Premium
On a Collection of Classes in Ackermann's Set Theory With the Axiomschema of Foundation
Author(s) -
Gloede Klaus
Publication year - 1980
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.19800263205
Subject(s) - ackermann function , foundation (evidence) , citation , set (abstract data type) , computer science , information retrieval , library science , mathematical economics , mathematics , linguistics , programming language , history , philosophy , inverse , archaeology
In 1956, ACKERMANN [l] introduced a system of axiomatic set theory A which was based on a principle seemingly different from the axioms of ZERMELO-FRAENKEL set theory ZF, yet LEVY [5] and REINHARDT [7] succeeded in showing that the theory A* (A with the axiom of foundation for sets) allows to prove the same theorems about sets as does the theory ZF. On the other hand, almost no general result can be proved in A* about the classes which are not sets (cp. LAKE [4]). Recently, ALKOR [2] generalized the notion of constructibility to the framework of ACKERMANN’S theory and proved that the constructible classes constitute an inner model of ZA in A + the axiomschema of foundation. We will use his ideas (which can be traced to back to GREWE [3]) to give an interpretation of ‘ZA in A** (A* + the minimum principle for the ordinals) which leaves the class V of all sets and the membership relation unchanged. This shows that one can define in A** a collection of classes which has the class V of all sets as an elementary submodel, and therefore this collection of classes satisfies the same properties as the sets in ZF. As a corollary, one obtains the result that in &k:I: the axioms of ZF are provable when relativized to V . This result is weaker than REINHARDT’S (he needs only A* in place of A**) but the proof is considerably simpler.