arXiv subjects

Ur Ya'ar

Publications and source records attributed to Ur Ya'ar.

6 recordsLinked to original sources

arXiv2025

Inner models from extended logics and the Delta-operation

If $\mathcal{L}$ is an abstract logic (a.k.a. model theoretic logic), we can define the inner model $C(\mathcal{L})$ by replacing first order logic with $\mathcal{L}$ in G\"odel's definition of the inner model $L$ of constructible sets. Set theoretic properties of such inner models $C(\mathcal{L})$ have been investigated recently and a spectrum of new inner models is emerging between $L$ and $\mathrm{HOD}$. The topic of this paper is the effect on $C(\mathcal{L})$ of a slight modification of $\mathcal{L}$ i.e. how sensitive is $C(\mathcal{L})$ on the exact definition of $\mathcal{L}$? The $\Delta$-extension $\Delta(\mathcal{L})$ of a logic is generally considered a "mild" extension of $\mathcal{L}$. We give examples of logics $\mathcal{L}$ for which the inner model $C(\mathcal{L})$ is consistently strictly smaller than the inner model $C(\Delta(\mathcal{L}))$, and in one case we show this follows from the existence of $0^{\sharp}$.

math.LO↗

arXiv2024

Models for short sequences of measures in the cofinality-$ω$ constructible model

We investigate the relation between $C^{*}$, the model of sets constructible using first order logic augmented with the "cofinality-$ω$" quantifier, and "short" sequences of measures - sequences of measures of order $1$, which are shorter than their minimum. We show that certain core models for short sequences of measures are contained in $C^{*}$; we compute $C^{*}$ in a model of the form $L\left[\mathcal{U}\right]$ where $\mathcal{U}$ is a short sequence of measures, and in models of the form $L\left[\mathcal{U}\right]\left[G\right]$ where $G$ is generic for adding Prikry sequences to some of the measurables of $\mathcal{U}$; and prove that if there is an inner model with a short sequence of measures of order type $χ$, then there is such an inner model in $C^{*}$.

math.LO↗

arXiv2022

Iterated club shooting and the stationary-logic constructible model

We investigate iterating the construction of $C(\mathtt{aa})$, the $L$-like inner model constructed using stationary-logic. We show that it is possible to force over generic extensions of $L$ to obtain a model of $V=C(\mathtt{aa})$, and to obtain models in which the sequence of iterated $C(\mathtt{aa})$s is decreasing of arbitrarily large order types. For this we prove distributivity and stationary-set preservation properties for countable iterations of club-shooting forcings using mutually stationary sets, and introduce the notion of mutually fat sets which yields better distributivity results even for uncountable iterations.

math.LO↗

arXiv2021

Absoluteness for the theory of the inner model constructed from finitely many cofinality quantifiers

We prove that the theory of the models constructible using finitely many cofinality quantifiers - $C_{λ_{1},...,λ_{n}}^{*}$ and $C_{<λ_{1},...,<λ_{n}}^{*}$ for $λ_{1},...,λ_{n}$ regular cardinals - is set-forcing absolute under the assumption of class many Woodin cardinals, and is independent of the regular cardinals used. Towards this goal we prove some properties of the generic embedding induced from the stationary tower restricted to $<μ$-closed sets.

math.LO↗

arXiv2021

Iterating the cofinality-$ω$ constructible model

We investigate iterating the construction of $C^{*}$, the $L$-like inner model constructed using first order logic augmented with the "cofinality $ω$" quantifier. We first show that $\left(C^{*}\right)^{C^{*}}=C^{*}\ne L$ is equiconsistent with ZFC, as well as having finite strictly decreasing sequences of iterated $C^{*}$s. We then show that in models of the form $L^μ$ we get infinite decreasing sequences of length $ω$, and that an inner model with a measurable cardinal is required for that.

math.LO↗

arXiv2019

The modal logic of $σ$-centered forcing and related forcing classes

We consider the modality "$φ$ is true in every $σ$-centered forcing extension", denoted $\squareφ$, and its dual "$φ$ is true in some $σ$-centered forcing extension", denoted $\lozengeφ$ (where $φ$ is a statement in set theory), which give rise to the notion of a "principle of $σ$-centered forcing". We prove that if ZFC is consistent, then the modal logic of $σ$-centered forcing, i.e. the ZFC-provable principles of $σ$-centered forcing, is exactly $\mathsf{S4.2}$. We also generalize this result to other related classes of forcing.

math.LO↗