Measures that violate the Generalized Continuum Hypothesis
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866917133241286656 |
|---|---|
| author | Benhamou, Tom Goldberg, Gabriel |
| author_facet | Benhamou, Tom Goldberg, Gabriel |
| contents | A simple \(P_λ\)-point on a regular cardinal \(κ\) is a uniform ultrafilter on \(κ\) with a mod-bounded decreasing generating sequence of length \(λ\).
We prove that if there is a simple $P_λ$-point ultrafilter over $κ>ω$, then $λ=\mathfrak{d}_κ=\mathfrak{b}_κ=\mathfrak{u}_κ=\mathfrak{r}_κ=\mathfrak{s}_κ$. We show that such ultrafilters appear in the models of \cite{SimonOmer,BROOKETAYLOR201737}. We improve the lower bound for the consistency strength of the existence of a $P_{κ^{++}}$-point to a $2$-strong cardinal. Finally, we apply our arguments to obtain non-trivial lower bounds for (1) the statement that the generalized tower number $\mathfrak{t}_κ$ is greater than $κ^+$ and $κ$ is measurable, (2) the preservation of measurability after the generalized Mathias forcing, and (3) variations of filter games of \cite{NIELSEN_WELCH_2019,HolySchlicht:HierarchyRamseyLikeCardinals,MagForZem} in the case $2^κ>κ^+$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_20094 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Measures that violate the Generalized Continuum Hypothesis Benhamou, Tom Goldberg, Gabriel Logic A simple \(P_λ\)-point on a regular cardinal \(κ\) is a uniform ultrafilter on \(κ\) with a mod-bounded decreasing generating sequence of length \(λ\). We prove that if there is a simple $P_λ$-point ultrafilter over $κ>ω$, then $λ=\mathfrak{d}_κ=\mathfrak{b}_κ=\mathfrak{u}_κ=\mathfrak{r}_κ=\mathfrak{s}_κ$. We show that such ultrafilters appear in the models of \cite{SimonOmer,BROOKETAYLOR201737}. We improve the lower bound for the consistency strength of the existence of a $P_{κ^{++}}$-point to a $2$-strong cardinal. Finally, we apply our arguments to obtain non-trivial lower bounds for (1) the statement that the generalized tower number $\mathfrak{t}_κ$ is greater than $κ^+$ and $κ$ is measurable, (2) the preservation of measurability after the generalized Mathias forcing, and (3) variations of filter games of \cite{NIELSEN_WELCH_2019,HolySchlicht:HierarchyRamseyLikeCardinals,MagForZem} in the case $2^κ>κ^+$. |
| title | Measures that violate the Generalized Continuum Hypothesis |
| topic | Logic |
| url | https://arxiv.org/abs/2503.20094 |