Tytuł artykułu
Autorzy
Wybrane pełne teksty z tego czasopisma
Identyfikatory
Warianty tytułu
Języki publikacji
Abstrakty
Let ZC - I (respectively, ZF - Ί ) be the theory obtained by deleting the axiom of infinity from the usual list of axioms for Zermelo set theory with choice (respectively, the usual list of axioms for Zermelo-Fraenkel set theory). In this note, we present a collection of sentences φ() for which (ZC - Ί) + φ() (respectively, (ZF - Ί)+φ()) proves the existence of an infinite set.
Czasopismo
Rocznik
Tom
Strony
43--56
Opis fizyczny
Bibliogr. 26 poz.
Twórcy
autor
- Department of Mathematics The University of Colorado Colorado Springs, CO 80918 USA
Bibliografia
- [1] S. Baratella, R. Ferro, A theory of sets with the negation of the axiom of infinity, Math. Logic Quart. 39:3 (1993), 338-352.
- [2] N. V. Belyakin, S.P. Odintsov, Nonstandard analysis and the axiom of determinacy, Algebra i Logika 32:6 (1993), 607-617, 711; translation in Algebra and Logic 32:6 (1994), 328-333.
- [3] P. Bernays, A system of axiomatic set theory. III. Infinity and enumerability. Analysis, J. Symbolic Logic 7 (1942), 65-89.
- [4] E. W. Beth, Axiomatique de la thfieorie des ensembles sans axiome de l'infini, Bull. Soc. Math. Belg. 16 (1964), 127-136.
- [5] J. Drabbe, Les axiomes de l'infini dans al theorie des ensembles sans axiome de substitution, C.R. Acad. Sci. Paris Sfier. A-B 268 (1969), A137-A138.
- [6] J. Degen, Some aspects and examples of infinity notions, Math. Logic. Quart. 40:1 (1994), 111-124.
- [7] A. Enayat, J. Schmerl, A. Visser, !-models of finite set theory, in: Set theory, arithmetic, and foundations of mathematics: theorems, philosophies, pp. 43-65, Lect. Notes Log., 36, Assoc. Symbol. Logic, La Jolla, CA, 2011.
- [8] H. Enderton, Elements of set theory, Academic Press [Harcourt Brace Jovanovish, Publishers], New York-London, 1977.
- [9] T. Jech, Set theory. The third millennium edition, revised and expanded, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003.
- [10] A. Kanamori, D. Gabbay, J. Woods (editors), Handbook of the history of logic: sets and extensions in the twentieth century, Elsevier, 2012.
- [11] R. Kay, T. Wong, On interpretations of arithmetic and set theory, Notre Dame J. Formal Logic 48:4 (2007), 497-510.
- [12] G. B. Keene, Abstract sets and finite ordinals. An introduction to the study of set theory, International Series of Monographs on Pure and Applied Mathematics, Vol. 23 Pergamon Press, New York-Oxford-London-Paris, 1961.
- [13] L. Kirby, Substandard models of finite set theory, Math. Log. Q. 56:6 (2010), 631-642.
- [14] A. Levi, The independence of various definitions of finiteness, Fundamenta Mathematicae 46 (1958), 1-13.
- [15] A. Mostowski, On the independence of definitions of finiteness in a system of logic, Ann. Soc. Math. Polon. Ser. II (1938), 1-54.
- [16] C. Parsons, Developing arithmetic in set theory without infinity: some historical remarks, Hist. Philos. Logic 8:2 (1987), 201-213.
- [17] G. Peano, Arithmetices principia, nova methodo exposita, 1889, 83-97.
- [18] L. Polkowski, Rough sets. Mathematical foundations. Advances in Soft Computing. Physica-Verlag, Heidelberg, 2002.
- [19] K. Sato, The strength of extensionality II - weak set theories without infinity, Ann. Pure Appl. Logic 162:8 (2011), 579-646.
- [20] A. Sochor, The alternative set theory. Set theory and hierarchy theory, Proc. Second Conf., Bierutowice, 1975), pp. 259-271. Lecture Notes in Math., Vol. 537, Springer, Berlin, 1976.
- [21] L. Spilsiak, P. Vojtas, Dependences between definitions of finiteness, Czech. Math. J. 38 (1988), 389-397.
- [22] A. Tarski, On finite sets, Fundamenta Mathematicae Volume 6H (1924), 45-95.
- [23] P. Vejjajiva, S. Panasawatwong, A note on weakly Dedekind finite sets, Notre Dame J. Form. Log. 55:3 (2014), 413-417.
- [24] P. Vopenka, Axiome der Theorie endlichter Mengen, Casopis Pest. Mat. 89 (1964), 312-317.
- [25] A. C. Walczak-Typke, The first-order theory of weakly Dedekind-finite sets, J. Symbolic Logic 70:4 (2005), 1161-1170.
- [26] E. Zermelo, Untersuchungen ber die Grundlagen der Mengenlehre I, Mathematische Annalen 65:2 (1908), 261-281.
Uwagi
PL
Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-101c27bb-337b-44ae-8cba-27ef021771a4