A Set-Theoretic Translation of Modal Logic via Forcing

@misc{chopoghloo2026settheoretic,
  title = {{A Set-Theoretic Translation of Modal Logic via Forcing}},
  author = {Somayeh Chopoghloo and Mohammad Golshani},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.07330},
  eprint = {2609.07330},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.07330},
}

TY  - UNPB
TI  - A Set-Theoretic Translation of Modal Logic via Forcing
AU  - Somayeh Chopoghloo
AU  - Mohammad Golshani
PY  - 2026
DA  - 2026/09/07
PB  - arXiv
DO  - 10.48550/arxiv.2609.07330
UR  - https://arxiv.org/abs/2609.07330
AB  - We develop a set-theoretic translation of a normal modal extension T_m of a recursively axiomatizable first-order theory T. We first pass to the Henkin expansion of the underlying language by adding witness constants and work with the sentence algebra of this expansion. The translation is constructed using the corresponding Lindenbaum-Tarski algebra, the Stone space of its ultrafilters, and quotient forcing by a σ-ideal of Borel sets; in particular, the meager ideal yields Cohen forcing and the null ideal yields the random forcing. In a Boolean-valued universe, we interpret the modal operator Box by membership of the Boolean value of the translated formula in a suitable filter name. We prove that this translation is sound and complete: a modal formula is provable in T_m if and only if its set-theoretic translation is forced in every associated interpretation. We then build Kripke frames and models from generic extensions and show that the forcing interpretation of Box agrees with quantification over the corresponding accessibility relation. As a consequence, we obtain completeness with respect to the resulting Kripke models.
ER  -