\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{Some remarks on axiomatized set theory}

\begin{itemize}%
\item [[Thoralf Skolem]], \emph{Einige Bemerkungen zur axiomatischen Begr\"u{}ndung der Mengenlehre}, 1922

\end{itemize}
is a talk on occasion of the Fith Congress of Scandinavian Mathematicians in which some major ideas of [[set theory]] appear and the idea of set theoretic [[foundation of mathematics|foundation]] is discussed.

\hypertarget{content}{}\subsection*{{Content}}\label{content}

Axiomatic set theory at the stage of this talk is not a formal theory, i.e. not conceived as being a manipulation of strings. Instead the axioms are considered to describe properties of a certain universe or range of things $B$ (German ``Bereich''). This $B$ would be called underlying set of a model in modern terminology.

\hypertarget{the_peculiar_fact_that_in_order_to_treat_sets_we_must_begin_with_domains_that_are_constituted_in_a_certain_way}{}\subsubsection*{{The peculiar fact that, in order to treat ``sets'', we must begin with ``domains'' that are constituted in a certain way}}\label{the_peculiar_fact_that_in_order_to_treat_sets_we_must_begin_with_domains_that_are_constituted_in_a_certain_way}

First, Skolem points out the problem of how to think of $B$ itself: if we conceive $B$ as a set itself, we can not do so as a thing of the theory we are developing. Second, it is pointed out that $B$ is not uniquely determined by the set of axioms.

\hypertarget{a_definition_much_to_be_desired_that_makes_zermelos_notion_definite_proposition_precise}{}\subsubsection*{{A definition, much to be desired, that makes Zermelo's notion ``definite proposition'' precise}}\label{a_definition_much_to_be_desired_that_makes_zermelos_notion_definite_proposition_precise}

In modern language a (well-formed) formula.

\hypertarget{the_fact_that_in_every_thoroughgoing_axiomatization_settheoretic_notions_are_unavoidably_relative}{}\subsubsection*{{The fact that in every thoroughgoing axiomatization set-theoretic notions are unavoidably relative}}\label{the_fact_that_in_every_thoroughgoing_axiomatization_settheoretic_notions_are_unavoidably_relative}

Downward version of [[Löwenheim-Skolem theorem]] is proved, i.e. existence of countable models. [[Skolem's paradox]] is explained.

\hypertarget{the_fact_that_zermelos_system_is_not_sufficient_to_provide_a_foundation_for_ordinary_set_theory}{}\subsubsection*{{The fact that Zermelo's system is not sufficient to provide a foundation for ordinary set theory}}\label{the_fact_that_zermelos_system_is_not_sufficient_to_provide_a_foundation_for_ordinary_set_theory}

[[axiom of replacement|Axioms of replacement]] is suggested.

\hypertarget{the_difficulties_caused_by_nonpredicative_stipulations_when_one_wants_to_prove_the_consistency_of_the_axioms}{}\subsubsection*{{The difficulties caused by nonpredicative stipulations when one wants to prove the consistency of the axioms}}\label{the_difficulties_caused_by_nonpredicative_stipulations_when_one_wants_to_prove_the_consistency_of_the_axioms}

Skolem argues that the consistency of set theory can not be proved (note that at that time G\"o{}del's [[incompleteness theorem]] was still unknown). Such a proof is in Skolem's opinion not possible as in set theory sets are formed in a ``non-predicative'' way, that is to say they are formed from all sets of $B$ and not just a finite number of sets, e.g. the intersection of all sets with some property.

Skolem remarks that a the ``non-predicative requirement of reproduction of sets'' also occurs in [[Principia Mathematica|Russell's type theory]] in form of the [[axiom of reducibility]].

\hypertarget{the_nonuniqueness_mehrdeutigkeit_of_the_domain_}{}\subsubsection*{{The nonuniqueness Mehrdeutigkeit of the domain $B$}}\label{the_nonuniqueness_mehrdeutigkeit_of_the_domain_}

In modern language: Based on a given model of set theory, a new model is constructed by extending the underlying set by a new element. In a footnote Skolem speculates that the [[continuum hypothesis]] can not decided in the set theory of his time.

\hypertarget{the_fact_that_mathematical_induction_is_necessary_for_the_logical_investigation_of_abstractly_given_systems_of_axioms}{}\subsubsection*{{The fact that mathematical induction is necessary for the logical investigation of abstractly given systems of axioms}}\label{the_fact_that_mathematical_induction_is_necessary_for_the_logical_investigation_of_abstractly_given_systems_of_axioms}

A skeptic view on [[Hilbert's program]], that was ongoing at that point, is explained. In modern language the argument amounts to saying that in the metalanguage of mathematical logic the induction axiom is indispensable.

\hypertarget{a_remark_on_the_principle_of_choice}{}\subsubsection*{{A remark on the principle of choice}}\label{a_remark_on_the_principle_of_choice}

Skolem criticizes opponents of the [[axiom of choice]] by rooting their opposition to a ``non-axiomatic'' understanding of sets: sets are for them not things subject to some axioms but just a collection on \emph{explicitely} given things.

\hypertarget{concluding_remark}{}\subsubsection*{{Concluding remark}}\label{concluding_remark}

\begin{quote}%
\ldots{} I believed that it was so clear that axiomatization in terms of sets was not a satisfactory ultimate foundation of mathematics that mathematicians would, for the most part, not be very much concerned with it. But in recent times I have seen to my surprise that so many mathematicians think that these axioms of set theory provide the ideal foundation for mathematics \ldots{}

\ldots{} glaubte ich, dass es so klar sei, dass diese Mengenaxiomatik keine befriedigende letzte Grundlage der Mathematik w\"a{}re, dass die Mathematiker gr\"o{}sstenteils sich nicht so sehr darum k\"u{}mmern w\"u{}rden. In der letzten Zeit habe ich aber zu meinem Erstauenen gesehen, dass sehr viele Mathematiker diese Axiome der Mengenlehre als ideale Begr\"u{}ndung der Mathematik betrachten; \ldots{}


\end{quote}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Originally published in German

\begin{itemize}%
\item [[Thoralf Skolem]], \emph{Einige Bemerkungen zu axiomatischen Begr\"u{}ndung der Mengenlehre}, Mathematikerkongressen i Helsingfors den 4--7 Juli 1922, Den femte skandinaviska matematikerkongressen, Redog\"o{}relse: 217--232

\end{itemize}
English translation

\begin{itemize}%
\item \emph{Some remarks on axiomatized set theory} in \emph{From Frege to G\"o{}del: A Source Book in Mathematical Logic, 1879-1931}, Cambridge, Massachusetts: Harvard University Press

\end{itemize}
[[!redirects Einige Bemerkungen zu axiomatischen Begründung der Mengenlehre]]

category: reference



\end{document}