A Relativization of Axioms of Strong-Infinity to ω_1
スポンサーリンク
概要
- 論文の詳細を見る
As we stated in [12] and [13], it is very interesting to think of the width of P(ω). The difficulty of this problem is typically seen if one considers of the continuum hypothesis. Three problems are involved here; 1) the size of ω_1, 2) the size of P(ω), and 3) the comparison of the sizes of ω_1 and P(ω). If ω_1 is very rich, then P(ω) must be very rich. Therefore merely the feeling that ω_1 or P(ω) is very rich does not say anything direct to the continuum hypothesis. The problem is a problem of comparison between richnesses of ω_1 and P(ω). Many interesting hypothesis like Axiom of Determinacy, Martin's Axiom, or the Real-valued Measurability of the Continuum are closely connected with this problem. We believe that many basic investigations in this area are needed. In this paper, we are interested in only the richness of ω_1. It is not very easy to express that ω_1 is very rich. Several attempts ( [11], [12] ) were done in the following way. There exists a transitive model M of ZF such that 1) M is aproper class, and 2) ω_1 is very big in M. However Silver's result [8] shows that if there exists a Ramsey cardinal and M is very narrow as L, then ω_1 is always extremly rich in M and it is rather meaningless to discuss the richness of ω_1 in M. We believe that the direction of this idea is right but the developement of the idea is not good enough. The meaningful improvement of this idea must be done by making M very wide. We shall discuss the following cases. There exists a transitive model M of ZF such that 1) M is a proper class, 2) P(ω) εM, and 3) ω_1 is very big in M. As for 3), we shall treat the following cases; ω_1 is semi-inaccessible, measurable, compact, or supercompact in M. We shall show that the consistency of the existence of M with the above stated conditions in which ω_1 is semi-inaccessible, measurable, compact, or super-compact follows from the consistency of the existence of an inaccessible cardinal, a measurable cardinal, a compact cardinal, or a super-compact cardinal respectively. To prove this, we shall develope a theory on Scott-Solovay's Boolean-valued models in [7] and generalize Levy-Solovay's theorem in [4] and McAloon's theorem in [5] on mild Cohen extension. A knowledge on [7]) is assumed.
- 科学基礎論学会の論文
- 1970-03-31