arXiv subjects

Andreas Lietz

Publications and source records attributed to Andreas Lietz.

5 recordsLinked to original sources

arXiv2025

Separating Maximality Principles

We investigate fragments of generic absoluteness principles known as Maximality Principles. We determine the consistency strength of $Σ_n$-$\mathsf{MP}(\mathbb R)$ and $Π_n$-$\mathsf{MP}(\mathbb R)$, the boldface Maximality Principle restricted respectively to $Σ_n$- and $Π_n$-formulas. Further, we show that no implication between $Σ_n$-$\mathsf{MP}(\mathbb R)$ and $Π_n$-$\mathsf{MP}(\mathbb R)$ is provable in $\mathsf{ZFC}$. We also establish the consistency, relative to a Woodin cardinal, of the Maximality Principle for $ω_1$-preserving posets with countable ordinal parameters and prove its consistency strength is bounded below by a Ramsey cardinal. Finally, we resolve questions of Ikegami-Trang and Goodman by separating the Maximality Principle for stationary set preserving posets restricted to $Σ_2$-formulas from $\mathsf{MM}^{++}$ in the presence of large cardinals.

math.LO↗

arXiv2024

Infinite Hat Problems and Large Cardinals

Picture countably many logicians all wearing a hat in one of $κ$-many colours. They each get to look at finitely many other hats and afterwards make finitely many guesses for their own hat's colour. For which $κ$ can the logicians guarantee that at least one of them guesses correctly? This will be the archetypical hat problem we analyse and solve here. We generalise this by varying the amount of logicians as well as the number of allowed guesses and describe exactly for which combinations the logicians have a winning strategy. We also solve these hat problems under the additional restriction that their vision is restrained in terms of a partial order. Picture e.g.~countably many logicians standing on the real number line and each logician is only allowed to look at finitely many others in front of them. In many cases, the least $κ$ for which the logicians start losing can be described by an instance of the free subset property which in turn is connected to large cardinals. In particular, $\mathrm{ZFC}$ can sometimes not decide whether or not the logicians can win for every possible set of colours.

math.LO↗

arXiv2024

The Axiom of Choice in the $κ$-Mantle

Usuba has asked whether the $κ$-mantle, the intersection of all grounds that extend to $V$ via a forcing of size ${<}κ$, is always a model of ZFC. We give a negative answers by constructing counterexamples where $κ$ is a Mahlo cardinal, $κ=ω_1$ and where $κ$ is the successor of a regular uncountable cardinal.

math.LO↗

arXiv2024

An Iteration Theorem for $ω_1$-preserving Forcings

We prove an iteration theorem which guarantees for a wide class of nice iterations of $ω_1$-preserving forcings that $ω_1$ is not collapse, at the price of needing large cardinals to burn as fuel. More precisely, we show that a nice iteration of $ω_1$-preserving forcings which force SRP at successor steps and preserves old stationary sets does not collapse $ω_1$.

math.LO↗

arXiv2024

Forcing "$\mathrm{NS}_{ω_1}$ is $ω_1$-dense" From Large Cardinals

We answer a question of Woodin by showing that assuming an inaccessible cardinal $κ$ which is a limit of ${<}κ$-supercompact cardinals exists, there is a stationary set preserving forcing $\mathbb{P}$ so that $V^{\mathbb P}\models``\mathrm{NS}_{ω_1}\text{ is }ω_1\text{-dense}"$. We also introduce a new forcing axiom $\mathrm{QM}$, show it is consistent assuming a supercompact limit of supercompact cardinals and prove that it implies $\mathbb{Q}_{\mathrm{max}}\text{-}(*)$. Consequently, $\mathrm{QM}$ implies ``$\mathrm{NS}_{ω_1}$ is $ω_1$-dense".

math.LO↗