SET A 一匹な 0101１３２人目の素数さん2022/01/13(木) 19:31:38.87ID:VN40DQWB>>85 >Hereditarily finite setは、 >”cardinality depend on the theory in context” >ってことです
英語翻訳できないの？中卒SET A
Discussion A symbol for the class of hereditarily finite sets is H_ｱﾚﾌ0, standing for the cardinality of each of its member being smaller than ｱﾚﾌ0. Whether H_ｱﾚﾌ0 is a set and statements about cardinality depend on the theory in context.
>つまり、”Theories of finite sets”＝有限集合理論 では、 >例えば、”In this context, the negation of the axiom of infinity may be added” >とあるように、無限公理の否定をあえて追加する議論もありってこと。
英語翻訳できないの？中卒SET A
Axiomatizations Theories of finite sets The set Φ also represents the first von Neumann ordinal number, denoted 0. And indeed all finite von Neumann ordinals are in ｱﾚﾌ0 and thus the class of sets representing the natural numbers, i.e it includes each element in the standard model of natural numbers. Robinson arithmetic can already be interpreted in ST, the very small sub-theory of Z^{-} with axioms given by Extensionality, Empty Set and Adjunction.
Their models then also fulfill the axioms consisting of the axioms of Zermelo-Fraenkel set theory without the axiom of infinity. In this context, the negation of the axiom of infinity may be added, thus proving that the axiom of infinity is not a consequence of the other axioms of set theory.
