caa a22 4500 445378980 CHVBK 20180317143011.0 cr unu---uuuuu 170323e20110401xx s 000 0 eng 10.1007/s10485-009-9212-5 doi (NATIONALLICENCE)springer-10.1007/s10485-009-9212-5 Herrlich Horst University of Bremen, Feldhäuser Str. 69, 28865, Lilienthal, Germany aut The Finite and the Infinite [Elektronische Daten] [Horst Herrlich] In ZF set theory finiteness classes are introduced and their stability under basic set theoretical constructions are being investigated. Typical results are: 1.The class of finite sets is the smallest finiteness class.2.The class of Dedekind-finite sets is the largest finiteness class.3.The class of almost finite sets is the largest summable finiteness class.4.Equivalent are: a.There is only one finiteness class.b.The union of each family of 1-element sets, indexed by a Dedekind-finite set, is Dedekind-finite.c.The axiom of choice, for countable families of Dedekind-finite sets.d.The shrinking principle for families (X i)i ∈ I of sets, indexed by a Dedekind-finite set I (i.e., there exists a family (Y i)i ∈ I of pairwise disjoint subsets Y i of X i with $\underset{i\in I}{\bigcup} Y_i = \underset{i\in I}{\bigcup} X_i$). 5.In suitable ZF-models there exist families $({\mathfrak{A}}_r)_{r\in{\mathbb{R}}}$ of finiteness classes such that $$r < s \Longrightarrow {\mathfrak{A}}_r \subsetneqq {\mathfrak{A}}_s.$$ Springer Science+Business Media B.V., 2009 Axiom of choice nationallicence Finiteness class nationallicence Finite nationallicence Dedekind-finite nationallicence Almost finite nationallicence Hereditary nationallicence Cohereditary nationallicence Summable nationallicence Productive nationallicence Power-closed nationallicence Full subcategories of Set nationallicence Applied Categorical Structures Springer Netherlands 19/2(2011-04-01), 455-468 0927-2852 19:2<455 2011 19 10485 https://doi.org/10.1007/s10485-009-9212-5 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s10485-009-9212-5 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Herrlich Horst University of Bremen, Feldhäuser Str. 69, 28865, Lilienthal, Germany aut NATIONALLICENCE 773 0- Applied Categorical Structures Springer Netherlands 19/2(2011-04-01), 455-468 0927-2852 19:2<455 2011 19 10485 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer