Boolean-valued second-order logic revisited
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866917987777249280 |
|---|---|
| author | Ikegami, Daisuke |
| author_facet | Ikegami, Daisuke |
| contents | Following the paper~[3] by Väänänen and the author, we continue to investigate on the difference between Boolean-valued second-order logic and full second-order logic. We show that the compactness number of Boolean-valued second-order logic is equal to $ω_1$ if there are proper class many Woodin cardinals. This contrasts the result by Magidor~[10] that the compactness number of full second-order logic is the least extendible cardinal. We also introduce the inner model $C^{2b}$ constructed from Boolean-valued second-order logic using the construction of Gödel's Constructible Universe L. We show that $C^{2b}$ is the least inner model of $\mathsf{ZFC}$ closed under $\mathrm{M}_n^{\#}$ operators for all $n < ω$, and that $C^{2b}$ enjoys various nice properties as Gödel's L does, assuming that Projective Determinacy holds in any set generic extension. This contrasts the result by Myhill and Scott~[14] that the inner model constructed from full second-order logic is equal to HOD, the class of all hereditarily ordinal definable sets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_12602 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Boolean-valued second-order logic revisited Ikegami, Daisuke Logic Following the paper~[3] by Väänänen and the author, we continue to investigate on the difference between Boolean-valued second-order logic and full second-order logic. We show that the compactness number of Boolean-valued second-order logic is equal to $ω_1$ if there are proper class many Woodin cardinals. This contrasts the result by Magidor~[10] that the compactness number of full second-order logic is the least extendible cardinal. We also introduce the inner model $C^{2b}$ constructed from Boolean-valued second-order logic using the construction of Gödel's Constructible Universe L. We show that $C^{2b}$ is the least inner model of $\mathsf{ZFC}$ closed under $\mathrm{M}_n^{\#}$ operators for all $n < ω$, and that $C^{2b}$ enjoys various nice properties as Gödel's L does, assuming that Projective Determinacy holds in any set generic extension. This contrasts the result by Myhill and Scott~[14] that the inner model constructed from full second-order logic is equal to HOD, the class of all hereditarily ordinal definable sets. |
| title | Boolean-valued second-order logic revisited |
| topic | Logic |
| url | https://arxiv.org/abs/2504.12602 |