File:NBG Evolution.pdf
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Summary[edit]
DescriptionNBG Evolution.pdf |
English: Historical evolution of the Von Neumann–Bernays–Gödel set theory |
Date | |
Source | Own work |
Author | Jochen Burghardt |
LaTeX source code |
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\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.
|
Licensing[edit]
I, the copyright holder of this work, hereby publish it under the following license:
This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International license.
- You are free:
- to share – to copy, distribute and transmit the work
- to remix – to adapt the work
- Under the following conditions:
- attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.
Original upload log[edit]
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/Time | Thumbnail | Dimensions | User | Comment | |
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current | 03:00, 22 September 2018 | 937 × 1,150 (94 KB) | OgreBot (talk | contribs) | (BOT): Reverting to most recent version before archival | |
03:00, 22 September 2018 | No thumbnail | 0 × 0 (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 2018 | No thumbnail | 0 × 0 (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 2018 | No thumbnail | 0 × 0 (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 2018 | No thumbnail | 0 × 0 (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 2018 | No thumbnail | 0 × 0 (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 2018 | No thumbnail | 0 × 0 (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 2018 | No thumbnail | 0 × 0 (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 2018 | 937 × 1,150 (94 KB) | Jochen Burghardt (talk | contribs) | User created page with UploadWizard |
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