The tree property at aleph_{omega+2} with a finite gap

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11210%2F17%3A10362111" target="_blank" >RIV/00216208:11210/17:10362111 - isvavai.cz</a>

Nalezeny alternativní kódy

RIV/00216208:11210/17:10362112

Výsledek na webu

<a href="http://www.winterschool.eu/files/995-The_tree_property_at_aleph_omega2_with_a_finite_gap1027058508.pdf" target="_blank" >http://www.winterschool.eu/files/995-The_tree_property_at_aleph_omega2_with_a_finite_gap1027058508.pdf</a>

DOI - Digital Object Identifier

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Jazyk výsledku

angličtina

Název v původním jazyce

The tree property at aleph_{omega+2} with a finite gap

Popis výsledku v původním jazyce

Let $kappa$ be an infinite regular cardinal. The tree property at $kappa$ is a compactness principle which says that every $kappa$-tree has a cofinal branch. Obtaining the tree property at the double successor of an infinite regular cardinal $kappa$ is relatively easy and only a weakly compact cardinal is required (&apos;&apos;Mitchell forcing&apos;&apos;). The situation is more complex when we wish to get this result at the double successor of a singular strong limit cardinal $kappa$ since we need to ensure the failure of SCH at $kappa$. In this talk we will discuss the important case of $aleph_omega$ and show that if $kappa$ is a certain large cardinal (not too large), and $1 &lt; n&lt;omega$ is fixed, then there is a forcing $P$ such that the following hold in $V^P$: begin{itemize} item $kappa = aleph_omega$ is strong limit, item $2^{alephomega}=aleph_{omega+n}$, and item The tree property holds at $aleph_{omega+2}$. end{itemize} The forcing $P$ is a combination of several subforcings which first prepare the universe and then use a combination of the Mitchell forcing and the Prikry forcing with collapses to force the tree property.

Název v anglickém jazyce

The tree property at aleph_{omega+2} with a finite gap

Popis výsledku anglicky

Let $kappa$ be an infinite regular cardinal. The tree property at $kappa$ is a compactness principle which says that every $kappa$-tree has a cofinal branch. Obtaining the tree property at the double successor of an infinite regular cardinal $kappa$ is relatively easy and only a weakly compact cardinal is required (&apos;&apos;Mitchell forcing&apos;&apos;). The situation is more complex when we wish to get this result at the double successor of a singular strong limit cardinal $kappa$ since we need to ensure the failure of SCH at $kappa$. In this talk we will discuss the important case of $aleph_omega$ and show that if $kappa$ is a certain large cardinal (not too large), and $1 &lt; n&lt;omega$ is fixed, then there is a forcing $P$ such that the following hold in $V^P$: begin{itemize} item $kappa = aleph_omega$ is strong limit, item $2^{alephomega}=aleph_{omega+n}$, and item The tree property holds at $aleph_{omega+2}$. end{itemize} The forcing $P$ is a combination of several subforcings which first prepare the universe and then use a combination of the Mitchell forcing and the Prikry forcing with collapses to force the tree property.

Druh

O - Ostatní výsledky

CEP obor

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OECD FORD obor

10101 - Pure mathematics

Projekt

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Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2017

Kód důvěrnosti údajů

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