The Finite and the Infinite

作者：Herrlich Horst^*

来源：Applied Categorical Structures, 2011, 19(2): 455-468.

DOI：10.1007/s10485-009-9212-5

• Axiom of choice
• Finiteness class
• Finite
• Dedekind-finite
• Almost finite
• Hereditary
• Cohereditary
• Summable
• Productive
• Power-closed
• Full subcategories of Set

In ZF set theory finiteness classes are introduced and their stability under basic set theoretical constructions are being investigated. Typical results are: The class of finite sets is the smallest finiteness class.
The class of Dedekind-finite sets is the largest finiteness class.
The class of almost finite sets is the largest summable finiteness class.
Equivalent are: There is only one finiteness class.
The union of each family of 1-element sets, indexed by a Dedekind-finite set, is Dedekind-finite.
The axiom of choice, for countable families of Dedekind-finite sets.
The shrinking principle for families (X (i) ) (i is an element of I) of sets, indexed by a Dedekind-finite set I (i.e., there exists a family (Y (i) ) (i is an element of I) of pairwise disjoint subsets Y (i) of X (i) with
In suitable ZF-models there exist families (21(r))(r subset of R) of finiteness classes such that r < s double right arrow 21(r) subset of(not equal) 21(s.)

• 出版日期2011-4