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"Mathematics is the Logic of the Infinite": Zermelo's Project of Infinitary Logic

Jerzy Pogonowski
2021 Studies in Logic, Grammar and Rhetoric  
In this paper I discuss Ernst Zermelo's ideas concerning the possibility of developing a system of infinitary logic that, in his opinion, should be suitable for mathematical inferences.  ...  I also stress the fact that the second axiomatization of set theory provided by Zermelo in 1930 involved the use of extremal axioms of a very specific sort.1  ...  An approach to his life and work. Springer-Verlag: Berlin and Heidelberg. Ebbinghaus, H.D., Fraser, C.G., Kanamori, A. 2010. Ernst Zermelo. Collected Works. Gesammelte Werke.  ... 
doi:10.2478/slgr-2021-0042 fatcat:nnowhogl3bhklm234xa7dion7u

Pro and Contra Hilbert: Zermelo's Set Theories

Volker Peckhaus
2005 Philosophia Scientiæ  
His work at that time was contra Hilbert. * I would like to thank Christian Thiel (Erlangen), Heinz-Dieter Ebbinghaus (Freiburg i.  ...  In the first period Hilbert elaborated his early axiomatic program, and Zermelo's work is clearly along the lines proposed by Hilbert, it is pro Hilbert.  ...  His main working fields were applied mathematics and theoretical physics, an interest which he kept all his life long.  ... 
doi:10.4000/philosophiascientiae.390 fatcat:6utjeiorrnbphepdncmmwswxai

Game, set, and graph

Norman Biggs
2018 BSHM Bulletin  
Here it will be argued that the transformation was the result of the work of mathematicians, such as Ernst Zermelo, John von Neumann and Dénes Kőnig, who also contributed to two other areas of mathematics  ...  It began as an amusing pastime, and ended as a major branch of mathematical research and a key paradigm of economic theory.  ...  Acknowledgement The author wishes to acknowledge helpful conversations with Professor Bernhard von Stengel and constructive comments from an anonymous referee. Bibliography  ... 
doi:10.1080/17498430.2018.1504458 fatcat:rythwlx7jzdgfdftmd54xqzvji

The Empty Set, The Singleton, and the Ordered Pair

Akihiro Kanamori
2003 Bulletin of Symbolic Logic  
These notions are the simplest building locks in the abstract, generative conception of sets advanced by the initial axiomatization of Ernst Zermelo [1908a] and are quickly assimilated long before the  ...  Circling back to the beginning, we shall see how central the empty set, the singleton, and the ordered pair were, after all.  ...  Pierce and Ernst Schröder, adopted and adapted Boole's "0".  ... 
doi:10.2178/bsl/1058448674 fatcat:k4hxto6nobgcnfpmr6vbcg3cce

In Memoriam: Ernst Specker 1920–2011

Erwin Engeler
2012 Bulletin of Symbolic Logic  
Born in Zurich, Switzerland, February 11, 1920, Ernst Paul Specker spent most of his life in that city, contributing decisively to the continuing international renown of the ETH in mathematics and logic  ...  -There were four international conferences dedicated to his work, on the occasion of his last round birthdays.  ... 
doi:10.2178/bsl/1344861890 fatcat:7p45kluzqbcw3eqegs54lv57ma

Küme Aksiyomları Kullanılarak Seçme Fonksiyonunun ve İyi Sıralı Sayılabilir Sonsuz Kümelerin Öğretimi

Burcu NİŞANCI TÜRKMEN-
2020 Turkish Studies - Educational Sciences  
In other words, a choice function definition application has been made for an infinite set. Rather than the real method using a very abstract approach, appropriate approaches have been used.  ...  This theory, which helps to reveal almost all mathematical problems and develop solution mathods with Zermelo-Fraenkel set theory, has a special importance in terms of building the base of set theory.  ...  Zermelo based his argument on a powerful new, and suspect, device, the axiom of choice.  ... 
doi:10.29228/turkishstudies.40280 fatcat:4eb6mcnjrvgbjimgkbwlayfqme

The new axiom of set theory and Bell inequality [article]

Kajetan Guz
2016 arXiv   pre-print
The new axiom emerges from the historical and philosophical analysis of set theory.  ...  The new axiom in set theory, axiom of the empty sets, allows another way to formulate Bell inequality in quantum mechanics.  ...  Until the end of his life, Cantor would struggle with the task of giving a philosophical justification of his theory based on mathematics.  ... 
arXiv:1603.08916v1 fatcat:ajvuf5yolbdclmdfyx5vz2tlqm

WE HOLD THESE TRUTHS TO BE SELF-EVIDENT: BUT WHAT DO WE MEAN BY THAT?

STEWART SHAPIRO
2009 The Review of Symbolic Logic  
This talk explores various notions of self-evidence, and the role they play in various foundational systems, notably those of Frege and Zermelo.  ...  At the beginning of Die Grundlagen der Arithmetik ( §2) [1884], Frege observes that "it is in the nature of mathematics to prefer proof, where proof is possible".  ...  A proposition can start life as a working hypothesis, and can later become an established belief or a known fact if it proves fruitful, serving an essential role in a successful system.  ... 
doi:10.1017/s175502030909011x fatcat:k2o72s4uljbgre3hyphz2uxuuu

Ludwig Edward Boltzmann [article]

S.Rajasekar, N.Athavan
2006 arXiv   pre-print
We point out his some other contributions in physics, characteristics of his work, his strong support on atomism, character of his personality and relationship with his students and final part of his life  ...  In this manuscript we present a brief life history of Ludwig Edward Boltzmann and his achivements.  ...  He went to the University of Leipzig in 1900 but returned to Vienna in 1902 because of his dislike of working with Ernst Mach.  ... 
arXiv:physics/0609047v1 fatcat:mpce5bsjqvgevo7vcnyqxj2e34

SET THEORY FROM CANTOR TO COHEN [chapter]

Akihiro Kanamori
2009 Philosophy of Mathematics  
There were two main junctures, the first being when Zermelo through his axiomatization shifted the notion of set from Cantor's range of inherently structured sets to sets solely structured by membership  ...  This view, from inside set theory and about itself, serves to shift the focus to  ...  and appears here with the permission of the Bulletin.  ... 
doi:10.1016/b978-0-444-51555-1.50014-6 fatcat:vnj2vdlpofdp7pg37tyiemdkiq

Set Theory from Cantor to Cohen [chapter]

Akihiro Kanamori
2012 Handbook of the History of Logic  
There were two main junctures, the first being when Zermelo through his axiomatization shifted the notion of set from Cantor's range of inherently structured sets to sets solely structured by membership  ...  This view, from inside set theory and about itself, serves to shift the focus to  ...  and appears here with the permission of the Bulletin.  ... 
doi:10.1016/b978-0-444-51621-3.50001-3 fatcat:ogdmpb6mlvgxbdvoh3jqq6umqe

Boltzmann's H-theorem, its limitations, and the birth of (fully) statistical mechanics [article]

Harvey R. Brown, Wayne Myrvold
2008 arXiv   pre-print
The little-cited 1896 (pre-recurrence) objection of Zermelo (similar to an 1889 argument due to Poincare) is also analysed.  ...  A comparison is made of the traditional Loschmidt (reversibility) and Zermelo (recurrence) objections to Boltzmann's H-theorem, and its simplified variant in the Ehrenfests' 1912 wind-tree model.  ...  at an exact location in his work seems somewhat arbitrary."  ... 
arXiv:0809.1304v1 fatcat:i7v53nd42zeqvehmn4hhx2y3ie

Boltzmann's H-theorem, its discontents, and the birth of statistical mechanics

Harvey R. Brown, Wayne Myrvold, Jos Uffink
2009 Studies in history and philosophy of modern physics  
at an exact location in his work seems somewhat arbitrary."  ...  considerations. 27 But it was the more detailed 1896 argument 28 by the young Ernst Zermelo (who would later achieve further fame through his work on the foundations of set theory) that drew Boltzmann's  ... 
doi:10.1016/j.shpsb.2009.03.003 fatcat:44kycfjyizb23j5wsftp6vubki

Peirce's Contributions to the 21st Century [chapter]

John Sowa
2006 Lecture Notes in Computer Science  
This talk surveys Peirce's approaches, their applications in modern systems, and the way his views of language and logic can clarify and support a more comprehensive view of natural language semantics.  ...  distance to stars. • Mathematics: Revised and extended his father's book on linear algebra. • Physics: Designed new instruments for measuring gravity. • Engineering: First to propose a wavelength of light  ...  It is one of the ways leading to an objective view of things and of human life. It is not an imitation but a discovery of reality.  ... 
doi:10.1007/11787181_5 fatcat:vzhp6dgvpjgelgyqqavknrjhia

Mathematical Knowledge : Motley and Complexity of Proof

Akihiro KANAMORI
2013 Annals of the Japan Association for Philosophy of Science  
Wit・h his Flyspeck Project Hales aspires to provide a complete formalizat・ion of a proof of the Kepler Conjecture and estimates it to be a 700e work-day, project・.  ...  The real precursor was Guiseppe Peano, who devised an eficient・ syrnbolic language and sought to provide, with his FbrTnulario Mathematico, an encyclopedia of all known ± brmulas and theorems of mathematics  ... 
doi:10.4288/jafpos.21.0_21 fatcat:2kq3e35l7bazhkdavqdr7xx474
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