On a Certain Notion of Finite and a Finiteness Class in Set Theory without Choice

• Autorzy: Herrlich, H. , Howard, P. , Tachtsis, E.
• Języki publikacji: EN

Abstrakty

EN

We study the deductive strength of properties under basic set-theoretical operations of the subclass E-Fin of the Dedekind finite sets in set theory without the Axiom of Choice (AC), which consists of all E-finite sets, where a set X is called E-finite if for no proper subset Y of X is there a surjection f:Y→X.

Słowa kluczowe

EN

axiom of choice; notions of finite; finiteness classes; E-finite; E-Fin finiteness class; ZFA- and ZF-models

• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2015
• Tom: Vol. 63, no. 2
• Strony: 89--112
• Opis fizyczny: Bibliogr. 19 poz.

Twórcy

autor

Herrlich, H.

• Department of Mathematics, University of the Aegean, Karlovassi, Samos 83200, Greece

autor

Howard, P.

• Department of Mathematics, Eastern Michigan University, Ypsilanti, MI 48197, U.S.A.,

autor

Tachtsis, E.

• Department of Mathematics, University of the Aegean, Karlovassi, Samos 83200, Greece,

Bibliografia

• [BS] B. Banaschewski and P. Schuster, The shrinking principle and the axiom of choice, Monatsh. Math. 151 (2007), 263-270.
• [B] A. Blass, Ramsey's theorem in the hierarchy of choice principles, J. Symbolic Logic 42 (1977), 387-390.
• [Co] P. J. Cohen, Set Theory and the Continuum Hypothesis, W. A. Benjamin, Reading, MA, 1966.
• [Cr] O. de la Cruz, Finiteness and choice, Fund. Math. 173 (2002), 57-76.
• [De] J. W. Degen, Some aspects and examples of infinity notions, Math. Logic Quart. 40 (1994), 111-124.
• [Di] J. H. Diel, Two definitions of finiteness, Notices Amer. Math. Soc. 21 (1974), 554-555.
• [F] U. Felgner, Models of ZF-Set Theory, Lecture Notes in Math. 223, Springer, Berlin, 1971.
• [HH] J. D. Halpern and P. E. Howard, Cardinals m such that 2m = m, Proc. Amer. Math. Soc. 26 (1970), 487-490.
• [He] H. Herrlich, Axiom of Choice, Lecture Notes in Math. 1876, Springer, Berlin, 2006.
• [Her] H. Herrlich, The finite and the infinite, Appl. Categor. Struct. 19 (2011), 455-468.
• [HHT] H. Herrlich, P. Howard and E. Tachtsis, Finiteness classes and small violations of choice, Notre Dame J. Formal Logic, to appear.
• [HT] H. Herrlich and E. Tachtsis, On the number of Russell's socks or 2+2+2+…=?, Comment. Math. Univ. Carolin. 47 (2006), 707-717.
• [HR] P. Howard and J. E. Rubin, Consequences of the Axiom of Choice, Math. Surveys Monogr. 59, Amer. Math. Soc., Providence, RI, 1998.
• [HS] P. Howard and L. Spišiak, Definitions of finite and the power set operation, submitted.
• [J] T. J. Jech, The Axiom of Choice, Stud. Logic Found. Math. 75, North-Holland, Amsterdam, 1973; reprint: Dover Publ., New York, 2008.
• [L] A. Lévy, The independence of various definitions of finiteness, Fund. Math. 46 (1958), 1-13.
• [LT] A. Lindenbaum et A. Tarski, Communication sur les recherches de la théorie des ensembles, Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie, Cl. III, Sciences Mathématiques et Physiques 1926, 299-330.
• [Ta] A. Tarski, Sur les ensembles finis, Fund. Math. 6 (1924), 45-95.
• [Tr] J. Truss, Classes of Dedekind finite cardinals, Fund. Math. 84 (1974), 187-208.

Identyfikatory

DOI

10.4064/ba63-2-1