The tree property below aleph_omega

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11210%2F17%3A10362108" target="_blank" >RIV/00216208:11210/17:10362108 - isvavai.cz</a>
Result on the web
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DOI - Digital Object Identifier
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Result language
angličtina
Original language name
The tree property below aleph_omega
Original language description
We say that a regular cardinal $kappa$, $kappa> aleph_0$, has the tree property if there are no $kappa$-Aronszajn trees; we say that $kappa$ has the weak tree property if there are no special $kappa$-Aronszajn trees. Starting with infinitely many weakly compact cardinals, we show that the tree property at every even cardinal $aleph_{2n}$, $0aleph_{2n+1}$, $n<omega$. Next, starting with infinitely many Mahlo cardinals, we show that the weak tree property at every cardinal $aleph_n$, $1 < n <omega$, is consistent with an arbitrary continuum function which satisfies $2^{aleph_n} > aleph_{n+1}$, $n<omega$. Thus the tree property has no provable effect on the continuum function below $aleph_omega$ except for the trivial requirement that the tree property at $kappa^{++}$ implies $2^kappa>kappa^+$ for every infinite $kappa$.
Czech name
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Czech description
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Project
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Continuities
S - Specificky vyzkum na vysokych skolach<br>I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2017
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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