\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}