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On $(Σ^2_1)^{uB}$ Absoluteness Between V and HOD

Gabriel Goldberg, Dan Hathaway

TL;DR

The paper investigates absoluteness between V and HOD for statements of the form (Σ^2_1)^{uB} under large-cardinal assumptions, notably a proper class of Woodin cardinals. It combines Woodin’s Σ^2_1 Basis Theorem with Vopěnka’s theorem to prove downward absoluteness to HOD under OD/uB scale hypotheses, and shows upward absoluteness when HOD itself has a proper class of Woodin cardinals. It also develops Ω-logic, discusses the Ω-conjecture, and analyzes the interaction with V = Ultimate L, showing that Ω-proofs transferring between V and HOD yield deep constraints on what can be absolved generically. The work further explores how universally Baire sets witnessed in HOD interact with forcing, symmetric extensions, and determinacy fragments in inner models, and concludes with results about reals lying in HOD under Ω-logic and determinacy assumptions. Collectively, the results illuminate the precise ways large-cardinal hypotheses constrain the transfer of projective and inner-model properties between V and HOD, and reveal limits of absoluteness under Ultimate L.

Abstract

We put together Woodin's $Σ^2_1$ basis theorem of AD$^+$ and Vopěnka's theorem to conclude the following: If there is a proper class of Woodin cardinals, then every $(Σ^2_1)^{\mbox{uB}}$ statement that is true in $V$ is true in $\mbox{HOD}$. Moreover, this is true even if we allow a parameter $C \subseteq \mathbb{R}$ such that $C$ and its complement have scales that are $\mbox{OD}$ and universally Baire. We also investigate whether $(Σ^2_1)^{\mbox{uB}}$ statements are upwards absolute from $\mbox{HOD}$ to $V$ under large cardinal hypotheses, observing that this is true if $\mbox{HOD}$ has a proper class of Woodin cardinals. Finally, we discuss $(\forall^{\mathbb{R}})\, (Σ^2_1)^{\mbox{uB}}$ absoluteness and conclude that this much absoluteness between $\mbox{HOD}$ and $V$ cannot be implied by any large cardinal axiom consistent with the axiom ``$V =$ Ultimate $L$''.

On $(Σ^2_1)^{uB}$ Absoluteness Between V and HOD

TL;DR

The paper investigates absoluteness between V and HOD for statements of the form (Σ^2_1)^{uB} under large-cardinal assumptions, notably a proper class of Woodin cardinals. It combines Woodin’s Σ^2_1 Basis Theorem with Vopěnka’s theorem to prove downward absoluteness to HOD under OD/uB scale hypotheses, and shows upward absoluteness when HOD itself has a proper class of Woodin cardinals. It also develops Ω-logic, discusses the Ω-conjecture, and analyzes the interaction with V = Ultimate L, showing that Ω-proofs transferring between V and HOD yield deep constraints on what can be absolved generically. The work further explores how universally Baire sets witnessed in HOD interact with forcing, symmetric extensions, and determinacy fragments in inner models, and concludes with results about reals lying in HOD under Ω-logic and determinacy assumptions. Collectively, the results illuminate the precise ways large-cardinal hypotheses constrain the transfer of projective and inner-model properties between V and HOD, and reveal limits of absoluteness under Ultimate L.

Abstract

We put together Woodin's basis theorem of AD and Vopěnka's theorem to conclude the following: If there is a proper class of Woodin cardinals, then every statement that is true in is true in . Moreover, this is true even if we allow a parameter such that and its complement have scales that are and universally Baire. We also investigate whether statements are upwards absolute from to under large cardinal hypotheses, observing that this is true if has a proper class of Woodin cardinals. Finally, we discuss absoluteness and conclude that this much absoluteness between and cannot be implied by any large cardinal axiom consistent with the axiom `` Ultimate ''.
Paper Structure (9 sections, 39 theorems, 83 equations)

This paper contains 9 sections, 39 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Upwards absoluteness
  4. Common symmetric extensions of V and HOD
  5. Local results
  6. Downward to $\textnormal{HOD}$ theorem (main theorem)
  7. Having scales that are OD and uB
  8. The $\Omega$ Conjecture
  9. $\Omega$ proofs that reals are in $\textnormal{HOD}$

Key Result

Corollary 1.3

If the existence of a proper class of some kind of large cardinal is consistent with the axiom "$V$ = Ultimate$L$", then the existence of such a proper class of large cardinals in both $V$ and $\textnormal{HOD}$ cannot imply $(\forall^{{^\omega}\omega})(\Sigma^2_1)^{\textnormal{uB}}$ absoluteness be

Theorems & Definitions (95)

  • Corollary 1.3
  • proof
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Theorem 2.4
  • proof
  • Definition 2.6
  • Theorem 2.7: Vopěnka
  • proof
  • ...and 85 more