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Some remarks on real-valued measurable cardinals
| Content Provider | Semantic Scholar |
|---|---|
| Author | Szymanski, Andrzej |
| Copyright Year | 1988 |
| Abstract | We consider the set [w}w and the cofinality of the set 'cA assuming that some cardinals are endowed in total measures. Introduction. A cardinal ,c is real-valued measurable if there exists a iccomplete atomless probability measure on P(c), the set of all subsets of the cardinal Kc. The status and the philosophy concerned e.g. real-valued measurable cardinals have been detailed and presented in a survey paper by A. Kanamori and M. Magidor [KM]. We shall concentrate mainly on two problems: the cofinality of sets of functions with respect to eventual domination and some combinatorics on w, both assuming the existence of some total measures. T. Jech and K. Prikry [JP] showed, assuming 2W is real-valued measurable, that the cofinality of the set of all functions from w1 into w equals 21. We extend and complete their result. We show (Theorems 1 and 3, ?2), under the same assumption about 2', that if Kc w. It is well known that under Martin's axiom, 2W cannot be real-valued measurable. The reason is, that under Martin's axiom the cofinality of the set of all functions from w into w equals 2W while assuming 2W is real-valued measurable, the cofinality is < 2W (see also Theorem 4, ?2). We give some other reasons for which some cardinals cannot carry total measures. We show (Corollary 2, ?1) that if there exists a maximal Kc-tower on w, then Kc is not real-valued measurable cardinal. It has been shown by S. Hechler [H] that each regular cardinal Kc, w < K < 2W can be (consistently) the length of some maximal Kc-tower on w. Throughout the paper we use standard set-theoretical notation. For example [w]W is used to denote the set of all infinite subsets of the least infinite ordinal W. All undefined terms can be found in [J]. 1. Let W denote the set of all functions from w into w. For two arbitrary functions f, g c WW we set f <* g iff f (n) < g(n) for all but finitely many n c w. Received by the editors July 30, 1986 and, in revised form, November 20, 1987. Presented at Spring Topology Conference, Lafayette, April 3-5, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 03E55, 04A20. |
| Starting Page | 596 |
| Ending Page | 602 |
| Page Count | 7 |
| File Format | PDF HTM / HTML |
| DOI | 10.1090/S0002-9939-1988-0962835-0 |
| Volume Number | 104 |
| Alternate Webpage(s) | http://www.ams.org/journals/proc/1988-104-02/S0002-9939-1988-0962835-0/S0002-9939-1988-0962835-0.pdf |
| Alternate Webpage(s) | https://doi.org/10.1090/S0002-9939-1988-0962835-0 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
