\documentclass[a4paper, 11pt]{scrartcl} \usepackage{jonne-phd} \linespread{1.3} \CompileMatrices \usepackage[mathletters]{ucs} \usepackage[utf8x]{inputenc} %\usepackage[breaklinks=true,unicode=true]{hyperref} \usepackage{hyperref} \hypersetup{colorlinks,% citecolor=black,% filecolor=black,% linkcolor=black,% urlcolor=black} % TODO: find a way to check if required %\usepackage{amsthm} \usepackage{philex} % TODO: find a way to check if required %\theoremstyle{definition} %\newtheorem{thm}{Theorem} %\newtheorem*{thm*}{Theorem} %\newtheorem{conj}[thm]{Conjecture} %\newtheorem{defn}[thm]{Definition} %\newtheorem{prop}[thm]{Proposition} %\newtheorem*{prop*}{Proposition} %\newtheorem*{fact*}{Fact} \usepackage{cancel} % Command for package xy (xypic) that speeds up compilation \CompileMatrices % Command for bussproofs to make available abbreviations \EnableBpAbbreviations %\setlength{\parindent}{0pt} %\setlength{\parskip}{6pt plus 2pt minus 1pt} \setcounter{secnumdepth}{0} \title{Leitgeb’s Theory of Grounded Classes is Interpretable in the Constructive Ordinals} \author{Jönne Speck} \begin{document} \maketitle

## Preliminary Observations about ⟨Φ lf, Γ lf⟩

### Universal Class

Since Leitgeb’s class theory has a universal class, the set axiom separation will not hold in it.

The universal class turns out to be grounded on Leitgeb’s definition of groundedness.

### An Example of an Ungrounded Class

Let φ be ‘x ∈ x’. To show this ersatz class to be ungrounded, assume for the sake of contradiction that φ ∈ Φ lf. Hence, for some α + 1, φ ∈ Φ α + 1 and φ ∉ Φ α. Hence, φ depends on Φ α × Φ α. That is, for any sets X and Y, if X ∩ Φ α × Φ α = Y ∩ Φ α × Φ α then $\text{\textit{Ext}}_X(\phi)=\text{\textit{Ext}}_Y(\phi)$. Now, since by assumption φ ∉ Φ α, ⟨φ, φ⟩ ∉ Φ α × Φ α. Hence {⟨φ, φ⟩} ∩ Φ α × Φ α = ∅ × ∅ ∩ Φ α × Φ α. However, $\text{\textit{Ext}}_{\{\langle\phi,\phi\rangle\}}(\phi)=\{\phi\}\neq\emptyset=\text{\textit{Ext}}_\emptyset\times\emptyset(\phi)$, contradiction. Therefore, φ ∉ Φ lf. In other words, φ is ungrounded.

## Interpretability in the Constructive Ordinals

Since ⟨ψ, φ⟩ ∈ Γ α is defined by a conjunction of a Π 11 predicate (‘φ ∈ D - 1(dm(X))’) and a Δ 11-predicate (‘$\text{\textit{Val}}_\Delta(\phi)=1$’), the operator that builds up the sequence (Γ )α is Π 11.

Hence, by Spector’s theorem ,1 Finally, by a general result due to Kleene we obtain

1. Do I need to define the monotone operator explicitly, in order to apply Spector’s theorem, or does a monotonically increasing sequence such as (Γ )α suffice? I need to ascertain that the operator is Π 11. Hence, I need to know a Π 11 definition of it. Q: Any monotonically increasing sequence is built up by a monotone operator?

%TODO: find a way to check if some citation in the document \bibliographystyle{apalike} \bibliography{/home/jonne/literatur} \end{document}