The Boolean Compactness Theorem for $\mathrm{L}_{\infty\infty}$

Juan M Santiago Suárez; Matteo Viale

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arXiv · 2507.21005

The Boolean Compactness Theorem for $\mathrm{L}_{\infty\infty}$

Juan M Santiago Suárez·Matteo Viale

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Abstract

We show that, contrary to the commonly held view, there is a natural and optimal compactness theorem for $\mathrm{L}_{\infty\infty}$ which generalizes the usual compactness theorem for first order logic. The key to this result is the switch from Tarski semantics to Boolean valued semantics. On the way to prove it, we also show that the latter is a (the?) natural semantics both for $\mathrm{L}_{\infty\infty}$ and for $\mathrm{L}_{\infty\omega}$.

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BibTeX RIS

Juan M Santiago Suárez, Matteo Viale. 2025-07-28. The Boolean Compactness Theorem for $\mathrm{L}_{\infty\infty}$. https://arxiv.org/abs/2507.21005

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