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Fodor space in generalized descriptive set theory

Ido Feldman, Miguel Moreno

Abstract

We study the continuous reducibility of isomorphism relations in the space of regresive functions in $κ^κ$. We show for inaccessible $κ$, that if $\mathcal{T}$ is a theory with less than $κ$ non-isomorphic models of size $κ$ and $\mathcal T'$ is unstable or superstable non-classifiable, then the isomorphism of models of $\mathcal{T}$ is continuous reducible to the isomorphism of models of $\mathcal{T}'$.

Fodor space in generalized descriptive set theory

Abstract

We study the continuous reducibility of isomorphism relations in the space of regresive functions in . We show for inaccessible , that if is a theory with less than non-isomorphic models of size and is unstable or superstable non-classifiable, then the isomorphism of models of is continuous reducible to the isomorphism of models of .
Paper Structure (7 sections, 9 theorems, 51 equations)

This paper contains 7 sections, 9 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Oganization of this paper
  4. Fodor space
  5. Trees
  6. Ehrenfeucht-Mostowski models
  7. Conclusions

Key Result

Proposition 2.2

Let $S\subseteq \kappa$ be a stationary set. Assume $\mathcal{T}$ is a countable complete classifiable theory over a countable vocabulary. Then $\cong_\mathcal{T} \ \hookrightarrow_c\ =^\kappa_S$. If $\cong_\mathcal{T}$ has less than $\kappa$ equivalence classes, then $\cong_\mathcal{T} \ \hookright

Theorems & Definitions (51)

  • Conjecture 1.1
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Definition 2.1: Fodor space
  • Proposition 2.2
  • Definition 2.3: The EF game
  • Definition 2.4
  • proof : Proof of Proposition \ref{['first_reduction']}
  • Definition 3.1: Coloured tree
  • ...and 41 more