Ultraproducts and Saturated Models

Keisler, H. Jerome

doi:10.1016/s1385-7258(64)50021-9

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Elementary Equivalence and Ultraproducts1

Theorem 73 (Stone-weierstrass Theorem)1

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1970

2023

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Cited by 50 publications

(2 citation statements)

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“…The proof in the classical case can be found in [35,40]. In the continuous case, a proof appears in [27] while a proof in the positive bounded case appears in [32].…”

Section: Elementary Equivalence and Ultraproductsmentioning

confidence: 99%

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Introduction

Hart

2023

The Perfect Puppy

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“…The proof in the classical case can be found in [35,40]. In the continuous case, a proof appears in [27] while a proof in the positive bounded case appears in [32].…”

Section: Elementary Equivalence and Ultraproductsmentioning

confidence: 99%

“…The ultrafilters used in the proof of this proposition are called good ultrafilters (see [22]). A proof of the existence of good ultrafilters can be found in [36] and the saturation of the ultraproduct can be found in [35]; see also [22,Chapter 8]. Proposition 7.10.…”

Section: Theorem 73 (Stone-weierstrass Theorem)mentioning

confidence: 99%

Introduction

Hart

2023

The Perfect Puppy

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Nonstandard Analysis

Geiser

1970

Mathematical Logic Qtrly

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Zcitechr. f. math. Logik und Grundlagen d. Math. Rd. 1 6 , S . 297-318 ( 1 9 7 0 ) NONSTANDARD ANALYSIS by JAMES R. GEISER in Hanover, New Hampshire (U.S.A.)l) l) This research was supported in part by a Carnegie Grant at Dartmouth. 2, This point is debatable e.g. see A. ROBINSON [Sg] p. 47. ROBINSON uses a modification of the finite type theory developed by L. HENKIN [3].20 Ztsohr. f. math. Logik

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Ramsey for ultrafilter mappings and their Dedekind cuts

Trujillo

2015

Mathematical Logic Qtrly

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Associated to each ultrafilter U on ω and each map p : ω → ω is a Dedekind cut in the ultrapower ω ω / p(U). Blass has characterized, under CH, the cuts obtainable when U is taken to be either a p-point ultrafilter, a weakly-Ramsey ultrafilter or a Ramsey ultrafilter. Dobrinen and Todorčević have introduced the topological Ramsey space R 1 . Associated to the space R 1 is a notion of Ramsey ultrafilter for R 1 generalizing the familiar notion of Ramsey ultrafilter on ω. We characterize, under CH, the cuts obtainable when U is taken to be a Ramsey for R 1 ultrafilter and p is taken to be any map. In particular, we show that the only cut obtainable is the standard cut, whose lower half consists of the collection of equivalence classes of constants maps. Forcing with R 1 using almost-reduction adjoins an ultrafilter which is Ramsey for R 1 . For such ultrafilters U 1 , Dobrinen and Todorčević have shown that the Rudin-Keisler types of the p-points within the Tukey type of U 1 consists of a strictly increasing chain of rapid p-points of order type ω. We show that for any Rudin-Keisler mapping between any two p-points within the Tukey type of U 1 the only cut obtainable is the standard cut. These results imply existence theorems for special kinds of ultrafilters.

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Ultraproducts and Saturated Models

Cited by 50 publications

References 8 publications

Introduction

Introduction

Nonstandard Analysis

Ramsey for ultrafilter mappings and their Dedekind cuts

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