Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On Some Infinitary Logics

Jouko Vaananen, Boban Velickovic

Abstract

We define a new class of infinitary logics $\mathscr L^1_{κ,α}$ generalizing Shelah's logic $\mathbb L^1_κ$ defined in \cite{MR2869022}. If $κ=\beth_κ$ and $α<κ$ is infinite then our logic coincides with $\mathbb L^1_κ$. We study the relation between these logics for different parameters $κ$ and $α$. We give many examples of classes of structures that can or cannot be defined in these logics. Finally, we give a different version of Lindström's Theorem for $\mathbb L^1_κ$ in terms of the $φ$-submodel relation.

On Some Infinitary Logics

Abstract

We define a new class of infinitary logics generalizing Shelah's logic defined in \cite{MR2869022}. If and is infinite then our logic coincides with . We study the relation between these logics for different parameters and . We give many examples of classes of structures that can or cannot be defined in these logics. Finally, we give a different version of Lindström's Theorem for in terms of the -submodel relation.
Paper Structure (18 sections, 33 theorems, 102 equations)

This paper contains 18 sections, 33 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The similarity game
  4. Komjath-Shelah lemma
  5. The Main Theorem
  6. Examples
  7. Cardinality
  8. Covering every small set by countably many sets from a fixed list
  9. The models $\Phi(A)$
  10. The irrationals vs. the reals
  11. The reals vs. the reals minus a point
  12. $\kappa^{\le \omega}$ and $\kappa^{\le \omega}\setminus\{b\}$
  13. Prikry forcing and the game $\rm DG^\beta_{<\kappa,\alpha}$
  14. $\oplus_n\Phi$ and $\Phi$
  15. A class of infinitary logics
  16. ...and 3 more sections

Key Result

Theorem 2.7

Let $\mathscr L$ be a regular logic. Then the following are equivalent:

Theorems & Definitions (77)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Theorem 2.7: MR244013
  • Definition 2.8
  • Definition 3.1
  • Definition 3.2
  • ...and 67 more