emphasizes the divergent trends in cryptocurrency transactions regulation in the region, when some countries move from a liberal approach to the use of cryptocurrencies to their total ban and vice versa. It is noted that almost all countries in the region give a legal assessment of the payment function of cryptocurrencies, using regulatory or prohibitive approaches, depending on the chosen policy, which indirectly confirms their understanding of the legal nature of cryptocurrencies primarily as a means of payment. At the same time, these countries not only categorically distinguish cryptocurrencies from fiat money issued by central banks, but also mostly avoid the official definition of cryptocurrency as private (decentralized) cash, preferring to qualify them as an intangible asset, virtual asset, digital asset, financial value and even a good or service, which is currently a kind of compromise between political expediency and economic realities. The author also notes that the Asian region is characterized by very active attempts to resolve the legal status of cryptocurrencies at the legislative level, and not just administrative or judicial response to the actual legal relationship, although the progress of different countries in this matter is different. As a result, the author concludes that in the Asian countries considered in the article, there is no same view on the legal nature of cryptocurrency, its qualification as an object of civil rights, and ways to regulate transactions with it (libertarian approach, positive-cryptocurrency approach but with detailed government regulation and control or a completely restrictive policy in relation to the cryptocurrency market).

scholarly journals The Tax Cuts and Jobs Act and the Middle Class

2019 ◽  
Vol 26 (01) ◽  
pp. 53
Author(s):  
Deborah Combs ◽  
Brian Nichols
Keyword(s):  
Middle Class ◽  
The United States ◽  
Census Bureau ◽  
Tax Rates ◽  
United States Census ◽  
United States Census Bureau ◽  
Definition Of ◽  
Almost All ◽  
Jobs Act ◽  
Tax Cuts

This paper explores how the tax cuts and jobs act of 2017 impacts middle-class taxpayers by calculating the tax liability at different levels of income and deductions in 2017 versus 2018. The results confirm the statements supporting the positive effect of the tax change for the middle class. The tax cut and jobs act eliminates personal exemptions, changes the standard deductions at various incomes and family sizes, and lowers marginal tax rates. After providing details of the act, this research examines the definition of the U.S. middle class by using prior research from the Pew Research Center, the United States Census Bureau, and the federal reserve to determine which income levels are attributable to the middle class. Then the tax liability for these income classes is calculated for single and married filing jointly taxpayers in both 2017 and 2018 to determine if the tax cuts and jobs act reduces the tax liability for the middle class. The results show that in almost all scenarios the tax liability in 2018 will be lower than in 2017, regardless of whether standard or itemized deductions are taken. The marriage penalty is no longer applicable, and the new tax act provides a substantial benefit to large families

scholarly journals Fuzzy Soft Multiset Theory

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Shawkat Alkhazaleh ◽  
Abdul Razak Salleh
Keyword(s):  
Decision Making ◽  
Set Theory ◽  
Fuzzy Set ◽  
Soft Set ◽  
Mathematical Tool ◽  
Basic Properties ◽  
Definition Of

In 1999 Molodtsov introduced the concept of soft set theory as a general mathematical tool for dealing with uncertainty. Alkhazaleh et al. in 2011 introduced the definition of a soft multiset as a generalization of Molodtsov's soft set. In this paper we give the definition of fuzzy soft multiset as a combination of soft multiset and fuzzy set and study its properties and operations. We give examples for these concepts. Basic properties of the operations are also given. An application of this theory in decision-making problems is shown.

Historical Account or Discourse on Identity? A Reexamination of Fulbe Hegemony and Autochthonous Submission in Banyo

1998 ◽  
Vol 25 ◽  
pp. 93-110 ◽  
Author(s):  
Quentin Gausset
Keyword(s):  
Nineteenth Century ◽  
The Other ◽  
Islamic State ◽  
Local Populations ◽  
Northern Cameroon ◽  
Definition Of ◽  
Historical Accounts ◽  
In The Beginning ◽  
Almost All ◽  
The Military

Traditional accounts of the nineteenth-century Fulbe conquest in northern Cameroon tell roughly the same story: following the example of Usman Dan Fodio in Nigeria, the Fulbe of Cameroon organized in the beginning of the nineteenth century a “jihad” or a “holy war” against the local pagan populations to convert them to Islam and create an Islamic state. The divisions among the local populations and the military superiority of the Fulbe allowed them to conquer almost all northern Cameroon. They forced those who submitted to give an annual tribute of goods and servants, and they raided the other groups. In these traditional accounts the Fulbe are presented as unchallenged masters, while the local populations are depicted as slaves who were powerless over their fate; their role in the conquest of the region and in the administration of the new political order is supposed to have been insignificant.I will show that, on the contrary, in the area of Banyo the Wawa and Bute played a crucial role in the conquest of the sultanate and in its administration. I will then re-examine the cliche that all members of the local populations were the slaves of the Fulbe by distinguishing the fate of the Wawa and Bute on one side from that of the Kwanja and Mambila on the other, and by showing the importance of the Fulbe's identity in shaping the definition of slavery. Finally I will argue that, if the historical accounts found in the scientific literature invariably insist on Fulbe hegemony and minimize the role played by the local populations, it is because those accounts are often based on Fulbe traditions, and because these traditions are remodeled by the Fulbe in order to correspond to their discourse on identity.

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