Second order arithmetic as the model companion of set theory

Venturi, Giorgio; Viale, Matteo

doi:10.1007/s00153-022-00831-9

Cited by 4 publications

(10 citation statements)

References 14 publications

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“…Then there is a recursive set 38 B ∈ spec AMC (S, ∈) with ∈ B containing ∈ Δ 0 , and such that for any Π 2 -sentence for ∈ B and any ∈-theory R ⊇ S the following are equivalent:…”

Section: Forcibility Versus Absolute Model Companionshipmentioning

confidence: 99%

“…37 The reader unaware of what MM ++ or a stationary set preserving forcing is can skip the second and third items of the theorem. 38 ∈ B can be for example ∈ NS 1 ,A as defined in [41, notation 5.3] where A consists of the sets of reals lightface definable in L(Ord ); see also (B) on page (B). 39 Here and elsewhere we write N to denote the relativization of to a definable class (or set) N; see [25, definition IV.2.1] for details.…”

Section: H Mmentioning

confidence: 99%

“…The present paper presents a new approach to the quest of a solution of the continuum problem and—more generally—to the search of new set theoretic truths; this approach stems from a model-theoretic perspective on set theory, and aims to clarify the philosophical import of the results in [38, 41]. As is standard, we recognize the intrinsic limitations of in capturing set-theoretic validities, but, to overcome these limitations, we bring forward the idea that a powerful criterion to validate a new axiom is given by the nice model theoretic properties of the first order presentations of set theory which include it.…”

Section: Introductionmentioning

confidence: 99%

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What Model Companionship Can Say About the Continuum Problem

Venturi¹,

VIALE²

2023

The Review of Symbolic Logic

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We present recent results on the model companions of set theory, placing them in the context of a current debate in the philosophy of mathematics. We start by describing the dependence of the notion of model companionship on the signature, and then we analyze this dependence in the specific case of set theory. We argue that the most natural model companions of set theory describe (as the signature in which we axiomatize set theory varies) theories of $H_{\kappa ^+}$ , as $\kappa $ ranges among the infinite cardinals. We also single out $2^{\aleph _0}=\aleph _2$ as the unique solution of the continuum problem which can (and does) belong to some model companion of set theory (enriched with large cardinal axioms). While doing so we bring to light that set theory enriched by large cardinal axioms in the range of supercompactness has as its model companion (with respect to its first order axiomatization in certain natural signatures) the theory of $H_{\aleph _2}$ as given by a strong form of Woodin’s axiom $(*)$ (which holds assuming $\mathsf {MM}^{++}$ ). Finally this model-theoretic approach to set-theoretic validities is explained and justified in terms of a form of maximality inspired by Hilbert’s axiom of completeness.

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Section: Forcibility Versus Absolute Model Companionshipmentioning

confidence: 99%

Section: H Mmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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What Model Companionship Can Say About the Continuum Problem

Venturi¹,

VIALE²

2023

The Review of Symbolic Logic

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“…Notation 2.1. Given a cba B ∈ V and a family A of elements of V B , 24 For a partial order P and p ∈ P , P ↾ p = {q ∈ P : q ≤ p}. Definition 2.2.…”

Section: Forcing With Forcingsmentioning

confidence: 99%

“…However, much stronger connections between generic absoluteness, forcing axioms, and model companionship can be obtained if one works with richer signatures. For instance, [24] shows that the theory of H V ω 1 is the model companion of the theory of V in the signature τ UB = τ ST ∪UB V where UB V is the class of universally Baire sets, and each B ∈ UB V defines a predicate symbol for τ UB . Using the result of the first author and Schindler establishing that Woodin's axiom (*) follows from MM ++ [2], the second author [25] has shown that in models of MM ++ where UB ♯ is invariant across forcing extensions, the theory of H ω 2 is the model companion of the theory of V in signature τ NSω 1 ,UB = τ UB ∪ {NS ω 1 } (where NS ω 1 is the nonstationary ideal on ω 1 , and is the canonical interpretation of its associated unary predicate symbol in τ NS ω 1 ,UB ).…”

Section: Introductionmentioning

confidence: 99%

Incompatible bounded category forcing axioms

Asperó

Viale

2022

J. Math. Log.

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We introduce bounded category forcing axioms for well-behaved classes [Formula: see text]. These are strong forms of bounded forcing axioms which completely decide the theory of some initial segment of the universe [Formula: see text] modulo forcing in [Formula: see text], for some cardinal [Formula: see text] naturally associated to [Formula: see text]. These axioms naturally extend projective absoluteness for arbitrary set-forcing — in this situation [Formula: see text] — to classes [Formula: see text] with [Formula: see text]. Unlike projective absoluteness, these higher bounded category forcing axioms do not follow from large cardinal axioms but can be forced under mild large cardinal assumptions on [Formula: see text]. We also show the existence of many classes [Formula: see text] with [Formula: see text] giving rise to pairwise incompatible theories for [Formula: see text].

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Are Large Cardinal Axioms Restrictive?

Barton

2023

Philosophia Mathematica

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The independence phenomenon in set theory, while pervasive, can be partially addressed through the use of large cardinal axioms. A commonly assumed idea is that large cardinal axioms are species of maximality principles. In this paper I question this claim. I show that there is a kind of maximality (namely absoluteness) on which large cardinal axioms come out as restrictive relative to a formal notion of restrictiveness. Within this framework, I argue that large cardinal axioms can still play many of their usual foundational roles.

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Second order arithmetic as the model companion of set theory

Cited by 4 publications

References 14 publications

What Model Companionship Can Say About the Continuum Problem

What Model Companionship Can Say About the Continuum Problem

Incompatible bounded category forcing axioms

Are Large Cardinal Axioms Restrictive?

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