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ic.tex
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% ic.tex
%
% driver file ic.tex to produce text
\preto\OLEndChapterHook{\IfFileExists{include/summary-\thechapter}
{\section*{Summary}\addcontentsline{toc}{section}{Summary}
\let\emph\textbf\input{include/summary-\thechapter}\let\emph\textit}{}}
\problemsperchapter
\allowdisplaybreaks
\frontmatter
\OLPfrontmatter
\input{include/preface}
\mainmatter
\olimport*[incompleteness/introduction]{introduction}
\olimport*[computability/recursive-functions]{recursive-functions}
\olimport*[incompleteness/arithmetization-syntax]{arithmetization-syntax}
\olimport*[incompleteness/representability-in-q]{representability-in-q}
\olimport*[incompleteness/incompleteness-provability]{incompleteness-provability}
\chapter{Models of Arithmetic}\label{mod:chap}
\olimport*[model-theory/models-of-arithmetic]{introduction}
\olimport*[model-theory/basics]{reducts-and-expansions}
\olimport*[model-theory/basics]{isomorphism}
\olimport*[model-theory/basics]{theory-of-m}
\olimport*[model-theory/models-of-arithmetic]{standard-models}
\olimport*[model-theory/models-of-arithmetic]{non-standard-models}
\olimport*[model-theory/models-of-arithmetic]{models-of-q}
\olimport*[model-theory/models-of-arithmetic]{models-of-pa}
\olimport*[model-theory/models-of-arithmetic]{computable-models}
\OLEndChapterHook
\chapter{Second-Order Logic}
\olimport*[second-order-logic/syntax-and-semantics]{introduction}
\let\oldolsection\olsection
\let\olsection\nosection
\olimport*[second-order-logic/sol-and-set-theory]{introduction}
\let\olsection\oldolsection
\olimport*[second-order-logic/syntax-and-semantics]{terms-formulas}
\olimport*[second-order-logic/syntax-and-semantics]{satisfaction}
\olimport*[second-order-logic/syntax-and-semantics]{semantic-notions}
\olimport*[second-order-logic/syntax-and-semantics]{expressive-power}
\olimport*[second-order-logic/syntax-and-semantics]{inf-count}
\olimport*[second-order-logic/metatheory]{second-order-arithmetic}
\olimport*[second-order-logic/metatheory]{undecidability-and-axiomatizability}
\olimport*[second-order-logic/metatheory]{compactness}
\olimport*[second-order-logic/metatheory]{loewenheim-skolem}
\olimport*[second-order-logic/sol-and-set-theory]{comparing-sets}
\olimport*[second-order-logic/sol-and-set-theory]{cardinalities}
\olimport*[second-order-logic/sol-and-set-theory]{power-of-continuum}
\OLEndChapterHook
\chapter{The Lambda Calculus}\label{lambda:chap}
\olimport*[lambda-calculus/introduction]{overview}
\olimport*[lambda-calculus/introduction]{syntax}
\olimport*[lambda-calculus/introduction]{reduction}
\olimport*[lambda-calculus/introduction]{church-rosser}
\olimport*[lambda-calculus/introduction]{currying}
\olsection{Lambda Definability}
\let\oldolsection\olsection
\def\olsection#1{}
\olimport*[lambda-calculus/lambda-definability]{introduction}
\let\olsection\oldolsection
\olimport*[lambda-calculus/lambda-definability]{arithmetical-functions}
\olimport*[lambda-calculus/lambda-definability]{pairs}
\olimport*[lambda-calculus/lambda-definability]{truth-values}
%\olimport{lists}
\olimport*[lambda-calculus/lambda-definability]{primitive-recursive-functions}
\olimport*[lambda-calculus/lambda-definability]{fixpoints}
\olimport*[lambda-calculus/lambda-definability]{minimization}
\olimport*[lambda-calculus/lambda-definability]{partial-recursive-functions}
\olimport*[lambda-calculus/lambda-definability]{lambda-definable-recursive}
\OLEndChapterHook
\appendix
\input{ic-derivations}\label{deriv:chap}
\let\intro\comment
\let\endintro\endcomment
\chapter{First-order Logic}\label{fol:chap}
\olimport*[first-order-logic/syntax-and-semantics]{first-order-languages}
\olimport*[first-order-logic/syntax-and-semantics]{terms-formulas}
\olimport*[first-order-logic/syntax-and-semantics]{free-vars-sentences}
\olimport*[first-order-logic/syntax-and-semantics]{substitution}
\olimport*[first-order-logic/syntax-and-semantics]{structures}
\olimport*[first-order-logic/syntax-and-semantics]{satisfaction}
\olimport*[first-order-logic/syntax-and-semantics]{assignments}
\olimport*[first-order-logic/syntax-and-semantics]{extensionality}
\olimport*[first-order-logic/syntax-and-semantics]{semantic-notions}
\section{Theories}
\begin{defn}
A set of !!{sentence}s~$\Gamma$ is \emph{closed} iff, whenever
$\Gamma \Entails !A$ then $!A \in \Gamma$. The \emph{closure} of a set
of !!{sentence}s~$\Gamma$ is $\Setabs{!A}{\Gamma \Entails !A}$.
We say that~$\Gamma$ is \emph{axiomatized by} a set of
sentences~$\Delta$ if $\Gamma$ is the closure of~$\Delta$.
\end{defn}
\begin{ex}
The theory of strict linear orders in the language~$\Lang L_<$ is
axiomatized by the set
\begin{align*}
& \lforall[x][\lnot x < x], \\
& \lforall[x][\lforall[y][((x < y \lor y <
x) \lor x = y)]], \\
& \lforall[x][\lforall[y][\lforall[z][((x < y
\land y < z) \lif x < z)]]]
\end{align*}
It completely captures the intended !!{structure}s: every strict
linear order is a model of this axiom system, and vice versa, if $R$
is a linear order on a set $X$, then the structure $\Struct M$ with
$\Domain M = X$ and $\Assign{<}{M} = R$ is a model of this theory.
\end{ex}
\begin{ex}
The theory of groups in the language $\Obj 1$ (!!{constant}), $\cdot$
(two-place !!{function}) is axiomatized by
\begin{align*}
& \lforall[x][(x \cdot \Obj 1) = x]\\
& \lforall[x][\lforall[y][\lforall[z][\eq[(x \cdot (y \cdot z))][((x
\cdot y) \cdot z)]]]]\\
& \lforall[x][\lexists[y][(x \cdot y) = \Obj 1]]
\end{align*}
\end{ex}
\begin{ex}
The theory of Peano arithmetic is axiomatized by the following
sentences in the language of arithmetic~$\Lang L_A$.
\begin{align*}
& \lnot\lexists[x][x' = \Obj 0]\\
& \lforall[x][\lforall[y][(x' = y' \lif x = y)]]\\
& \lforall[x][\lforall[y][(x < y \liff \lexists[z][\eq[(z' + x)][y]])]]\\
& \lforall[x][\eq[(x + \Obj 0)][x]]\\
& \lforall[x][\lforall[y][\eq[(x + y')][(x + y)']]]\\
& \lforall[x][\eq[(x \times \Obj 0)][\Obj 0]]\\
& \lforall[x][\lforall[y][\eq[(x \times y')][((x \times y) + x)]]]\\
\intertext{plus all sentences of the form}
& (!A(\Obj 0) \land \lforall[x][(!A(x) \lif !A(x'))]) \lif \lforall[x][!A(x)]
\end{align*}
Since there are infinitely many sentences of the latter form, this
axiom system is infinite. The latter form is called the
\emph{induction schema}. (Actually, the induction schema is a bit more
complicated than we let on here.)
The third axiom is an \emph{explicit definition} of~$<$.
\end{ex}
\OLEndChapterHook
\chapter{Natural Deduction}\label{nd:chap}
\olimport*[first-order-logic/proof-systems]{natural-deduction}[\nosection]
\olimport*[first-order-logic/natural-deduction]{rules-and-proofs}
\olimport*[first-order-logic/natural-deduction]{propositional-rules}
\olimport*[first-order-logic/natural-deduction]{quantifier-rules}
\olimport*[first-order-logic/natural-deduction]{derivations}
\olimport*[first-order-logic/natural-deduction]{proving-things}
\olimport*[first-order-logic/natural-deduction]{proving-things-quant}
\olimport*[first-order-logic/natural-deduction]{identity}
\olimport*[first-order-logic/natural-deduction]{proof-theoretic-notions}
\olimport*[first-order-logic/natural-deduction]{soundness}
\OLEndChapterHook
\stopproblems
\def\ifproblems#1{}
\def\figurename{Fig.}
\chapter{Biographies}\label{bios:chap}
\olimport*[history/biographies]{alonzo-church}
\olimport*[history/biographies]{kurt-goedel}
\olimport*[history/biographies]{rozsa-peter}
\olimport*[history/biographies]{julia-robinson}
\olimport*[history/biographies]{alfred-tarski}
\backmatter
\clearpage
\photocredits
\bibliographystyle{\olpath/bib/natbib-oup}
\bibliography{\olpath/bib/open-logic.bib}
\olimport*{\olpath/content/open-logic-about}