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Description
English: Historical evolution of the Von Neumann–Bernays–Gödel set theory
Date
Source Own work
Author Jochen Burghardt
LaTeX source code
\documentclass[12pt]{article}
\usepackage[latin1]{inputenc}
\setlength{\unitlength}{1mm}
\usepackage[pdftex]{color}
\usepackage[paperwidth=159\unitlength,paperheight=195\unitlength]{geometry}
\setlength{\topmargin}{-36mm}
\setlength{\textwidth}{154\unitlength}
\setlength{\textheight}{188\unitlength}
\setlength{\oddsidemargin}{-23mm}
\setlength{\parindent}{0cm}
\pagestyle{empty}

% colors
\definecolor{cApp}      {rgb}{0.00,0.00,0.00}   % approach name
\definecolor{cPrm}      {rgb}{0.00,0.50,0.70}   % primitives
\definecolor{cAxm}      {rgb}{0.00,0.00,0.70}   % axioms
\definecolor{cAxO}      {rgb}{0.40,0.20,0.00}   % deleted axioms
\definecolor{cAxI}      {rgb}{0.20,0.40,0.00}   % new axioms expressible

\newcommand{\approach}[1]{\textcolor{cApp}{\bf #1}}
\newcommand{\primitives}[1]{\textcolor{cPrm}{\sf #1}}
\newcommand{\axioms}[1]{\textcolor{cAxm}{\sl #1}}
\newcommand{\axmOut}[1]{\textcolor{cAxO}{\rm #1}}
\newcommand{\axmIn}[1]{\textcolor{cAxI}{\rm #1}}

\newlength{\appWd}
\setlength{\appWd}{82mm}

% switch off hyphenation
\hyphenation{Infinity}
\hyphenation{existence}
\hyphenation{Regularity}
\hyphenation{Replacement}
\hyphenation{Extensionality}
\hyphenation{Neumann}

\begin{document}
\begin{tabular}{@{}r@{}c@{}c@{}}

        \begin{tabular}[b]{|p{55mm}|}
        \hline
        \approach{Fraenkel 1922,}       \\
        \approach{Skolem 1922}  \\  
        \\
        Stated: \axioms{Replacement}    \\  
        \hline
        \end{tabular}
        &&
        \begin{tabular}[b]{|p{\appWd}|}
        \hline
        \approach{Zermelo 1908 (Zermelo set theory)}    \\  
        \primitives{Sets}       \\
        \axioms{%
                Extensionality (sets),
                Elementary sets,
                Union,
                Power set,
                Infinity,
                Separation,
                Choice%
        }       \\
        \hline
        \end{tabular}
        \\

& $\searrow$ & $\downarrow$     \\  

        \begin{tabular}[t]{r}
        \axmOut{Elementary sets}        \\
        \axmOut{Separation, Choice}     \\  
        [1ex]
        \axmIn{Pairing (ordered)}       \\
        \axmIn{Function existence axioms}       \\
        \axmIn{Limitation of size}      \\
        (implies Replacement, Choice)   \\
        \end{tabular}
        &
        \begin{tabular}[t]{c}
        \axmOut{$\leftarrow$}   \\
        \\
        [1ex]
        \axmIn{$\rightarrow$}   \\
        \end{tabular}
        &
        \begin{tabular}[t]{|p{\appWd}|}
        \hline
        \approach{Von Neumann 1925, 1928}       \\
        \primitives{Functions, Arguments}       \\
        \axioms{%
                %Replacement,
                Extensionality (functions),
                Pairing (ordered),
                Function existence axioms,
                Union,
                Power set,
                Infinity,
                Limitation of size
        }       \\
        Stated but not adopted: \axioms{Regularity}     \\
        \hline
        \end{tabular}
        \\
        \\[-2ex]

&& $\downarrow$ \\

        \begin{tabular}[t]{r}
        \axmOut{Limitation of size}     \\
        [1ex]
        \axmIn{Replacement}     \\
        \axmIn{Von Neumann choice}      \\
        \end{tabular}
        &
        \begin{tabular}[t]{c}
        \axmOut{$\leftarrow$}   \\
        [1ex]
        \axmIn{$\rightarrow$}   \\
        \end{tabular}
        &
        \begin{tabular}[t]{|p{\appWd}|}
        \hline
        \approach{Von Neumann 1929}     \\
        \primitives{Functions, Arguments}       \\
        \axioms{%
                Extensionality (functions),
                Pairing (ordered),
                Function existence axioms,
                Union,
                Power set,
                Infinity,
                Replacement,
                Von Neumann choice}     \\
        Proved relatively consistent: \axioms{Regularity}       \\
        \hline
        \end{tabular}
        \\
        \\[-2ex]

&& $\downarrow$ \\

        \begin{tabular}[t]{r}
        \axmOut{Pairing (ordered)}      \\
        \axmOut{Function existence axioms}      \\
        [1ex]
        \axmIn{Pairing (unordered)}     \\
        \axmIn{Class existence axioms}  \\
        \axmIn{Separation, Regularity}  \\
        \end{tabular}
        &
        \begin{tabular}[t]{c}
        \axmOut{$\leftarrow$}   \\
        \\
        %\\
        [1ex]
        \axmIn{$\rightarrow$}   \\
        \end{tabular}
        &
        \begin{tabular}[t]{|p{\appWd}|}
        \hline
        \approach{Bernays 1931 [letter to Gödel],}      \\
        \approach{\hspace*{4.1mm} 1937, 1941 [axioms published]}        \\
        \primitives{Classes, Sets (two sorts)}  \\
        \axioms{%
                Extensionality (classes),
                Pairing (unordered),
                Class existence axioms,
                Union,
                Power set,
                Infinity,
                Separation,
                Replacement,
                Von Neumann choice,
                Regularity}     \\
        \hline
        \end{tabular}
        \\
        \\[-2ex]

&& $\downarrow$ \\

        \begin{tabular}[t]{r}
        \axmOut{Separation}     \\
        \axmOut{Von Neumann choice}     \\
        [1ex]
        \axmIn{Global choice}   \\
        [0.6ex]
        \multicolumn{1}{@{}l}{
                \scriptsize
                \begin{tabular}[t]{|l|@{}}
                \multicolumn{1}{l}{\it Legend:} \\
                [1ex]
                \hline
                \approach{Approach}     \\
                \primitives{Primitives} \\
                \axioms{Axioms} \\
                \hline
                \end{tabular}
                \hspace*{36mm}
        }
        \end{tabular}
        &
        \begin{tabular}[t]{c}
        \axmOut{$\leftarrow$}   \\
        \\
        [1ex]
        \axmIn{$\rightarrow$}   \\
        \end{tabular}
        &
        \begin{tabular}[t]{|p{\appWd}|}
        \hline
        \approach{Gödel 1940 (NBG)}     \\
        \primitives{Classes, Sets (one sort)}   \\
        \axioms{%
                Extensionality (classes),
                Pairing (unordered),
                Class existence axioms,
                Union,
                Power set,
                Infinity,
                Replacement,
                Global choice,
                Regularity}     \\
        \hline
        \end{tabular}
        \\

\end{tabular}
\end{document}
Note: Except for Bernays 1931, all dates are publication dates.

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Original upload logEdit

The original description page was here. All following user names refer to en.wikipedia.
  • 2017-11-19 16:26 Jochen Burghardt 938×1151× (96656 bytes)
  • 2017-11-15 21:30 Jochen Burghardt 938×1151× (96503 bytes) implemented Robert's suggestions of 15 Nov 2017 at [[User_talk:RJGray/Sandboxcantor1#History diagram]]
  • 2017-10-29 18:38 Jochen Burghardt 927×1151× (96456 bytes) implemented Robert's suggestions of 6+7 Aug 2017 at [[User talk:RJGray#New History section]]
  • 2017-08-05 17:33 Jochen Burghardt 885×1210× (97160 bytes) added "Ordered pair", as suggested on 26 July at [[User_talk:RJGray#NBG History section]]; used "Unordered pair" instead of "Pairing" for better distinction (to demonstrate it still fits into the space)
  • 2017-07-24 20:38 Jochen Burghardt 885×1210× (50508 bytes) adapted to comments of 24 Jul at [[User_talk:RJGray#NBG_History_section]]
  • 2017-07-23 13:42 Jochen Burghardt 885×1181× (49924 bytes) adapted according to {{u|RJGray}}'s comments of 12 July 2017 at [[User_talk:RJGray#New_revision_of_my_article_rewrite]]
  • 2017-07-11 18:32 Jochen Burghardt 1181×1299× (51488 bytes) sketches for [[User:RJGray/Sandboxcantor1#Evolution of NBG (diagram)]] ([[Von Neumann–Bernays–Gödel set theory]])

File history

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Date/TimeThumbnailDimensionsUserComment
current03:00, 22 September 2018Thumbnail for version as of 03:00, 22 September 2018938 × 1,151 (94 KB)OgreBot (talk | contribs)(BOT): Reverting to most recent version before archival
03:00, 22 September 2018Thumbnail for version as of 03:00, 22 September 2018938 × 1,151 (94 KB)OgreBot (talk | contribs)(BOT): Uploading old version of file from en.wikipedia; originally uploaded on 2017-11-19 16:26:15 by Jochen Burghardt
03:00, 22 September 2018Thumbnail for version as of 03:00, 22 September 2018938 × 1,151 (94 KB)OgreBot (talk | contribs)(BOT): Uploading old version of file from en.wikipedia; originally uploaded on 2017-11-15 21:30:33 by Jochen Burghardt
03:00, 22 September 2018Thumbnail for version as of 03:00, 22 September 2018927 × 1,151 (94 KB)OgreBot (talk | contribs)(BOT): Uploading old version of file from en.wikipedia; originally uploaded on 2017-10-29 18:38:59 by Jochen Burghardt
03:00, 22 September 2018Thumbnail for version as of 03:00, 22 September 2018885 × 1,210 (95 KB)OgreBot (talk | contribs)(BOT): Uploading old version of file from en.wikipedia; originally uploaded on 2017-08-05 17:33:09 by Jochen Burghardt
03:00, 22 September 2018Thumbnail for version as of 03:00, 22 September 2018885 × 1,210 (49 KB)OgreBot (talk | contribs)(BOT): Uploading old version of file from en.wikipedia; originally uploaded on 2017-07-24 20:38:11 by Jochen Burghardt
03:00, 22 September 2018Thumbnail for version as of 03:00, 22 September 2018885 × 1,181 (49 KB)OgreBot (talk | contribs)(BOT): Uploading old version of file from en.wikipedia; originally uploaded on 2017-07-23 13:42:33 by Jochen Burghardt
03:00, 22 September 2018Thumbnail for version as of 03:00, 22 September 20181,181 × 1,299, 2 pages (50 KB)OgreBot (talk | contribs)(BOT): Uploading old version of file from en.wikipedia; originally uploaded on 2017-07-11 18:32:24 by Jochen Burghardt
09:15, 20 September 2018Thumbnail for version as of 09:15, 20 September 2018938 × 1,151 (94 KB)Jochen Burghardt (talk | contribs)User created page with UploadWizard
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