Axioms of infinity

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Axioms of infinity Document created on October 5, 2018 Most recent update: October 5, 2018.

Articles on axioms of infinity Equivalent Forms of the Axiom of Infinity Paul Corazza http://pcorazza.lisco.com/drwu/excerpts/infinity-equivalents.pdf

Equivalents of Infinite Theorem, page 4. The theory ZFC−Infinity proves that the following statements are equivalent. (1) There is a set that contains every finite ordinal. (2) There is an inductive set. (3) There is a set that cannot be placed in 1-1 correspondence with any finite ordinal. (4) There is a set that has the same size as one of its proper subsets http://pcorazza.lisco.com/ http://pcorazza.lisco.com/mathPublications.html https://www.mum.edu/computer-science/faculty-profiles/paul-corazza/ Strong Axioms of Infinity and the Search for V Hugh Woodin, Professor Emeritus, University of California at Berkeley Plenary lecture International Congress of Mathematicians Hyderabad, August 19–27, 2010 Free copy: http://logic.harvard.edu/EFI_Woodin_StrongAxiomsOfInfinity https://math.berkeley.edu/people/faculty/w-hugh-woodin https://www.mathunion.org/fileadmin/IMU/ICM2010/offline/www.icm2010.in/wp-content/icmfiles/abstracts/InvitedAbstracts.pdf See pages 15--16 The Search for Mathematical Truth Hugh Woodin November 17, 2010. http://logic.harvard.edu/old-colloquia.php Abstract. The evident ubiquity of (formally) unsolvable problems in Set Theory argues for a re-examination of the conception of truth at least for Set Theory. The mathematical investigation of this apparently non-mathematical problem has led to a surprising series of mathematical conjectures, problems and successes. The stage is now set for either: (1) The validation of a single additional axiom which if added to the traditional ZFC axioms

would entirely eliminate the phenomena of unsolvability based on Cohen's method of forcing while preserving the fundamental strong axioms of infinity; or (2) The refutation from the same strong axioms of infinity of any such possibility based on anything like that currently imagined. In other words, Set Theory is facing a critical dichotomy in possible futures. Choice principles in elementary topology and analysis Horst Herrlich Comment. Math. Univ. Carolin. 38,3(1997)545–552 http://www.emis.de/journals/CMUC/pdf/cmuc9703/herrli.pdf The Equivalence of Some Axioms of Strong Infinity Luiz Paulo de Alcantara Studies in Logic and the Foundations of Mathematics Volume 89, pages 169—177 DOI: 10.1016/S0049-237X(08)70651-7 December 1977 https://www.sciencedirect.com/science/article/pii/S0049237X08706517 https://www.researchgate.net/publication/265446894_The_Equivalence_of_Some_Axioms_of_Strong_Infinity Axiom of infinity and construction of N F. Portal https://hal.archives-ouvertes.fr/hal-01162075/document

Related Web pages https://en.wikipedia.org/wiki/Dedekind-infinite_set https://en.wikipedia.org/wiki/Axiom_of_infinity#Formal_statement https://en.wikipedia.org/wiki/Axiom_of_infinity#An_apparently_weaker_version http://mathworld.wolfram.com/AxiomofInfinity.html http://mathworld.wolfram.com/InductiveSet.html https://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory https://en.wikipedia.org/wiki/Equinumerosity https://en.wikipedia.org/wiki/Bijection https://en.wikipedia.org/wiki/Surjective_function#Cardinality_of_the_domain_of_a_surjec tion https://en.wikipedia.org/wiki/Axiom_of_choice https://en.wikipedia.org/wiki/Axiom_of_countable_choice Major books on logic and set theory Introduction to Mathematical Logic Elliott Mendelson ISBN-13: 9781482237726 6th edition, 489 + xxiv pages, CRC Press, 2015 https://www.crcpress.com/Introduction-to-Mathematical-Logic-Sixth-Edition/Mendelson/p/book/9781482237726

Set Theory The Third Millennium Edition Thomas Jech Springer Monographs in Mathematics DOI: 10.1007/3-540-44761-X 3rd edition, 772 + xiv pages, Springer, 2003 http://www.springer.com/us/book/9783540440857 https://www.researchgate.net/publication/266248528_Set_Theory_The_Third_Millennium_Edition_Revised_and_Ex panded (The most accessible in spite of its very strong content, but I do NOT say nice to read.) Logic, Mathematics, and Computer Science Modern Foundations with Practical Applications Yves Nievergelt DOI: 10.1007/978-1-4939-3223-8 2nd edition, 391 + xii pages, Springer, 2015 https://www.springer.com/us/book/9781493932221 Theory of Sets Nicolas Bourbaki DOI: 10.1007/978-3-642-59309-3 ISBN-13: 9783540225256 414 + viii pages, Springer, 2004 https://www.springer.com/us/book/9783540225256 https://www.researchgate.net/publication/316801091_Theory_of_Sets Translation from French into English Théorie des ensembles, chapitres 1 à 4, 352 pages, 1970 352 pages, Éditions Hermann, 1970 https://en.wikipedia.org/wiki/Éléments_de_mathématique

Major book on the history of set theory Labyrinth of Thought A History of Set Theory and Its Role in Modern Mathematic José Ferreirós DOI: 10.1007/978-3-7643-8350-3 ISBN-13: 9783764383497 2nd edition, 466 + xxvi pages, Springer-Birkhäuser, 2007 https://www.springer.com/us/book/9783764383497 https://www.researchgate.net/publication/259983079_Labirynth_of_Thought_A_History_of_Set_Theory_and_its_Ro le_in_Modern_Mathematics The author had the courage to find the access to old archives, including old letters, and understand old German, old terminology, old notation. I cannot recommend this book enough.

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