Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userCofinalities of elementary substructures of structures on aleph_omega
167
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Let 0<n^*< omega and f:X-> n^*+1 be a function where X subseteq omega backslash (n^*+1) is infinite. Consider the following set S_f= {x subset aleph_omega : |x| <= aleph_{n^*} & (for all n in X)cf(x cap alpha_n)= aleph_{f(n)}}. The question, first posed by Baumgartner, is whether S_f is stationary in [alpha_omega]^{< aleph_{n^*+1}}. By a standard result, the above question can also be rephrased as certain transfer property. Namely, S_f is stationary iff for any structure A=< aleph_omega, ... > theres a B prec A such that |B|= aleph_{n^*} and for all n in X we have cf(B cap aleph_n)= aleph_{f(n)}. In this paper, we are going to prove a few results concerning the above question.
rate research
Read More
We discuss a system of strengthenings of $aleph_omega$ is Jonsson indexed by real numbers, and identify a strongest one. We give a proof of a theorem of Silver and show that there is a barrier to weakening its hypothesis.
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, $varphiin V$ and $Vmodels varphi text{ is an } L_{omega_1,omega}text{-sentence}$, 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.
When classes of structures are not first-order definable, we might still try to find a nice description. There are two common ways for doing this. One is to expand the language, leading to notions of pseudo-elementary classes, and the other is to allow infinite conjuncts and disjuncts. In this paper we examine the intersection. Namely, we address the question: Which classes of structures are both pseudo-elementary and $mathcal{L}_{omega_1 omega}$-elementary? We find that these are exactly the classes that can be defined by an infinitary formula that has no infinitary disjunctions.
We study the saturation properties of several classes of $C^*$-algebras. Saturation has been shown by Farah and Hart to unify the proofs of several properties of coronas of $sigma$-unital $C^*$-algebras; we extend their results by showing that some coronas of non-$sigma$-unital $C^*$-algebras are countably degree-$1$ saturated. We then relate saturation of the abelian $C^*$-algebra $C(X)$, where $X$ is $0$-dimensional, to topological properties of $X$, particularly the saturation of $CL(X)$.
In several classes of countable structures it is known that every hyperarithmetic structure has a computable presentation up to bi-embeddability. In this article we investigate the complexity of embeddings between bi-embeddable structures in two such classes, the classes of linear orders and Boolean algebras. We show that if $mathcal L$ is a computable linear order of Hausdorff rank $n$, then for every bi-embeddable copy of it there is an embedding computable in $2n-1$ jumps from the atomic diagrams. We furthermore show that this is the best one can do: Let $mathcal L$ be a computable linear order of Hausdorff rank $ngeq 1$, then $mathbf 0^{(2n-2)}$ does not compute embeddings between it and all its computable bi-embeddable copies. We obtain that for Boolean algebras which are not superatomic, there is no hyperarithmetic degree computing embeddings between all its computable bi-embeddable copies. On the other hand, if a computable Boolean algebra is superatomic, then there is a least computable ordinal $alpha$ such that $mathbf 0^{(alpha)}$ computes embeddings between all its computable bi-embeddable copies. The main technique used in this proof is a new variation of Ash and Knights pairs of structures theorem.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
