Farmer Schlutzenberg

activity

20122026

most citedDeterminacy and Jónsson cardinals in

L (R)

21 citations · 60 across the 34 of their papers we have counts for

collaborators

Showing math.LOShow all

38 papers · 1 filter

math.LO2026

Diamonds in mice

Farmer Schlutzenberg

Let $M$ be a $(0, ω_{1} + 1)$ -iterable mouse with no largest cardinal. Let $γ \leq κ$ be uncountable cardinals of $M$ , with $κ$ regular in $M$ . Then $M$ satisfies $♢_{κγ}^{+}$ , a…

math.LO2026

$⋆$ -translation for Varsovian models

Farmer Schlutzenberg

The technique of $⋆$ -translation is important in arguments calibrating the strengths of determinacy theories against large cardinals, for example in [9] and [1] in the paper's…

math.LO2025

Full normalization for $κ^{+}$ -supercompactness

Farmer Schlutzenberg

We extend the normalization results of the author's paper "Full normalization for transfinite stacks" [5] to mice at the level of $κ^{+}$ -supercompactness: given a normal iteration s…

math.LO2025

Power $Σ_{1}$ in Card with two Woodin cardinals

Farmer Schlutzenberg

Väänänen and Welch asked in the paper "When cardinals determine the power set: inner models and Härtig quantifier logic" which large cardinals are consistent with the power set ope…

math.LO2025★ 1 cited

Analysis of HOD for Admissible Structures

Jan Kruschewski, Farmer Schlutzenberg

Let $n \geq 1$ and assume that there is a Woodin cardinal. For $x \in R$ let $α_{x}$ be the least $β$ such that \[ L_β[x] \models Σ_n \text{-KP} + \exists κ(``κ\text{ is ina…

math.LO2024

Ladder mice

Farmer Schlutzenberg

Assume ZF + AD + $V = L (R)$ . We prove some "mouse set" theorems, for definability over $J_{α} (R)$ where $[α, α]$ is a projective-like gap (of $L (R)$ ) and $α$ …

math.LO2024

Stationary tower free homogeneously Suslin scales

Farmer Schlutzenberg, John R. Steel

Let $λ$ be a limit of Woodin cardinals. It was shown by the second author that the pointclass of $< λ$ -homogeneously Suslin sets has the scale property. We give a new proof of thi…

math.LO2024

Mice with Woodin cardinals from a Reinhardt

Farmer Schlutzenberg

Suppose there is a Reinhardt cardinal. Then (1) $M_{n} (X)$ exists and is fully iterable (above $X$ ) for every transitive set $X$ and every $n < ω$ (here $M_{n} (X)$ denotes the canonical…

math.LO2024

Low level definability above large cardinals

Farmer Schlutzenberg

We study connections between definability in generalized descriptive set theory and large cardinals, under ZFC. We show that if $κ$ is a limit of measurables then there is no wello…

math.LO2023

Mouse scales

Farmer Schlutzenberg

We give a construction of scales (in the descriptive set theoretic sense) directly from mouse existence hypotheses, without using any determinacy arguments. The construction is rel…

math.LO2023

$Σ_{1}$ gaps as derived models and correctness of mice

Farmer Schlutzenberg, John Steel

Assume ZF + AD + V=L(R). Let $[α, β]$ be a $Σ_{1}$ gap with $J_{α} (R)$ admissible. We analyze $J_{β} (R)$ as a natural form of "derived model" of a premouse $P$ , where $P$ is found in a ge…

math.LO2023

The initial segment condition for $κ^{+}$ -supercompactness

Farmer Schlutzenberg

We give a development of the fine structure of mice with long extenders, to the level of $κ^{+}$ -supercompact cardinals $κ$ . We do this using a hierarchy with features more analogous…

math.LO2022

Varsovian models $ω$

Farmer Schlutzenberg

For $n < ω$ , let $N_{n}$ be the minimal iterable proper class mouse $M$ such that $M ⊨$ "there are ordinals $δ_{0} < κ_{0} < \dots < δ_{n - 1} < κ_{n - 1}$ such that each $δ_{i}$ is a Woodin cardi…

math.LO2022★ 1 cited

On a Conjecture Regarding the Mouse Order for Weasels

Jan Kruschewski, Farmer Schlutzenberg

We investigate Steel's conjecture in 'The Core Model Iterability Problem', that if $W$ and $R$ are $Ω + 1$ -iterable, $1$ -small weasels, then $W \leq^{*} R$ iff there is a club $C\subse…

math.LO2021★ 1 cited

Varsovian models II

Grigor Sargsyan, Ralf Schindler, Farmer Schlutzenberg

Assume the existence of sufficent large cardinals. Let $M_{sw n}$ be the minimal iterable proper class $L [E]$ model satisfying "there are $δ_{0} < κ_{0} < \dots < δ_{n - 1} < κ_{n - 1}$ s…

math.LO2021★ 1 cited

Mutually embeddable models of ZFC

Monroe Eskew, Sy-David Friedman, Yair Hayut +1

We investigate systems of transitive models of ZFC which are elementarily embeddable into each other and the influence of definability properties on such systems.

math.LO2021★ 1 cited

Local mantles of $L [x]$

Farmer Schlutzenberg

Assume ZFC. Let $κ$ be a cardinal. Recall that a $< κ$ -ground is a transitive proper class $W$ modelling ZFC such that $V$ is a generic extension of $W$ via a forcing $\mathbb{P}\…

math.LO2021★ 2 cited

Full normalization for transfinite stacks

Farmer Schlutzenberg

We describe the extension of normal iteration strategies with appropriate condensation properties to strategies for stacks of normal trees, with full normalization. Given a regular…

math.LO2021

Background construction for $λ$ -indexed mice

Farmer Schlutzenberg

Let $M$ be a $λ$ -indexed (that is, Jensen indexed) premouse. We prove that $M$ is iterable with respect to standard $λ$ -iteration rules iff $M$ is iterable with respect to a natura…

math.LO2020★ 2 cited

Ordinal definability in $L [E]$

Farmer Schlutzenberg

Let $M$ be a tame mouse modelling ZFC. We show that $M$ satisfies " $V = HOD_{x}$ for some real $x$ ", and that the restriction $E ↾ [ω_{1}^{M}, OR^{M})$ …

math.LO2020★ 2 cited

Fine structure from normal iterability

Farmer Schlutzenberg

We show that (i) the standard fine structural properties for premice follow from normal iterability (whereas the classical proof relies on iterability for stacks of normal trees),…

math.LO2020★ 2 cited

Extenders under ZF and constructibility of rank-to-rank embeddings

Farmer Schlutzenberg

Assume ZF (without the Axiom of Choice). Let $j : V_{ε} \to V_{δ}$ be a non-trivial $\in$ -cofinal $Σ_{1}$ -elementary embedding, where $ε, δ$ are limit ordinals. We prove…

math.LO2020

Choice principles in local mantles

Farmer Schlutzenberg

Assume ZFC. Let $κ$ be a cardinal. A $< κ$ -ground is a transitive proper class $W$ modelling ZFC and such that $V$ is a generic extension of $W$ via a forcing $P \in W$ of…

math.LO2020★ 1 cited

Periodicity in the cumulative hierarchy

Gabriel Goldberg, Farmer Schlutzenberg

We investigate the structure of rank-to-rank elementary embeddings, working in ZF set theory without the Axiom of Choice. Recall that the levels $V_{α}$ of the cumulative hierarchy a…

Diamonds in mice

⋆-translation for Varsovian models

Full normalization for κ+-supercompactness

Power Σ1​ in Card with two Woodin cardinals

Analysis of HOD for Admissible Structures

Ladder mice

Stationary tower free homogeneously Suslin scales

Mice with Woodin cardinals from a Reinhardt

Low level definability above large cardinals

Mouse scales

Σ1​ gaps as derived models and correctness of mice

The initial segment condition for κ+-supercompactness

Varsovian models ω

On a Conjecture Regarding the Mouse Order for Weasels

Varsovian models II

Mutually embeddable models of ZFC

Local mantles of L[x]

Full normalization for transfinite stacks

Background construction for λ-indexed mice

Ordinal definability in L[E]

Fine structure from normal iterability

Extenders under ZF and constructibility of rank-to-rank embeddings

Choice principles in local mantles

Periodicity in the cumulative hierarchy

$⋆$ -translation for Varsovian models

Full normalization for $κ^{+}$ -supercompactness

Power $Σ_{1}$ in Card with two Woodin cardinals

$Σ_{1}$ gaps as derived models and correctness of mice

The initial segment condition for $κ^{+}$ -supercompactness

Varsovian models $ω$

Local mantles of $L [x]$

Background construction for $λ$ -indexed mice

Ordinal definability in $L [E]$