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Boolean-valued second-order logic revisited

Daisuke Ikegami

TL;DR

This work compares Boolean-valued second-order logic (BVSOL) with full second-order logic under strong large-cardinal hypotheses, showing that BVSOL can be much simpler to analyze. It proves that the BVSOL compactness number is $ω_1$ once there are proper class many Woodin cardinals, and introduces the inner model $C^{2b}$ built by applying Gödel’s constructible process to $L$ in the Boolean-valued setting; under Projective Determinacy in any set-generic extension, $C^{2b}$ is the least inner model of ZFC closed under all $M_n^{\#}$, enjoys a fine structure, and is invariant under forcing, mirroring Gödel’s $L$ properties. In contrast, Myhill–Scott showed that the inner model from full SOL is $HOD$, which lacks these structural features; the paper thus clarifies a regime where BVSOL is substantially more tractable than full SOL. It also links the $L^{2b}$-compactness number to the least weakly generically extendible cardinal and proves equiconsistencies with generically extendible notions, yielding a coherent inner-model picture for Boolean-valued logics. Open questions probe the boundaries of these notions, including the Löwenheim number $\ell^{2b}$ and further equiconsistency results.

Abstract

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.

Boolean-valued second-order logic revisited

TL;DR

This work compares Boolean-valued second-order logic (BVSOL) with full second-order logic under strong large-cardinal hypotheses, showing that BVSOL can be much simpler to analyze. It proves that the BVSOL compactness number is once there are proper class many Woodin cardinals, and introduces the inner model built by applying Gödel’s constructible process to in the Boolean-valued setting; under Projective Determinacy in any set-generic extension, is the least inner model of ZFC closed under all , enjoys a fine structure, and is invariant under forcing, mirroring Gödel’s properties. In contrast, Myhill–Scott showed that the inner model from full SOL is , which lacks these structural features; the paper thus clarifies a regime where BVSOL is substantially more tractable than full SOL. It also links the -compactness number to the least weakly generically extendible cardinal and proves equiconsistencies with generically extendible notions, yielding a coherent inner-model picture for Boolean-valued logics. Open questions probe the boundaries of these notions, including the Löwenheim number and further equiconsistency results.

Abstract

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 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 constructed from Boolean-valued second-order logic using the construction of Gödel's Constructible Universe L. We show that is the least inner model of closed under operators for all , and that 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.

Paper Structure

This paper contains 5 sections, 18 theorems, 3 equations, 1 table.

Table of Contents

  1. Introduction
  2. Basic definitions and theorems
  3. Compactness numbers
  4. The construction of Gödel's Constructible Universe
  5. Questions

Key Result

Lemma 4

Let $\mathcal{L}$ be as above and $M = ( A, \mathbb{B} , \{R_i^M\}_{i \in I }, \{ c_j^M\}_{j \in J})$ be a Boolean-valued $\mathcal{L}$-structure. Let $G$ be a $\mathbb{B}$-generic filter over $V$. Then for each second-order $\mathcal{L}$-formula, $\vec{a} \in A^{< \omega}$, and $\vec{f} \in (\mathb where $\vec{f^G} = \{f_1^G, \ldots , f_n^G\}$ and for each $i$ with $1\le i \le n$,

Theorems & Definitions (47)

  • Definition 1: Boolean-valued structures
  • Definition 2
  • Definition 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • ...and 37 more