Can You Take Komjath's Inaccessible Away?

In this paper we aim to compare Kurepa trees and Aronszajn trees. Moreover, we analyze the affect of large cardinal assumptions on this comparison. Using the the method of walks on ordinals, we will show it is consistent with ZFC that there is a Kurepa tree and every Kurepa tree contains a Souslin subtree, if there is an inaccessible cardinal. This is stronger than Komjath's theorem that asserts the same consistency from two inaccessible cardinals. We will show that our large cardinal assumption is optimal, i.e. if every Kurepa tree has an Aronszajn subtree then $\omega_2$ is inaccessible in the constructible universe \textsc{L}. Moreover, we prove it is consistent with ZFC that there is a Kurepa tree $T$ such that if $U \subset T$ is a Kurepa tree with the inherited order from $T$, then $U$ has an Aronszajn subtree. This theorem uses no large cardinal assumption. Our last theorem immediately implies the following: assume $\textrm{MA}_{\omega_2}$ holds and $\omega_2$ is not a Mahlo cardinal in \textsc{L}. Then there is a Kurepa tree with the property that every Kurepa subset has an Aronszajn subtree. Our work entails proving a new lemma about Todorcevic's $\rho$ function which might be useful in other contexts.Comment: 20 page

Non-Absoluteness of Model Existence at ℵω\aleph_\omega

In [FHK13], the authors considered the question whether model-existence of $L_{\omega_1,\omega}$-sentences is absolute for transitive models of ZFC, in the sense that if $V \subseteq W$ are transitive models of ZFC with the same ordinals, $\varphi\in V$ and $V\models "\varphi \text{ is an } L_{\omega_1,\omega}\text{-sentence}"$, then $V \models "\varphi \text{ has a model of size } \aleph_\alpha"$ if and only if $W \models "\varphi \text{ has a model of size } \aleph_\alpha"$. From [FHK13] we know that the answer is positive for $\alpha=0,1$ and under the negation of CH, the answer is negative for all $\alpha>1$. Under GCH, and assuming the consistency of a supercompact cardinal, the answer remains negative for each $\alpha>1$, except the case when $\alpha=\omega$ which is an open question in [FHK13]. We answer the open question by providing a negative answer under GCH even for $\alpha=\omega$. Our examples are incomplete sentences. In fact, the same sentences can be used to prove a negative answer under GCH for all $\alpha>1$ assuming the consistency of a Mahlo cardinal. Thus, the large cardinal assumption is relaxed from a supercompact in [FHK13] to a Mahlo cardinal. Finally, we consider the absoluteness question for the $\aleph_\alpha$-amalgamation property of $L_{\omega_1,\omega}$-sentences (under substructure). We prove that assuming GCH, $\aleph_\alpha$-amalgamation is non-absolute for $1<\alpha<\omega$. This answers a question from [SS]. The cases $\alpha=1$ and $\alpha$ infinite remain open. As a corollary we get that it is non-absolute that the amalgamation spectrum of an $L_{\omega_1,\omega}$-sentence is empty

The non-absoluteness of model existence in uncountable cardinals for Lw1,w

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