Generalized cardinal invariants for an inaccessible κ with compactness at κ++

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11210%2F25%3A10507449" target="_blank" >RIV/00216208:11210/25:10507449 - isvavai.cz</a>

Výsledek na webu

<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=g6PzL2qy37" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=g6PzL2qy37</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s00153-025-00977-2" target="_blank" >10.1007/s00153-025-00977-2</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Generalized cardinal invariants for an inaccessible κ with compactness at κ++

Popis výsledku v původním jazyce

We study the relationship between non-trivial values of generalized cardinal invariants at an inaccessible cardinal kappa and compactness principles at kappa(+) and kappa(++). Let TP(kappa(++)), SR(kappa(++)) and &lt;not sign&gt;wKH(kappa(+)) denote the tree property and stationary reflection on kappa++ and the negation of the weak Kurepa Hypothesis on kappa(+), respectively. We show that if the existence of a supercompact cardinal kappa with a weakly compact cardinal lambda above kappa is consistent, then the following are consistent as well (where t(kappa) and u(kappa) are the tower number and the ultrafilter number, respectively): (i) There is an inaccessible cardinal kappa such that kappa(+ )&lt; t(kappa) = u(kappa) &lt; 2 kappa and SR(kappa(++)) hold, and (ii) There is an inaccessible cardinal kappa such that kappa(+ )= t(kappa) &lt; u(kappa) &lt; 2(kappa) and SR(kappa(++)), TP(kappa(++)) and &lt;not sign&gt;wKH(kappa(+)) hold. The cardinals u(kappa) and 2 kappa can have any reasonable values in these models. We obtain these results by combining the forcing construction from [4] due to Brooke-Taylor, Fischer, Friedman and Montoya with the Mitchell forcing and with (new and old) indestructibility results related to TP(kappa(++)), SR(kappa(++)) and &lt;not sign&gt;wKH(kappa(+)). Apart from u(kappa) and t(kappa) we also compute the values of b(kappa), d(kappa), s(kappa), r(kappa), a(kappa), cov (M-kappa), add(M-kappa), non(M-kappa), cof(M-kappa) which will all be equal to u(kappa). In (ii), we compute p(kappa) = t(kappa) = kappa(+) by observing that the kappa(+)-distributive quotient of the Mitchell forcing adds a tower of size kappa(+). Finally, as a corollary of the construction, we observe that items (i) and (ii) hold also for the traditional invariants on kappa = omega, using Mitchell forcing up to a weakly compact cardinal; in this case we also obtain the disjoint stationary sequence property DSS(omega(2)), which implies the negation of the approachability property &lt;not sign&gt;AP(omega(2)).

Název v anglickém jazyce

Generalized cardinal invariants for an inaccessible κ with compactness at κ++

Popis výsledku anglicky

We study the relationship between non-trivial values of generalized cardinal invariants at an inaccessible cardinal kappa and compactness principles at kappa(+) and kappa(++). Let TP(kappa(++)), SR(kappa(++)) and &lt;not sign&gt;wKH(kappa(+)) denote the tree property and stationary reflection on kappa++ and the negation of the weak Kurepa Hypothesis on kappa(+), respectively. We show that if the existence of a supercompact cardinal kappa with a weakly compact cardinal lambda above kappa is consistent, then the following are consistent as well (where t(kappa) and u(kappa) are the tower number and the ultrafilter number, respectively): (i) There is an inaccessible cardinal kappa such that kappa(+ )&lt; t(kappa) = u(kappa) &lt; 2 kappa and SR(kappa(++)) hold, and (ii) There is an inaccessible cardinal kappa such that kappa(+ )= t(kappa) &lt; u(kappa) &lt; 2(kappa) and SR(kappa(++)), TP(kappa(++)) and &lt;not sign&gt;wKH(kappa(+)) hold. The cardinals u(kappa) and 2 kappa can have any reasonable values in these models. We obtain these results by combining the forcing construction from [4] due to Brooke-Taylor, Fischer, Friedman and Montoya with the Mitchell forcing and with (new and old) indestructibility results related to TP(kappa(++)), SR(kappa(++)) and &lt;not sign&gt;wKH(kappa(+)). Apart from u(kappa) and t(kappa) we also compute the values of b(kappa), d(kappa), s(kappa), r(kappa), a(kappa), cov (M-kappa), add(M-kappa), non(M-kappa), cof(M-kappa) which will all be equal to u(kappa). In (ii), we compute p(kappa) = t(kappa) = kappa(+) by observing that the kappa(+)-distributive quotient of the Mitchell forcing adds a tower of size kappa(+). Finally, as a corollary of the construction, we observe that items (i) and (ii) hold also for the traditional invariants on kappa = omega, using Mitchell forcing up to a weakly compact cardinal; in this case we also obtain the disjoint stationary sequence property DSS(omega(2)), which implies the negation of the approachability property &lt;not sign&gt;AP(omega(2)).

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

60301 - Philosophy, History and Philosophy of science and technology

Projekt

Výsledek vznikl pri realizaci vícero projektů. Více informací v záložce Projekty.

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2025

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Archive for Mathematical Logic

ISSN

0933-5846

e-ISSN

1432-0665

Svazek periodika

64

Číslo periodika v rámci svazku

7-8

Stát vydavatele periodika

DE - Spolková republika Německo

Počet stran výsledku

26

Strana od-do

1077-1102

Kód UT WoS článku

001489776300001

EID výsledku v databázi Scopus

2-s2.0-105005110200