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\begin{document}

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\section*{elementary function arithmetic}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{type_theory}{}\paragraph*{{Type theory}}\label{type_theory}

[[!include type theory - contents]]

\hypertarget{foundations}{}\paragraph*{{Foundations}}\label{foundations}

[[!include foundations - contents]]

\hypertarget{elementary_function_arithmetic}{}\section*{{Elementary function arithmetic}}\label{elementary_function_arithmetic}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Elementary function arithmetic, or EFA, is a [[first-order theory]] of [[natural numbers]], one of the weakest fragments of [[arithmetic]] strong enough to do nontrivial mathematics. It is strictly weaker than [[Peano arithmetic]].

Regarding the strength of EFA, [[Harvey Friedman]] has put forth the following ``grand conjecture'':

``Every theorem published in the Annals of Mathematics whose statement involves only finitary mathematical objects (i.e., what logicians call an arithmetical statement) can be proved in EFA. EFA is the weak fragment of Peano Arithmetic based on the usual quantifier-free axioms for 0, 1, +, $\times$, exp, together with the scheme of induction for all formulas in the language all of whose quantifiers are bounded.''

Here ``bounded quantifier'' refers to a [[quantifier]] of shape $\forall_{m \lt n}$; more formally, if a [[variable]] $n$ does not occur in a formula $\phi$, then $\forall_{m \lt n} \phi(m)$ means $\forall_m (m \lt n) \Rightarrow \phi(m)$.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

The [[theory|language]] of EFA is one-sorted with

\begin{itemize}%
\item Two constants $0, 1$


\item Three binary function symbols $+, \cdot, \exp$ (where $\exp(x, y)$ is usually written $x^y$).



\end{itemize}
The [[axioms]] of EFA are

\begin{itemize}%
\item Those of \href{http://ncatlab.org/nlab/show/second+order+arithmetic#firstorder_axioms_10}{Robinson arithmetic} (where $s$ is translated as $1 + (-)$, i.e., as $+(1, -)$);


\item Exponentiation axioms, viz. $x^0 = 1$ and $x^y \cdot x = x^{y+1}$;


\item The \href{http://ncatlab.org/nlab/show/second+order+arithmetic#secondorder_axioms_11}{induction axiom} for formulas all of whose quantifiers are bounded.



\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Peano arithmetic]]


\item [[second-order arithmetic]]



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

\begin{itemize}%
\item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Elementary_function_arithmetic}{Elementary function arithmetic}}.

\end{itemize}
``Grand conjectures'' by Harvey Friedman may be found here:

\begin{itemize}%
\item Post from [[Harvey Friedman]] to FOM, April 16, 1999. (\href{http://cs.nyu.edu/pipermail/fom/1999-April/003014.html}{web})

\end{itemize}
A MathOverflow discussion on Friedman's grand conjecture about EFA may be found here,

\begin{itemize}%
\item \href{http://mathoverflow.net/users/6043}{Charles}, \emph{Status of Harvey Friedman's grand conjecture?}, \href{http://mathoverflow.net/questions/39452}{http://mathoverflow.net/questions/39452} (version: 2011-03-24), (\href{http://mathoverflow.net/questions/39452/status-of-harvey-friedmans-grand-conjecture}{link})

\end{itemize}
[[!redirects elementary function arithmetic]] [[!redirects Elementary Function Arithmetic]] [[!redirects EFA]]



\end{document}