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\begin{document}

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\section*{reflection principle}

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

\hypertarget{universes}{}\paragraph*{{Universes}}\label{universes}

[[!include universe - contents]]

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

[[!include foundations - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{settheoretic_reflection_principles}{Set-theoretic reflection principles}\dotfill \pageref*{settheoretic_reflection_principles} \linebreak
\noindent\hyperlink{universes_in_type_theory}{Universes in type theory}\dotfill \pageref*{universes_in_type_theory} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{related_page}{Related page}\dotfill \pageref*{related_page} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \textbf{reflection principle} is roughly a statement or formula (scheme) that expresses the idea that a certain logical object contains copies of itself. They play an important role in different guises in several fields of [[mathematical logic]] e.g. in [[set theory]], [[proof theory]], [[type theory]] and, recently, [[constructive analysis]].

G\"o{}del's encoding of an unprovability predicate into [[Peano arithmetic]] $PA$ can be viewed as an \emph{`anti-reflection'} principle for $PA$. Conversely, this suggests that the validity of reflection principles for some theory $T$ can be informally understood as expressing or approximating the (internal) \emph{consistency} and \emph{completeness} of $T$.

\hypertarget{settheoretic_reflection_principles}{}\subsubsection*{{Set-theoretic reflection principles}}\label{settheoretic_reflection_principles}

A \emph{reflection principle} in [[set theory]] states that it is possible to find [[sets]] that resemble the [[class]] of all sets that are constructed in some ways; in other words, any statement true of the universe $V$ already holds at some intial segment $V_\alpha$. This can be interpreted as saying that no first-order property can distinguish the set-theoretic universe from the extensions of all its member sets thereby suggesting the \emph{expansiveness} or strong infinity of the universe: $V$ gets `arbitrarily large'.

The first reflection principles go back to [[Richard Montague|R. Montague]] and A. L\'e{}vy around 1960. It emerged from their work that reflection principles can supplant the traditional axioms of [[Zermelo-Fraenkel set theory]], e.g. the [[axiom of replacement]] (cf. \hyperlink{BellMach77}{Bell-Machover 1977}, p.495). This capacity and the intuition that they express `completeness' make them interesting candidates in the quest for new axioms of set theory (cf. \hyperlink{Koell09}{Koellner 2009}).

\hypertarget{universes_in_type_theory}{}\subsubsection*{{Universes in type theory}}\label{universes_in_type_theory}

In [[type theory]] this principle is embodied by a [[type of types]] (\hyperlink{MartinLoef74}{Martin-L\"o{}f 74, p. 6}).

\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[universe]]


\item [[type of types]]


\item [[cumulative hierarchy]]


\item [[continuous truth]]



\end{itemize}
\hypertarget{related_page}{}\subsection*{{Related page}}\label{related_page}

\begin{itemize}%
\item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Reflection_principle}{Reflection principle}} .

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

\begin{itemize}%
\item J. L. Bell, M. Machover, \emph{A Course in Mathematical Logic} , North-Holland Amsterdam 1977. (ch. 10,\S{}5)


\item [[Michael Fourman|M. P. Fourman]], \emph{Continuous Truth II: Reflections} , LNCS \textbf{8071} (2013) pp.153-167. (\href{http://homepages.inf.ed.ac.uk/mfourman/research/publications/pdf/fourman2013-continuous-truth-II.pdf}{preprint})


\item P. Koellner, \emph{On Reflection Principles} , APAL \textbf{157} (2009) pp.209-219. (\href{http://logic.harvard.edu/koellner/ORP_final.pdf}{preprint})


\item [[Georg Kreisel]], Mathematical Logic , pp. 95-195 in Saaty (ed.) , Lectures on Modern Mathematics III , Wiley New York 1965.


\item [[Georg Kreisel]], A. L\'e{}vy, \emph{Reflection Principles and their Use for Establishing the Complexity of Axiomatic Systems}, Mathematical Logic Quarterly \textbf{14} (7-12) pp.97--142, 1968.


\item A. L\'e{}vy, \emph{Axiom Schemata of Strong Infinity in Axiomatic Set Theory} , Pacific J. Math. \textbf{10} (1960) pp.223-238. (\href{http://projecteuclid.org/download/pdf_1/euclid.pjm/1103038638}{pdf})


\item [[Per Martin-Löf]], section 1.9, p. 7 of \emph{An intuitionistic theory of types: predicative part}, In Logic Colloquium (1973), ed. H. E. Rose and J. C. Shepherdson (North-Holland, 1974), pp.73-118. (\href{http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.131.926}{web})



\end{itemize}


\end{document}