# Countable structures, Ehrenfeucht strategies, and wadge reductions

Journal of Symbolic Logic 56 (4):1325-1348 (1991)
Abstract
For countable structures U and B, let $\mathfrak{U}\overset{\alpha}{\rightarrow}\mathfrak{B}$ abbreviate the statement that every Σ0 α (Lω1,ω) sentence true in U also holds in B. One can define a back and forth game between the structures U and B that determines whether $\mathfrak{U}\overset{\alpha}{\rightarrow}\mathfrak{B}$ . We verify that if θ is an Lω,ω sentence that is not equivalent to any Lω,ω Σ0 n sentence, then there are countably infinite models U and B such that $\mathfrak{U} \vDash \theta, \mathfrak{B} \vDash \neg \theta$ , and $\mathfrak{U}\overset{n}{\rightarrow}\mathfrak{B}$ . For countable languages L there is a natural way to view L structures with universe ω as a topological space, XL. Let [U] = {B ∈ XL∣B ≅ U} denote the isomorphism class of U. Let U and B be countably infinite nonisomorphic L structures, and let $C \subseteq \omega^\omega$ be any Π0 α subset. Our main result states that if $\mathfrak{U}\overset{\alpha}{\rightarrow}\mathfrak{B}$ , then there is a continuous function f: ωω → XL with the property that $x \in C \Rightarrow f(x) \in \lbrack\mathfrak{U}\rbrack$ and $x \notin C \Rightarrow f(x) \in \lbrack\mathfrak{B}\rbrack$ . In fact, for α ≤ 3, the continuous function f can be defined from the $\overset{\alpha}{\rightarrow}$ relation
Keywords No keywords specified (fix it)
Categories (categorize this paper)
DOI 10.2307/2275478
Options
 Save to my reading list Follow the author(s) My bibliography Export citation Find it on Scholar Edit this record Mark as duplicate Revision history Request removal from index

Download options
 PhilPapers Archive Upload a copy of this paper     Check publisher's policy on self-archival     Papers currently archived: 23,217 External links Setup an account with your affiliations in order to access resources via your University's proxy server Configure custom proxy (use this if your affiliation does not provide a proxy) Through your library Sign in / register to customize your OpenURL resolver.Configure custom resolver
References found in this work BETA
Citations of this work BETA

No citations found.

Similar books and articles

2009-01-28

### Total downloads

5 ( #579,823 of 1,932,455 )

### Recent downloads (6 months)

1 ( #456,114 of 1,932,455 )

How can I increase my downloads?

My notes

Discussion
 Order: Most recently started first Most recently active first There  are no threads in this forum
Nothing in this forum yet.