File:NBG Evolution.pdf
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DescriptionNBG Evolution.pdf 
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.

LicensingEdit
I, the copyright holder of this work, hereby publish it under the following license:

This file is licensed under the Creative Commons AttributionShare Alike 4.0 International license.  
https://creativecommons.org/licenses/bysa/4.0 CC BYSA 4.0 Creative Commons AttributionShare Alike 4.0 truetrue 
Original upload logEdit
The original description page was here. All following user names refer to en.wikipedia.
 20171119 16:26 Jochen Burghardt 938×1151× (96656 bytes)
 20171115 21:30 Jochen Burghardt 938×1151× (96503 bytes) implemented Robert's suggestions of 15 Nov 2017 at [[User_talk:RJGray/Sandboxcantor1#History diagram]]
 20171029 18:38 Jochen Burghardt 927×1151× (96456 bytes) implemented Robert's suggestions of 6+7 Aug 2017 at [[User talk:RJGray#New History section]]
 20170805 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)
 20170724 20:38 Jochen Burghardt 885×1210× (50508 bytes) adapted to comments of 24 Jul at [[User_talk:RJGray#NBG_History_section]]
 20170723 13:42 Jochen Burghardt 885×1181× (49924 bytes) adapted according to {{uRJGray}}'s comments of 12 July 2017 at [[User_talk:RJGray#New_revision_of_my_article_rewrite]]
 20170711 18:32 Jochen Burghardt 1181×1299× (51488 bytes) sketches for [[User:RJGray/Sandboxcantor1#Evolution of NBG (diagram)]] ([[Von Neumann–Bernays–Gödel set theory]])
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current  03:00, 22 September 2018  938 × 1,151 (94 KB)  OgreBot (talk  contribs)  (BOT): Reverting to most recent version before archival  
03:00, 22 September 2018  938 × 1,151 (94 KB)  OgreBot (talk  contribs)  (BOT): Uploading old version of file from en.wikipedia; originally uploaded on 20171119 16:26:15 by Jochen Burghardt  
03:00, 22 September 2018  938 × 1,151 (94 KB)  OgreBot (talk  contribs)  (BOT): Uploading old version of file from en.wikipedia; originally uploaded on 20171115 21:30:33 by Jochen Burghardt  
03:00, 22 September 2018  927 × 1,151 (94 KB)  OgreBot (talk  contribs)  (BOT): Uploading old version of file from en.wikipedia; originally uploaded on 20171029 18:38:59 by Jochen Burghardt  
03:00, 22 September 2018  885 × 1,210 (95 KB)  OgreBot (talk  contribs)  (BOT): Uploading old version of file from en.wikipedia; originally uploaded on 20170805 17:33:09 by Jochen Burghardt  
03:00, 22 September 2018  885 × 1,210 (49 KB)  OgreBot (talk  contribs)  (BOT): Uploading old version of file from en.wikipedia; originally uploaded on 20170724 20:38:11 by Jochen Burghardt  
03:00, 22 September 2018  885 × 1,181 (49 KB)  OgreBot (talk  contribs)  (BOT): Uploading old version of file from en.wikipedia; originally uploaded on 20170723 13:42:33 by Jochen Burghardt  
03:00, 22 September 2018  1,181 × 1,299, 2 pages (50 KB)  OgreBot (talk  contribs)  (BOT): Uploading old version of file from en.wikipedia; originally uploaded on 20170711 18:32:24 by Jochen Burghardt  
09:15, 20 September 2018  938 × 1,151 (94 KB)  Jochen Burghardt (talk  contribs)  User created page with UploadWizard 
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