On the existence of nonregular ultrafilters and the cardinality of ultrapowers
On the existence of nonregular ultrafilters and the cardinality of ultrapowers
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关于非常规超滤器的存在性和超幂的基数
DOI：
10.1090/s0002-9947-1979-0526312-2
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发表时间：
1979
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影响因子：
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通讯作者：
M. Magidor
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文献类型：
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作者：
M. Magidor
假设大基数的一致性，我们证明了W3可以携带一个超过滤器D，使得w13 /D具有基数W3。（因此D不是（W3, ol）规则的。）类似地，W2可以携带一个超滤波器D，使得w'2/D具有基数W2。（因此D不是（CA2, o）规则的。）0. 介绍。当引入超滤波器和超功率（见[4]）时，在给定基结构和索引集的基数的情况下确定超功率的基数的问题变得有趣了（见[8]）。正如[4]中已经指出的那样，如果假设所讨论的超过滤器是规则的（定义见下文），这个问题可以完全解决。在正则性假设下，超滤机的基数性具有最大可能的基数性，即如果M是基数性a > X的结构，D是基数性为8的指标集I上的一致规则超滤机，则M′/D具有基数性a。由Jensen在可构造宇宙中得到部分答案，随后由Prikry[14]改进，他证明如果V = L，则w，，上的每一个均匀超滤子都是正则的。Benda和Ketonen[2]、Kanamori[6]以及最后的Ketonen[10]的结果表明，该问题与大基数的存在密切相关。特别是（见[10]），如果K +携带非（K， K）正则的均匀超滤，则0#存在。另一方面，大基数立即暗示存在不规则超滤子，见下面的定义。请注意，K+上的正则超滤是（K+， K）正则，但反之不一定成立。如果D是基数K上的WI完全非主超过滤器，则D是非正则的：注意，K必须至少与第一个可测量值一样大，才能携带一个w完全超过滤器。Keisler的问题对于较小的基数，如(1，W2等，当然更有趣，我们研究这个版本的问题。关于连续体的相同问题，有一个事实是已知的；如果连续体携带一个w，饱和理想，那么它携带一个不规则的超滤。请注意，在这个假设下，连续体是非常大的，这是1977年8月1日和1977年9月26日修订后的编辑收到的。AMS （MOS）学科分类（1970年）。主要02 k05;二级02 k35。
Assuming the consistency of huge cardinals, we prove that W3 can carry an ultrafilter D such that W1 3/D has cardinality W3. (Hence D is not (W3, ol) regular.) Similarly W2 can carry an ultrafilter D such that w'2/D has cardinality w2. (Hence D is not (CA2, o) regular.) 0. Introduction. When ultrafilters and ultrapowers (see [4]) were introduced, the problem of determining the cardinality of the ultrapower given the cardinality of the basis structure and of the index set became of interest (see [8]). As was already noted in [4], the problem can be completely settled, if one assumes that the ultrafilter in question is regular (see below for definition). Under the assumption of regularity the cardinality of the ultrapower has largest cardinality possible, i.e. if M is a structure of cardinality a > X and D is a uniform regular ultr4filter on an index set I of cardinality ,8, then M'/D has cardinality a. The problem whether every ultrafilter is regular, was posed by Keisler in [9], and partial answers were obtained in the constructible universe by Jensen and then improved by Prikry [14] who showed that if V = L then every uniform ultrafilter on w,, is regular. Results of Benda and Ketonen [2], Kanamori [6] and finally Ketonen [10] indicate the problem is closely connected with the existence of large cardinals. In particular (see [10]) if K + carries a uniform ultrafilter which is not (K , K) regular, then 0# exists. On the other hand large cardinals immediately imply the existence of nonregular ultrafilters, see definition below. Note that a regular ultrafilter on K+ iS (K+, K) regular though the converse does not necessarily hold. If D is an WI complete, nonprincipal ultrafilter on a cardinal K, then D is not regular: Note that K has to be at least as large as the first measurable in order to carry an w, complete ultrafilter. Keisler's problem is of course much more interesting for smaller cardinals like (O1, W2 etc., and we attend this version of the problem. There is one fact known about the same problem for the continuum; if the continuum carries an w, saturated ideal then it carries a nonregular ultrafilter. Note that under this assumption the continuum is very large and this Received by the editors August 1, 1977 and, in revised form, September 26, 1977. AMS (MOS) subject classifications (1970). Primary 02K05; Secondary 02K35.