% !TEX TS-program = pdflatex
% !TEX encoding = UTF-8 Unicode
% This file is a template using the "beamer" package to create slides for a talk or presentation
% - Talk at a conference/colloquium.
% - Talk length is about 20min.
% - Style is ornate.
% MODIFIED by Jonathan Kew, 2008-07-06
% The header comments and encoding in this file were modified for inclusion with TeXworks.
% The content is otherwise unchanged from the original distributed with the beamer package.
\documentclass{beamer}
% Copyright 2004 by Till Tantau .
%
% In principle, this file can be redistributed and/or modified under
% the terms of the GNU Public License, version 2.
%
% However, this file is supposed to be a template to be modified
% for your own needs. For this reason, if you use this file as a
% template and not specifically distribute it as part of a another
% package/program, I grant the extra permission to freely copy and
% modify this file as you see fit and even to delete this copyright
% notice.
\mode
{
\usetheme{Warsaw}
% or ...
\setbeamercovered{transparent}
% or whatever (possibly just delete it)
}
\usepackage[english]{babel}
% or whatever
\usepackage[utf8]{inputenc}
% or whatever
\usepackage{times}
\usepackage[T1]{fontenc}
% Or whatever. Note that the encoding and the font should match. If T1
% does not look nice, try deleting the line with the fontenc.
\title[Reading Report: Possible Collapse] % (optional, use only with long paper titles)
{A Reading Report on: \\ ``Warning Signs of a Possible Collapse \\ of Contemporary Mathematics''}
\subtitle
{ by Edward Nelson, Math Professor, Princeton University\\ d. September 10, 2014}
\author
%[Author, Another] % (optional, use only with lots of authors)
{Bruce N. ~Lundberg \inst{1}
%\and S.~Another\inst{2}
}
% - Give the names in the same order as the appear in the paper.
% - Use the \inst{?} command only if the authors have different
% affiliation.
\institute[Colorado State University - Pueblo] % (optional, but mostly needed)
{
\inst{1}%
Department of Mathematics and Physics\\
Colorado State University - Pueblo}
% \and
% \inst{2}%
% Department of Theoretical Philosophy\\
% University of Elsewhere}
% - Use the \inst command only if there are several affiliations.
% - Keep it simple, no one is interested in your street address.
\date[CFP 2003] % (optional, should be abbreviation of conference name)
{Math 495 Seminar Presentations}
% - Either use conference name or its abbreviation.
% - Not really informative to the audience, more for people (including
% yourself) who are reading the slides online
\subject{Set Theory, Peano Arithmetic}
% This is only inserted into the PDF information catalog. Can be left
% out.
% If you have a file called "university-logo-filename.xxx", where xxx
% is a graphic format that can be processed by latex or pdflatex,
% resp., then you can add a logo as follows:
% \pgfdeclareimage[height=0.5cm]{university-logo}{university-logo-filename}
% \logo{\pgfuseimage{university-logo}}
% Delete this, if you do not want the table of contents to pop up at
% the beginning of each subsection:
%\AtBeginSubsection[]
%{
%\begin{frame}{Outline}
% \tableofcontents[currentsection,currentsubsection]
% \end{frame}
%}
% If you wish to uncover everything in a step-wise fashion, uncomment
% the following command:
%\beamerdefaultoverlayspecification{<+->}
\begin{document}
\begin{frame}
\titlepage
\end{frame}
\begin{frame}{Overview}
\tableofcontents
% You might wish to add the option [pausesections]
\end{frame}
% Structuring a talk is a difficult task and the following structure
% may not be suitable. Here are some rules that apply for this
% solution:
% - Exactly two or three sections (other than the summary).
% - At *most* three subsections per section.
% - Talk about 30s to 2min per frame. So there should be between about
% 15 and 30 frames, all told.
% - A conference audience is likely to know very little of what you
% are going to talk about. So *simplify*!
% - In a 20min talk, getting the main ideas across is hard
% enough. Leave out details, even if it means being less precise than
% you think necessary.
% - If you omit details that are vital to the proof/implementation,
% just say so once. Everybody will be happy with that.
\section{Approach to the Thought of a Technically Competent ``Ultra-Finitist'' in Mathematics}
\subsection{``Is infinity real? E.g. Are there infinitely many numbers? Yes indeed.''}
\subsection{Etymology of infinity: ``unfinished'', always potential, never actual or completed.}
\subsection{Contrasting idea of {\it completed infinity} -- arouses strong `loathing and feeling of oppression ...It is the antithesis of life, of newness, of becoming -- it is finished.'}
\section{Peano Arithmetic (P)}
\begin{frame}
{\bf Peano Arithmetic P:} \ \ \ \ S means ``Successor of''
\begin{enumerate}
\item $\neg Sx = 0$
\item $Sx = Sy \rightarrow x = y$
\item $x + 0 = x $
\item $x + Sy = S(x + y)$
\item $x \cdot 0 = 0 $
\item $x \cdot Sy = (x \cdot y) + x$
\item $\phi(0) \& \forall x[\phi(x) \rightarrow \phi(Sx)] \rightarrow \forall x [\phi(x)]$
\end{enumerate}
\end{frame}
\section{Nelson's Three Warning Signs}
\begin{frame}{Nelson's Three ``Warning Signs''}
%{Subtitles are optional.}
% - A title should summarize the slide in an understandable fashion
% for anyone how does not follow everything on the slide itself.
\begin{enumerate}
\item
The intuitive notion of infinite sets leads to a contradiction (e.g. Russell's paradox).
\pause
\item
G\"{o}del's Second Incompleteness Theorem: P (Peano Arithmetic) cannot be proven consistent by means expressible in P, {\bf provided P is consistent}. \\ (So, ..., ``perhaps P is inconsistent'')
\pause
\item ``Most serious'': ``The belief that exponentiation, superexponentiation, and so forth, applied to numerals yield numerals is just that -- a belief.'' (based on circular reasoning--assuming numerals exist to form them).
\end{enumerate}
\end{frame}
\section*{Postscript}
\begin{frame}{Postscript}
% Keep the summary *very short*.
\begin{itemize}
\item
\alert{Nelson's view of infinity only being potential is common in the history of philosophy.}
\item
\alert{Nelson anounced writing on a proof of P's inconsistency Sept. 26, 2011} which he later withdrew.
\end{itemize}
\end{frame}
% The following outlook is optional.
% \vskip0pt plus.5fill
\begin{frame}
{\bf Outlook}
\begin{itemize}
\item
Need more study of Nelson's Peano Arithmetic proofs
\item
Need to ponder the ways he exploits exponentiation not being associative, e.g. $2^{(2^3)} \neq (2^2)^3$
\end{itemize}
\end{frame}
% All of the following is optional and typically not needed.
\appendix
\section*{\appendixname}
\subsection*{For Further Reading}
\begin{frame}[allowframebreaks]
\frametitle{For Further Reading}
\begin{thebibliography}{10}
\beamertemplatebookbibitems
% Start with overview books.
\bibitem{Nelson2011}
Edward Nelson
\newblock {``Warning Signs of a Possible Collapes of Contemporary Mathematics.''}
\newblock {in \em Infinity: New Research Frontiers}.
\newblock {ed. M. Heller and W. H. Woodin.}
\newblock Cambridge University Press, 2011. p. 76-85.
\bibitem{Aristotle}
\newblock {Aristotle's \em Physics:}
\newblock {A Guided Study}.
\newblock {Joe Sachs}
\newblock Rutgers University Press, 2001.
\beamertemplatearticlebibitems
% Followed by interesting articles. Keep the list short.
%\bibitem{Someone2000}
% S.~Someone.
% \newblock On this and that.
% \newblock {\em Journal of This and That}, 2(1):50--100,
% 2000.
\end{thebibliography}
\end{frame}
\end{document}