## Notre Dame Journal of Formal Logic

### Syntax and Semantics of the Logic $\mathcal{L}^\lambda_{\omega\omega}$

Carsten Butz

#### Abstract

In this paper we study the logic $\mathcal{L}^\lambda_{\omega\omega}$, which is first-order logic extended by quantification over functions (but not over relations). We give the syntax of the logic as well as the semantics in Heyting categories with exponentials. Embedding the generic model of a theory into a Grothendieck topos yields completeness of $\mathcal{L}^\lambda_{\omega\omega}$ with respect to models in Grothendieck toposes, which can be sharpened to completeness with respect to Heyting-valued models. The logic $\mathcal{L}^\lambda_{\omega\omega}$ is the strongest for which Heyting-valued completeness is known. Finally, we relate the logic to locally connected geometric morphisms between toposes.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 38, Number 3 (1997), 374-384.

Dates
First available in Project Euclid: 12 December 2002

https://projecteuclid.org/euclid.ndjfl/1039700744

Digital Object Identifier
doi:10.1305/ndjfl/1039700744

Mathematical Reviews number (MathSciNet)
MR1624950

Zentralblatt MATH identifier
0904.03005

Subjects
Secondary: 03C75: Other infinitary logic

#### Citation

Butz, Carsten. Syntax and Semantics of the Logic $\mathcal{L}^\lambda_{\omega\omega}$. Notre Dame J. Formal Logic 38 (1997), no. 3, 374--384. doi:10.1305/ndjfl/1039700744. https://projecteuclid.org/euclid.ndjfl/1039700744

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