\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*{explicit mathematics}

\hypertarget{explicit_mathematics}{}\section*{{Explicit mathematics}}\label{explicit_mathematics}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{the_basic_theory_of_operations_and_numbers}{The basic theory of operations and numbers}\dotfill \pageref*{the_basic_theory_of_operations_and_numbers} \linebreak
\noindent\hyperlink{elementary_comprehension}{Elementary comprehension}\dotfill \pageref*{elementary_comprehension} \linebreak
\noindent\hyperlink{join_and_inductive_generation}{Join and inductive generation}\dotfill \pageref*{join_and_inductive_generation} \linebreak
\noindent\hyperlink{universes_in_explicit_mathematics}{Universes in explicit mathematics}\dotfill \pageref*{universes_in_explicit_mathematics} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

\textbf{Explicit mathematics} is a programme initiated by [[Solomon Feferman]] which is related to [[constructive mathematics]].

It is also a range of foundational logical systems based on the [[logic of partial terms]]. Starting from a first-order logic of partial terms axiomatizing [[combinatory algebra]], a second sort of \emph{classes} is added. Sometimes, the alternative names \emph{classifications} or \emph{types} are used.

The classes of explicit mathematics are not like the classes of [[set theory]], but also not like types of [[type theory]]. They are like the former in that they represent certain collections of the underlying first-order model, but they are like the latter in that they have \emph{names} that can be operated on to construct new ones.

The systems come in flavors with either intuitionistic or [[classical logic]]. On benefit is that they allow for smooth axiomatization of strong notions of [[universes]], see below.

The primary systems are called $T_0$ (the classical variant) and $T_0^i$ (the intuitionistic variant).

In the 1980s there was a proof assistant and program extraction tool called PX based on a version of $T_0$ relative to the Lisp programming language.

\hypertarget{the_basic_theory_of_operations_and_numbers}{}\subsection*{{The basic theory of operations and numbers}}\label{the_basic_theory_of_operations_and_numbers}

The first-order part of the systems treat an applicative universe in the sense of combinatory algebra. It includes constants $\mathrm{k}$ and $\mathrm{s}$ (combinators), $\mathrm{p}$, $\mathrm{p}_0$, and $\mathrm{p}_1$ (pairing and projection), $0$, $\mathrm{s}_N$, and $\mathrm{p}_N$ (zero, successor, and predecessor), and $\mathrm{d}_N$ (definition by cases on the natural numbers), and possibly further applicative constants.

There is one binary function symbol ${\cdot}$ for application (usual just indicated by juxtaposition), one unary relation symbol $\downarrow$ for being defined, one unary relation symbol $\mathrm{N}$ for natural numbers, as well the binary equality relation.

In the logic of partial terms, we use the usual abbreviation $s \simeq t \leftrightarrow ({s\downarrow} \wedge {t\downarrow} \to s=t)$. The basic axioms for the operations and natural numbers are:

\begin{enumerate}%
\item $\mathrm{k} a b = a$,
\item $\mathrm{s} a b \downarrow \wedge \mathrm{s} a b c \simeq a c(b c)$,
\item $\mathrm{p}_0(a,b)=a \wedge \mathrm{p}_1(a,b) = b$,
\item $0\in N \wedge (\forall x\in N)(\mathrm{s}_N x \in N)$,
\item $(\forall x\in N)(\mathrm{s}_N x \ne 0 \wedge \mathrm{p}_N(\mathrm{s}_N x)=x)$,
\item $(\forall x\in N)(x\ne0 \to \mathrm{s}_N(\mathrm{p}_N x) = x)$,
\item $a\in N \wedge b\in N \wedge a=b \to \mathrm{d}_N u v a b = u$,
\item $a\in N \wedge b\in N \wedge a\ne b \to \mathrm{d}_N u v a b = v$.

\end{enumerate}
From the first two axioms we get lambda-definability and a fixed-point combinator.

To these axioms we add the scheme of induction over the natural numbers to obtain the theory $BON + (\mathrm{L}{-}Ind_N)$. This is proof-theoretically equivalent to first-order Peano arithmetic, $PA$, via interpretations both ways. We can interpret $BON + (\mathrm{L}{-}Ind_N)$ in $PA$ by coding operations as indices for recursive functions using Kleene's $T$ and $U$ predicates.

\hypertarget{elementary_comprehension}{}\subsection*{{Elementary comprehension}}\label{elementary_comprehension}

When we go from the purely applicative theory to the theories of classes and names, we add a second sort (for classes) as well as several new constants, a binary relation symbol $\in$ (for membership) and a binary relation symbol for names $\Re$, where $\Re(s, U)$ means that $s$ is a name for the class $U$. We use the abbreviation $\Re(s) :\leftrightarrow \exists U \Re(s,U)$, indicating that $s$ is a name.

The basic axioms regarding classes and names state that every class has a name, that there are no homonyms, and that $\Re$ respects extensional equality.

The names of classes corresponding to elementary comprehension are generated from $nat$ (natural numbers) and $id$ (the equality relation) using $co$ (complements), $int$ (intersections), $dom$ (domains), and $inv$ (inverse images). These are for the classical systems. The intuitionistic systems are slightly different. Todo: look this up

Using the abbreviation $s\dot\in t :\leftrightarrow \exists U(\Re(t,U) \wedge s\in U)$, the axioms are:

\begin{enumerate}%
\item $\Re(nat) \wedge \forall x(x \dot\in nat \leftrightarrow N(x))$,
\item $\Re(id) \wedge \forall x(x \dot\in id \leftrightarrow \exists y(x = (y,y)))$,
\item $\Re(a) \to \Re(co(a)) \wedge \forall x(x \dot\in co(a) \leftrightarrow \neg x\dot\in a)$,
\item $\Re(a) \wedge \Re(b) \to \Re(int(a,b)) \wedge \forall x(x \dot\in int(a,b) \leftrightarrow x \dot\in a \wedge x \dot\in b)$,
\item $\Re(a) \to \Re(dom(a)) \wedge \forall x(x \dot\in dom(a) \leftrightarrow \exists y((x,y) \dot\in a))$,
\item $\Re(a) \to \Re(inv(a,f)) \wedge \forall x(x \dot\in inv(a,f) \leftrightarrow f x \dot\in a)$.

\end{enumerate}
Again we add some induction over the natural numbers to obtain theories $EC + (\mathrm{C}{-}Ind_N)$ (induction axiom for classes) and $EC + (\mathrm{L}{-}Ind_N)$ (induction schema for all formulas). These correspond proof-theoretically to the subsystems of second-order arithmetic $ACA_0$ and $ACA$, respectively.

\hypertarget{join_and_inductive_generation}{}\subsection*{{Join and inductive generation}}\label{join_and_inductive_generation}

The \emph{join} of classes gives disjoint unions of classes, and thus corresponds to the $\Sigma$-types of type theory. The inductive generation operation provides accesible parts of classes coding binary relations.

todo: the axioms for join and inductive generation.

The theory $T_0$ has elementary comprehension, join, inductive generation, and full induction over the natural numbers. It has the same proof-theoretic strength as [[CZF]] plus $REA$, the [[regular extension axiom]]. Further equivalences are listed at [[ordinal analysis]].

\hypertarget{universes_in_explicit_mathematics}{}\subsection*{{Universes in explicit mathematics}}\label{universes_in_explicit_mathematics}

todo

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

\begin{itemize}%
\item \href{https://math.stanford.edu/~feferman/papers/DraftIntroFEM.pdf}{Introduction (draft)}


\item Solomon Feferman, \href{http://home.inf.unibe.ch/~ltg/em_bibliography/feferman13.pdf}{How a Little Bit goes a Long Way: Predicative Foundations of Analysis}, notes prepared 1977-1981 with a new introduction from 2013.


\item \href{http://www.ltg.unibe.ch/research/Foundations%20of%20Explicit%20Mathematics}{Foundations of explicit mathematics}


\item \href{http://home.inf.unibe.ch/~ltg/em_bibliography/}{Bibliography of explicit mathematics}


\item Reinhard Kahle and Anton Setzer, \href{http://www.cs.swan.ac.uk/~csetzer/articles/kahleSetzerExtendedPredicativeMahloPohlersFestschrift.pdf}{An extended predicative definition of the Mahlo universe}, In: Ways of Proof Theory, pp. 315–340. Ontos Mathematical Logic, vol. 2, 2010.


\item Susumu Hayashi and Hiroshi Nakano, \href{https://dl.acm.org/citation.cfm?id=62010}{PX: a computational logic}, MIT Press, 1988.



\end{itemize}


\end{document}