The Boolean Compactness Theorem for $\mathrm{L}_{\infty\infty}$
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866916867390570496 |
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| author | Suárez, Juan M Santiago Viale, Matteo |
| author_facet | Suárez, Juan M Santiago Viale, Matteo |
| contents | 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 $\mathrm{L}_{\inftyω}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_21005 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Boolean Compactness Theorem for $\mathrm{L}_{\infty\infty}$ Suárez, Juan M Santiago Viale, Matteo Logic 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 $\mathrm{L}_{\inftyω}$. |
| title | The Boolean Compactness Theorem for $\mathrm{L}_{\infty\infty}$ |
| topic | Logic |
| url | https://arxiv.org/abs/2507.21005 |