𝔖 Bobbio Scriptorium
✦   LIBER   ✦

On a problem of Gillman and Keisler

✍ Scribed by Karel Prikry

Publisher
Elsevier Science
Year
1970
Weight
363 KB
Volume
2
Category
Article
ISSN
0003-4843

No coin nor oath required. For personal study only.

✦ Synopsis

We shall obtain a result concerning the question 3B of Keisler [6]. If S is a set, then ISl is the cardinality of S. The axiom of choice is assumed throughout. Eor the notion of an ultrafilter over a set S we refer e.g. to Keisler [6], p. 115. Suppose now that ISI =m is infinite and og is an ultrafilter over S. c-g is said to be uniform if for every X ~ ~, IXl = m. Keisler [6] calls an ultrafilter~ over S regular if there is a set ~c_ cg such that Iq~t -m and the intersection of any infinitely many sets from q~ is empty. Now we can state the question 3B, [6]. Is it true that for any infinite S, every uniform ultrafiltcr o,Oer S is regular? Keisler [6] states that the answer is knowl~, only in the case when ISI = ~0-The answer is then trivially positive. In the present paper we shall consider the next case, namely ISI = s 1 • Our main result is then as follows. If G6del's axiom of constructibility, V = L, is true, then every uniform ultrafilter over S is regular. Using the methods of Cohen [21, models of set theory can be constructed in which "V = L" is false and "it ISI = S l, then every uniform ultrafilter over S is regular" is true.

The question, "'if ISl = ~ 1, is every uniform ultrafilter over S regular?", also appears in Erd6s, Hajnal [3] as Problem 82.1he wording there is different, The question is attributed to L.Gillman. No reference is given. Thus we answer Gillman's question under the assumption tlaat V=L.

📜 SIMILAR VOLUMES

A generalization of the Keisler-Morley t
A generalization of the Keisler-Morley theorem to recursively saturated ordered structures
✍ Shahram Mohsenipour 📂 Article 📅 2007 🏛 John Wiley and Sons 🌐 English ⚖ 138 KB

## Abstract We prove a model theoretic generalization of an extension of the Keisler‐Morley theorem for countable recursively saturated models of theories having a __κ__ ‐like model, where __κ__ is an inaccessible cardinal. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On a Problem of Solynin
On a Problem of Solynin
✍ James Jenkins 📂 Article 📅 2001 🏛 John Wiley and Sons 🌐 English ⚖ 104 KB 👁 1 views
On a Problem of Erdős and Sárközy
On a Problem of Erdős and Sárközy
✍ Tomasz Schoen 📂 Article 📅 2001 🏛 Elsevier Science 🌐 English ⚖ 89 KB

Let A=[a 1 , a 2 , ...] N and put A(n)= a i n 1. We say that A is a P-set if no element a i divides the sum of two larger elements. It is proved that for every P-set A with pairwise co-prime elements the inequality A(n)<2n 2Â3 holds for infinitely many n # N. ## 2001 Academic Press where A(n)= a i