Abstract
Shelah has shown that there are no chains of length ω3 increasing modulo finite in (Formula presented.). We improve this result to sets. That is, we show that there are no chains of length ω3 in (Formula presented.) increasing modulo finite. This contrasts with results of Koszmider who has shown that there are, consistently, chains of length ω2 increasing modulo finite in (Formula presented.) as well as in (Formula presented.). More generally, we study the depth of function spaces (Formula presented.) quotiented by the ideal (Formula presented.) where (Formula presented.) are infinite cardinals.
| Original language | English |
|---|---|
| Pages (from-to) | 286-301 |
| Number of pages | 16 |
| Journal | Mathematika |
| Volume | 69 |
| Issue number | 1 |
| DOIs | |
| State | Published - Jan 2023 |
Bibliographical note
Funding Information:The debt to [32] should be clear to any reader who makes it to this point. My interest in the results of Koszmider began when my teachers David Asperó and Mirna Džamonja suggested reading [14] to start my doctoral studies. Some of this research was done when I was a postdoctoral researcher at the Instituto Tecnológico Autónomo de México. After listening to a talk on Theorem A in February 2020, Assaf Rinot emphasised isolating the cardinals from Section 2.1 and focusing on the space (κμ,<θ)$({}^\kappa \mu, <_\theta)$ and also informed me of Lemma 2.3. The emphasis on (κμ,<θ)$({}^\kappa \mu, <_\theta)$ makes it clear that the results of Section 3 are more general than the results in [32]. This article has benefited greatly from his suggestions, his notes and his interest. A conversation with Ido Feldman helped inspire the discovery of Theorem 4.2. While working on the article at Bar-Ilan University I was supported by a postdoctoral fellowship funded by the Israel Science Foundation (grant agreement 2066/18).
Funding Information:
The debt to [ 32 ] should be clear to any reader who makes it to this point. My interest in the results of Koszmider began when my teachers David Asperó and Mirna Džamonja suggested reading [ 14 ] to start my doctoral studies. Some of this research was done when I was a postdoctoral researcher at the Instituto Tecnológico Autónomo de México. After listening to a talk on Theorem A in February 2020, Assaf Rinot emphasised isolating the cardinals from Section 2.1 and focusing on the space and also informed me of Lemma 2.3 . The emphasis on makes it clear that the results of Section 3 are more general than the results in [ 32 ]. This article has benefited greatly from his suggestions, his notes and his interest. A conversation with Ido Feldman helped inspire the discovery of Theorem 4.2 . While working on the article at Bar‐Ilan University I was supported by a postdoctoral fellowship funded by the Israel Science Foundation (grant agreement 2066/18).
Publisher Copyright:
© 2023 The Authors. Mathematika is copyright © University College London and published by the London Mathematical Society on behalf of University College London.