Boolean valued semantics for infinitary logics
| Autor: | Santiago, Juan M., Viale, Matteo |
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| Rok vydání: | 2021 |
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| Druh dokumentu: | Working Paper |
| Popis: | It is well known that the completeness theorem for $\mathrm{L}_{\omega_1\omega}$ fails with respect to Tarski semantics. Mansfield showed that it holds for $\mathrm{L}_{\infty\infty}$ if one replaces Tarski semantics with boolean valued semantics. We use forcing to improve his result in order to obtain a stronger form of boolean completeness (but only for $\mathrm{L}_{\infty\omega}$). Leveraging on our completeness result, we establish the Craig interpolation property and a strong version of the omitting types theorem for $\mathrm{L}_{\infty\omega}$ with respect to boolean valued semantics. We also show that a weak version of these results holds for $\mathrm{L}_{\infty\infty}$ (if one leverages instead on Mansfield's completeness theorem). Furthermore we bring to light (or in some cases just revive) several connections between the infinitary logic $\mathrm{L}_{\infty\omega}$ and the forcing method in set theory. |
| Databáze: | arXiv |
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