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On the Mathematical Content of the Theory of Classes KM
Author(s) -
Jansana Ramón
Publication year - 1989
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.19890350505
Subject(s) - citation , reading (process) , computer science , content (measure theory) , information retrieval , library science , mathematics , philosophy , linguistics , mathematical analysis
In his important paper [4] HARVEY FRIEDMAN presents some propositions concerning Bore1 functions with special combinatorial properties, and proves them to be independent of ZFC by arguments different from the standard forcing ones used to prove independence results. One of these independent propositions, proposition R for n = 4 and ni < OJ, holds in KMC (the theory of classes of Kelley-Morse plus the axiom of choice for sets [8], 191). Hence KMC goes beyond ZFC not only in its metamathematical content (KMC proves the consistency of ZFC) but also in its mathematical content, and still, as FRIEDMAN says, formalizes only commonly accepted principles of mathematical reasoning. Proposition R, as I said, is proved independent of ZFC by arguments different from the standard ones. It is known that virtually all independence results obtained by forcing are decided in the constructible universe. The independence results presented by FRIEDMAN in the aformentioned paper are shown not to be decided in the constructible universe. Since one of them is decided in KMC, i t seems natural to ask the following question : Is every sentence proved independent from ZFC by forcing, when properly translated, independent of KMC? One can suspect that the answer is Yes, and in fact that is the case as is shown by Theorem 6.3 of the present paper, when the collection of forcing conditions is a set. The aim of the paper can be seen as twofold. In the first part i t presents an adaptation of the method of Boolean valued models for KMG (KM plus global choice). And in the second one uses this adaptation to answer the previously stated question. The reason why we work in KM plus global choice is that in the context of an impredicative theory of classes i t seems more natural to suppose the existence of a well-ordering of the class of all sets. In fact that seems to be one of the elements in the conception that CANTOR had of sets (see HALLETT [5], pp. 171-174). R. CHUAQUI in [2] and [3] presents a version of SHOENFIELD’S unramified forcing for KM and in [2] he uses i t to prove the consistency of GCH relative to KMG and to prove EASTON’S result. I n [3] he proves the independence of local choice from KM, and the independence of global choice from KM plus local choice. CHUAQUI in the introduction to 121 explicitly states that he is not sure the method of Boolean valued models is suitable for KM. The results of the first part of the present paper ( I 1-5) show that i t is, a t least when one uses a Boolean algebra that is a set. The results of the first part of this paper ( 5 1 5) were already published in Catalan, without proofs, in [6].