Generalized cardinal invariants for an inaccessible κ with compactness at κ++
The result's identifiers
Result code in 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>
Result on the web
<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>
Alternative languages
Result language
angličtina
Original language name
Generalized cardinal invariants for an inaccessible κ with compactness at κ++
Original language description
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 <not sign>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(+ )< t(kappa) = u(kappa) < 2 kappa and SR(kappa(++)) hold, and (ii) There is an inaccessible cardinal kappa such that kappa(+ )= t(kappa) < u(kappa) < 2(kappa) and SR(kappa(++)), TP(kappa(++)) and <not sign>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 <not sign>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 <not sign>AP(omega(2)).
Czech name
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Czech description
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Classification
Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
60301 - Philosophy, History and Philosophy of science and technology
Result continuities
Project
Result was created during the realization of more than one project. More information in the Projects tab.
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)
Others
Publication year
2025
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Archive for Mathematical Logic
ISSN
0933-5846
e-ISSN
1432-0665
Volume of the periodical
64
Issue of the periodical within the volume
7-8
Country of publishing house
DE - GERMANY
Number of pages
26
Pages from-to
1077-1102
UT code for WoS article
001489776300001
EID of the result in the Scopus database
2-s2.0-105005110200