Relevant bibliographies by topics / Transfinite numbers

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  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers

Academic literature on the topic 'Transfinite numbers'

Author: Grafiati
Published: 4 June 2021
Last updated: 1 August 2024

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Journal articles on the topic "Transfinite numbers"

1

Agustito, Denik, Krida Singgih Kuncoro, Istiqomah Istiqomah, and Agus Hendriyanto. "Construction of Ordinal Numbers and Arithmetic of Ordinal Numbers." JTAM (Jurnal Teori dan Aplikasi Matematika) 7, no. 3 (2023): 781. http://dx.doi.org/10.31764/jtam.v7i3.15039.

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The purpose of this paper is to introduce the idea of how to construct transfinite numbers and study transfinite arithmetic. The research method used is a literature review, which involves collecting various sources such as scientific papers and books related to Cantorian set theory, infinity, ordinal or transfinite arithmetic, as well as the connection between infinity and theology. The study also involves constructing the objects of study, namely ordinal numbers such as finite ordinals and transfinite ordinals, and examining their arithmetic properties. The results of this research include t
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Srivastava, S. M. "Transfinite numbers." Resonance 2, no. 3 (1997): 58–68. http://dx.doi.org/10.1007/bf02838969.

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Weinert, Thilo. "Transfinite Ramsey Numbers." Electronic Notes in Discrete Mathematics 43 (September 2013): 231–34. http://dx.doi.org/10.1016/j.endm.2013.07.038.

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Álvarez J., Carlos. "De la determinación del infinito a la inaccesibilidad en los cardinales transfinitos." Crítica (México D. F. En línea) 26, no. 78 (1994): 27–71. http://dx.doi.org/10.22201/iifs.18704905e.1994.956.

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In this paper I deal with two problems in mathematical philosophy: the (very old) question about the nature of infinity, and the possible answer to this question after Cantor’s theory of transfinite numbers. Cantor was the first to consider that his transfinite numbers theory allows to speak, within mathematics, of an actual infinite and allows to leave behind the Aristotelian statement that infinity exists only as potential infinity. In the first part of this paper I discuss Cantorian theory of transfinite numbers and his particular point of view about this matter. But the development of the
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Velev, Milen V. "Infinite multisets: Basic properties and cardinality." Notes on Number Theory and Discrete Mathematics 30, no. 2 (2024): 335–56. http://dx.doi.org/10.7546/nntdm.2024.30.2.335-356.

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This research work presents the topic of infinite multisets, their basic properties and cardinality from a somewhat different perspective. In this work, a new property of multisets, ‘m-cardinality’, is defined using multiset functions. M-cardinality unifies and generalizes the definitions of cardinality, injection, bijection, and surjection to apply to multisets. M-cardinality takes into account both the number of distinct elements in a multiset and the number of copies of each element (i.e., the multiplicity of the elements). Based on m-cardinality, ‘m-cardinal numbers’ are defined as a gener
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WEBER, ZACH. "TRANSFINITE NUMBERS IN PARACONSISTENT SET THEORY." Review of Symbolic Logic 3, no. 1 (2010): 71–92. http://dx.doi.org/10.1017/s1755020309990281.

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This paper begins an axiomatic development of naive set theory—the consequences of a full comprehension principle—in a paraconsistent logic. Results divide into two sorts. There is classical recapture, where the main theorems of ordinal and Peano arithmetic are proved, showing that naive set theory can provide a foundation for standard mathematics. Then there are major extensions, including proofs of the famous paradoxes and the axiom of choice (in the form of the well-ordering principle). At the end I indicate how later developments of cardinal numbers will lead to Cantor’s theorem, the exist
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Beyer, W. A., and J. D. Louck. "Transfinite Function Iteration and Surreal Numbers." Advances in Applied Mathematics 18, no. 3 (1997): 333–50. http://dx.doi.org/10.1006/aama.1996.0513.

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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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Rioux, Jean W. "Cantor’s Transfinite Numbers and Traditional Objections to Actual Infinity." Thomist: A Speculative Quarterly Review 64, no. 1 (2000): 101–25. http://dx.doi.org/10.1353/tho.2000.0003.

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Pąk, Karol. "The Ring of Conway Numbers in Mizar." Formalized Mathematics 31, no. 1 (2023): 215–28. http://dx.doi.org/10.2478/forma-2023-0020.

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Summary Conway’s introduction to algebraic operations on surreal numbers with a rather simple definition. However, he combines recursion with Conway’s induction on surreal numbers, more formally he combines transfinite induction-recursion with the properties of proper classes, which is diffcult to introduce formally. This article represents a further step in our ongoing e orts to investigate the possibilities offered by Mizar with Tarski-Grothendieck set theory [4] to introduce the algebraic structure of Conway numbers and to prove their ring character.
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Dissertations / Theses on the topic "Transfinite numbers"

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Carey, Patrick Hatfield. "Beyond Infinity: Georg Cantor and Leopold Kronecker's Dispute over Transfinite Numbers." Thesis, Boston College, 2005. http://hdl.handle.net/2345/481.

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Thesis advisor: Patrick Byrne<br>In the late 19th century, Georg Cantor opened up the mathematical field of set theory with his development of transfinite numbers. In his radical departure from previous notions of infinity espoused by both mathematicians and philosophers, Cantor created new notions of transcendence in order to clearly described infinities of different sizes. Leading the opposition against Cantor's theory was Leopold Kronecker, Cantor's former mentor and the leading contemporary German mathematician. In their lifelong dispute over the transfinite numbers emerge philosophical di
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Nestra, Härmel. "Iteratively defined transfinite trace semantics and program slicing with respect to them /." Online version, 2006. http://dspace.utlib.ee/dspace/bitstream/10062/1109/5/nestraharmel.pdf.

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JUNIOR, WALTER GOMIDE DO NASCIMENTO. "THE INFINITE COUNTED BY GOD: A DEDEKINDIAN INTERPRETATION OF CANTOR S TRANSFINITE ORDINAL NUMBER CONCEPT." PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 2006. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=9031@1.

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CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICO<br>Subjacente à teoria dos números ordinais transfinitos de Cantor, há uma perspectiva finitista. Segundo tal perspectiva, Deus pode bem ordenar o infinito usando, para tanto, de procedimentos similares ao ato de contar, entendido como o ato de bem ordenar o finito. Desta maneira, um diálogo natural entre Cantor e Dedekind torna-se possível, dado que Dedekind foi o primeiro a tratar o ato de contar como sendo, em sua essência, uma forma de bem ordenar o mundo espáciotemporal pelos números naturais. Nesta tese, o conceito de nú
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Segura, Lorena. "Consideraciones epistemológicas sobre algunos ítems de los fundamentos de las matemáticas." Doctoral thesis, Universidad de Alicante, 2018. http://hdl.handle.net/10045/80507.

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Tomando como punto de partida el proceso revisión de los fundamentos matemáticos llevado a cabo durante el siglo XIX, este estudio se centra en uno de los conceptos matemáticos más importantes: el infinito. Es innegable la importancia de este concepto en el avance de las Matemáticas y es fácil encontrar ejemplos matemáticos en los que interviene (definición de límite, definición de derivada, definición de integral de Riemann, entre otras). Debido a que algunas de las paradojas y contradicciones originadas por la falta de rigor en las Matemáticas están relacionadas con este concepto, se comienz
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"Finitism and the Cantorian theory of numbers." 2008. http://library.cuhk.edu.hk/record=b5896870.

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Lie, Nga Sze.<br>Thesis (M.Phil.)--Chinese University of Hong Kong, 2008.<br>Includes bibliographical references (leaves 103-111).<br>Abstracts in English and Chinese.<br>Abstract --- p.i<br>Chapter 1 --- Introduction and Preliminary Discussions --- p.1<br>Chapter 1.1 --- Introduction --- p.1<br>Chapter 1.1.1 --- Overview of the Thesis --- p.2<br>Chapter 1.1.2 --- Background --- p.3<br>Chapter 1.1.3 --- About Chapter 3: Details of the Theory --- p.4<br>Chapter 1.1.4 --- About Chapter 4: Defects of the Theory --- p.7<br>Chapter 1.2 --- Preliminary Discussions --- p.12<br>Chapter 1.2.1 -
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Books on the topic "Transfinite numbers"

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Shakunle, Lere. Logic numbers and the continuum hypothesis. Journal of Transfigural Mathematics, 1991.

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Reischer, Corina. Nombres finis & nombres transfinis. Presses de l'Université du Québec, 2002.

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Huntington, E. V. The continuum, and other types of serial order: With an introduction to Cantor's transfinite numbers. Dover Publications, 2003.

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Lauria, Philippe. Cantor et le transfini: Mathématique et ontologie. Harmattan, 2004.

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Hallett, Michael. Cantorian set theory and limitation of size. Clarendon, 1986.

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Sinkevich, G. I. Georg Kantor & polʹskai︠a︡ shkola teorii mnozhestv: Monografii︠a︡. Sankt-Peterburgskiĭ gosudarstvennyĭ arkhitekturno-stroitelʹnyĭ universitet, 2012.

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Grantham, S. B. Galvin's "racing pawns" game and a well-ordering of trees. American Mathematical Society, 1985.

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Katasonov, V. N. Borovshiĭsi͡a︡ s beskonechnym: Filosofsko-religioznye aspekty genezisa teorii mnozhestv G. Kantora. Martis, 1999.

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Bachmann, Heinz. Transfinite Zahlen. Springer, 2012.

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Zemanian, Armen H. Graphs and Networks: Transfinite and Nonstandard. Birkhäuser Boston, 2004.

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Book chapters on the topic "Transfinite numbers"

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Honig, William M. "QM Axiom Representations with Imaginary & Transfinite Numbers and Exponentials." In The Concept of Probability. Springer Netherlands, 1989. http://dx.doi.org/10.1007/978-94-009-1175-8_32.

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Honig, William M. "Logical Meanings in Quantum Mechanics for Axioms and for Imaginary and Transfinite Numbers and Exponentials." In NATO ASI Series. Springer US, 1987. http://dx.doi.org/10.1007/978-1-4684-5386-7_15.

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"Transfinite Cardinal Numbers." In Introductory Concepts for Abstract Mathematics. Chapman and Hall/CRC, 2018. http://dx.doi.org/10.1201/9781315273761-36.

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"Ordinal Numbers and the Transfinite." In Set Theory and Foundations of Mathematics: An Introduction to Mathematical Logic. WORLD SCIENTIFIC, 2020. http://dx.doi.org/10.1142/9789811201936_0005.

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Giaquinto, M. "Numbers and Classes." In The Search for Certainty. Oxford University PressOxford, 2002. http://dx.doi.org/10.1093/oso/9780198752448.003.0003.

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Abstract The conceptual clarification described in the previous chapter dispenses with infinitesimals. But that was not felt to be good enough: some account of the real numbers was also needed. This chapter first explains that need and presents the two best known accounts of real numbers. That is followed by a sketch of the way in which the ideas for the transfinite ordinal and cardinal number systems grew out of the study of classes of points, and the rudiments of those number systems are presented. Finally, we look at accounts of the natural numbers. This will complete an account of the nineteenth-century movement to clarify mathematical concepts and to place the accepted practices and theorems of mathematics on a sound footing.
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Kline, Morris. "The Foundations of the Real and Transfinite Numbers." In Mathematical Thought from Ancient to Modern Times. Oxford University PressNew York, NY, 1990. http://dx.doi.org/10.1093/oso/9780195061376.003.0008.

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Abstract One of the most surprising facts in the history of mathematics is that the logical foundation of the real number system was not erected until the late nineteenth century. Up to that time not even the simplest properties of positive and negative rational numbers and irrational numbers were logically established, nor were these numbers defined. Even the logical foundation of complex numbers had not been long in existence (Chap. 32, sec. 1), and that foundation presupposed the real number system. In view of the extensive development of algebra and analysis, all of which utilized the real numbers, the failure to consider the precise structure and properties of the real numbers shows how illogically mathematics progresses. The intuitive understanding of these numbers seemed adequate and mathematicians were content to operate on this basis.
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Giaquinto, M. "Cantor’s Approach to the 2 Class Paradoxes." In The Search for Certainty. Oxford University PressOxford, 2002. http://dx.doi.org/10.1093/oso/9780198752448.003.0006.

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Abstract Cantor was the first to discover class paradoxes, but they did not strike him as paradoxical. Rather, he took them to confirm a view he had come to years before, that some totalities cannot be treated as objects of mathematical study. In the course of developing the theory of infinite sets, in particular the theory of infinite ordinal and cardinal numbers that he had created,1 he naturally reflected on the totality of ordinal numbers and the totality of cardinal numbers, and he came to view them as importantly unlike other infinite totalities. It had been traditional to distinguish between finite and infinite, the former being increasable by addition, the latter not. Cantor challenged this tradition. Instead of a simple division into finite and infinite, Cantor’s theory led him to a tripartite classification: finite, transfinite, and absolutely infinite. A transfinite number is unlike a finite number in having infinitely many predecessors; but it is like a finite number in being increasable.
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Stewart, Ian. "7. Counting infinity." In Infinity: A Very Short Introduction. Oxford University Press, 2017. http://dx.doi.org/10.1093/actrade/9780198755234.003.0008.

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‘Counting infinity’ returns to the mathematics of infinity, discussing Cantor’s remarkable theory of how to count infinite sets, and the discovery that there are different sizes of infinity. For example, the set of all integers is infinite, and the set of all real numbers (infinite decimals) is infinite, but these infinities are fundamentally different, and there are more real numbers than integers. The ‘numbers’ here are called transfinite cardinals. For comparison, another way to assign numbers to infinite sets is mentioned, by placing them in order, leading to transfinite ordinals. It ends by asking whether the old philosophical distinction between actual and potential infinity is still relevant to modern mathematics, and examining the meaning of mathematical existence.
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"5. Subversion Infinite Series and Transfinite Numbers in Borges's Fictions." In The Cosmic Web. Cornell University Press, 2018. http://dx.doi.org/10.7591/9781501722974-007.

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Tait, William. "Constructing Cardinals from Below." In The Provenance of Pure Reason. Oxford University PressNew York, NY, 2005. http://dx.doi.org/10.1093/oso/9780195141924.003.0007.

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Abstract The totality n of transfinite numbers was first introduced in Cantor (1883) by means of the principle If the initial segment E of n is a set, then it has a least strict upper bound S(E) E n. Thus, for numbers a = S(E) and /3 = S(E’), a &amp;lt; /3 iff a E E’; a = /3 iff E = E’; 8(0) is the least number O (although Cantor himself took the least number to be 1); if E has a greatest element ‘Y, then a is its successor ‘Y +1; and if E is nonnull and has no greatest element, then a is the least upper bound of E. The problem with the definition, of course, is in determining what it means for an initial segment to be a set. Obviously, not all of them are; for the totality n of all numbers is an initial segment, but to admit it as a set would yield S(O) &amp;lt; S(O), contradicting the assumption that n is well-ordered by &amp;lt;. Cantor himself understood this already in 1883. In his earlier writings [e.g., (1882)], he had essentially defined a set “in some conceptual sphere” such as arithmetic or geometry, to be the extension of a well-defined property. But in these cases, he was considering sets of objects of some type A, where being an object of type A is itself not defined in terms of the notion of a set of objects of type A. But with his definition of the transfinite numbers, an entirely novel situation arises: the definition of n depends on the notion of a subset of n. Accordingly, he abandoned his earlier definition of set in (1883), and in his later writings, he distinguished between those initial segments which are sets and those which are not in terms of his concept of “consistent multiplicity”; but that is just naming the problem, not solving it.
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Conference papers on the topic "Transfinite numbers"

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Usab, W. J., and J. M. Verdon. "Advances in the Numerical Analysis of Linearized Unsteady Cascade Flows." In ASME 1990 International Gas Turbine and Aeroengine Congress and Exposition. American Society of Mechanical Engineers, 1990. http://dx.doi.org/10.1115/90-gt-011.

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This paper describes two new developments in the numerical analysis of linearized unsteady cascade flows, that have been motivated by the need for an accurate analytical procedure for predicting the onset of flutter in highly loaded compressors. In previous work, results were determined using a two-step or single-pass procedure in which a solution was first determined on a rectilinear-type cascade mesh to determine the unsteady flow over an extended blade-passage solution domain and then on a polar-type local mesh to resolve the unsteady flow in high-gradient regions. In the present effort a c
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