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Cartagena Logic

Siiri Kivimäki, Jouko Väänänen, Andrés Villaveces

TL;DR

The main result is that Cartagena logic is a good syntactically defined approximation to Shelah's infinitary ${\mathcal L}^1_\kappa$.

Abstract

We introduce a new kind of infinitary logic that we call Boolean expansion of ${\mathcal L}_{κκ}$. This logic involves a new kind of variable, that we call generalised Boolean variable. These variables range over the powerset of a cardinal number in a way reminiscent of random variables. From this Boolean expansion, we extract a traditional infinitary logic, called Cartagena logic. We prove several model-theoretic properties of Cartagena logic, and give multiple examples of its expressive power. The main result is that Cartagena logic is a good syntactically defined approximation to Shelah's infinitary ${\mathcal L}^1_κ$. The latter is not known to have a generative syntax, while Cartagena logic does have a very clear one.

Cartagena Logic

TL;DR

The main result is that Cartagena logic is a good syntactically defined approximation to Shelah's infinitary .

Abstract

We introduce a new kind of infinitary logic that we call Boolean expansion of . This logic involves a new kind of variable, that we call generalised Boolean variable. These variables range over the powerset of a cardinal number in a way reminiscent of random variables. From this Boolean expansion, we extract a traditional infinitary logic, called Cartagena logic. We prove several model-theoretic properties of Cartagena logic, and give multiple examples of its expressive power. The main result is that Cartagena logic is a good syntactically defined approximation to Shelah's infinitary . The latter is not known to have a generative syntax, while Cartagena logic does have a very clear one.

Paper Structure

This paper contains 18 sections, 20 theorems, 112 equations.

Table of Contents

  1. Boolean expansion $\mathcal{L}^{\mathop{\mathrm{\mathsf{Bool}}}\nolimits}_{\kappa\kappa}$
  2. Formulas with Boolean variables
  3. Free Boolean variables and substitution
  4. Projection to $\mathcal{L}_{\kappa\kappa}$
  5. Definition of Cartagena logic
  6. Cartagena game
  7. Syntax of Cartagena logic
  8. Upwards and downwards correctness
  9. Cartagena formulas
  10. Game-syntax equivalence
  11. Model theory of Cartagena logic
  12. Union Lemma
  13. Löwenheim-Skolem-Tarski Theorem
  14. Strong undefinability of well order
  15. Expressive power of Cartagena logic
  16. ...and 3 more sections

Key Result

Proposition 1.14

Every global valuation $\bar{A}$ uniquely determines a mapping \begin{tikzcd} \LL^{\Bool}_{\kappa\kappa}\text{-formulas}\arrow{r}{\pi_{\bar{A}}}&\LL_{\kappa\kappa}\text{-formulas} \end{tikzcd}via the substitution \begin{tikzcd} \phi(\x,\X)\arrow[|->]{r} & \phi(\x,\A). \end{tikzcd}

Theorems & Definitions (77)

  • Definition 1.1
  • Definition 1.3: Boolean extension $\mathcal{L}^{\mathop{\mathrm{\mathsf{Bool}}}\nolimits}_{\kappa\kappa}$
  • Example 1.4
  • Remark 1.5
  • Definition 1.6
  • Definition 1.7
  • Remark 1.8
  • Definition 1.9
  • Remark 1.10
  • Definition 1.11
  • ...and 67 more