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\section*{axiom of replacement}

\hypertarget{axioms_of_replacement_and_collection}{}\section*{{Axioms of replacement and collection}}\label{axioms_of_replacement_and_collection}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{statements}{Statements}\dotfill \pageref*{statements} \linebreak
\noindent\hyperlink{lawvere_on_replacement}{Lawvere on replacement}\dotfill \pageref*{lawvere_on_replacement} \linebreak
\noindent\hyperlink{related_discussion}{Related discussion}\dotfill \pageref*{related_discussion} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

\begin{quote}%
\emph{Zwei \"a{}quivalente Vielheiten sind entweder beide `Mengen' oder beide inkonsistent.} [[Georg Cantor]] (1899)\footnote{\emph{`Two equivalent multiplicities either are both ``sets'' or are both inconsistent'}, letter to Dedekind from 28th July 1899 (\hyperlink{Cantor99}{Cantor 1932}, p.444). This is suggested as an early formulation of the axiom of replacement by van Heijenoort (1967, p.113). A categorical formalization of Cantor's idea as an extension for [[ETCS]] is given in \hyperlink{McLarty}{McLarty (2004)}.} 


\end{quote}
\textbf{Axioms of collection} and replacement are axiom schemata in [[set theory]] that permit to construct new sets from other already given sets thereby contributing substantially to the size of the set-theoretic universe and hence are seen as `strong' axioms.

The most famous of these schemata is the \textbf{axiom of replacement}\footnote{The term `replacement', or `Ersetzungsaxiom' in German, is apparently due to \hyperlink{Fraenkel22}{Fraenkel (1922)} and was intended as a provisory terminology until the final formalization of Zermelo's notion of a `definite property' which was identified with a first-order formula in the language of set theory by Skolem in the same year (and independently earlier by [[Hermann Weyl|H. Weyl]]).}  of [[Zermelo-Fraenkel set theory]] that was suggested by A. Fraenkel and formulated by [[Thoralf Skolem|T. Skolem]] in 1922. Given a unary operation $F$ and a set $x$ it permits to collect all $F(y)$ for $y\in x$ into a new set.

The resulting expansiveness of the set-theoretic universe is somewhat peripheral to the practice of `ordinary' mathematics and therefore a [[structural set theory]] like [[ETCS]] can omit replacement without incurring a great loss\footnote{It is possible, however, to augment a categorical set theory with a version of replacement if necessary as shown in (\hyperlink{Osius74}{Osius 1974}, section 9) resulting in a system with the full strength of ZF. According to \hyperlink{McLarty04}{McLarty (2004)}, Osius' ideas go back to discussions between Lawvere and the Berkeley logicians on reflection principles in 1963. McLarty's paper proposes another equivalent way to flesh out replacement categorically!} . Even in the context of a ZF-equivalent material set theory the axiom of replacement can be traded in for a [[reflection principle]]\footnote{See \hyperlink{BellMach77}{Bell-Machover 1977}, p.495.} .

Axioms of replacement and collection become useful, however, whenever [[recursion|recursively]] constructing a set that is `larger' than any set known before:

\begin{quote}%
what the axiom of replacement is mainly needed for in mathematical practice is to define families of sets indexed by some set I carrying some inductive structure as, typically, the set $N$ of natural numbers.\footnote{[[Thomas Streicher]] (\hyperlink{Streicher05}{2005, p.79}). See there for further discussion of the role of replacement for \emph{mathematics beyond $V_{\omega +\omega}$} and the handling of similar iterated collection processes in toposes by universes.} 


\end{quote}
There are many variations on these axiom schemata, but any given system should only need one.

\hypertarget{statements}{}\subsection*{{Statements}}\label{statements}

In general, these axioms apply to a [[binary relation]] that relates elements of one [[set]] $A$ to arbitrary sets. However, we do \emph{not} expect that the relation itself be an object in the theory; really, we have an axiom schema with one axiom for every binary [[predicate]] of the proper form. We will write this predicate as $\phi(x,Y)$, where $x$ stands for an elment of $A$ and $Y$ stands for any set. (Note that there may well be other free variables in the predicate.)

Generally, $\phi$ will need to be an [[entire relation]] for the axiom to apply; that is, the axiom has as a hypothesis that, for every $x \in A$, there is some $Y$ such that $\phi(x,Y)$ holds. In versions called `replacement' instead of `collection', $\phi$ also needs to be [[functional relation|functional]]; that is, the axiom has the hypothesis that, for every $x \in A$, there is a \emph{unique} $Y$ such that $\phi(x,Y)$ holds. Thus, most of the `replacement' versions only make sense if the language has a notion of [[equality]] of sets.

So much for the hypothesis of the axiom; the conclusion asserts the existence of a [[family of sets]] to which appropriate $Y$s belong. In a material set theory, we can simply state the existence of set $\mathcal{F}$ such that certain $Y \in \mathcal{F}$. In a structural set theory, we state the existence of an index set $I$, a total set $E$, and a function $f\colon E \to I$ such that each [[fibre]] $f^*(x)$ for $x \in I$ is equal to (or at least isomorphic to) certain $Y$. (Often we can take $I$ to be $A$, but that does not come into the statement of the axioms.)

Who wants to write out some of these?

\hypertarget{lawvere_on_replacement}{}\subsection*{{Lawvere on replacement}}\label{lawvere_on_replacement}

\begin{quote}%
A question that has been much of a ``foundational'' interest, though of hardly any significance for the practice of algebra, topology, functional analysis, etc. is whether, for a given $T$, all imaginable families of sets parametrized by $T$ can be represented by $E\to T$ for some $E$ and some mapping; if ``imaginable'' is interpreted to mean ``definable'', an affirmative answer to this question is essentially equivalent (for abstract, constant sets) to the postulation of the so-called ``replacement schema'', whereas if $\mathcal{S}$ is considered as an object in some larger realm, an affirmative answer means that $\mathcal{S}$ itself has ``inaccessible cardinality''. However, in view of practice and in view of the role of $\mathcal{S}$ as a limiting case of the general notion of continuously variable sets, it seems appropriate to simply define ``an internal-to-$\mathcal{S}$ $T$-parametrized family of objects of $\mathcal{S}$'' to mean just a morphism of $\mathcal{S}$ with domain $T$. \hyperlink{Lawvere76}{Lawvere (1976, p.121)}


\end{quote}
See also the remarks on pages 721 and 727 of (\hyperlink{Lawvere00}{Lawvere 2000}).

\hypertarget{related_discussion}{}\subsection*{{Related discussion}}\label{related_discussion}

\begin{itemize}%
\item [[Thomas Streicher]], [[William Lawvere]], [[Colin McLarty]] et al, \emph{Categorical formulations of replacement}, March 2008. (\href{http://www.mta.ca/~cat-dist/archive/2008/08-3}{link})


\item MO-discussion: \emph{Who needs Replacement anyway ?} (\href{http://mathoverflow.net/questions/208711/who-needs-replacement-anyway}{link})


\item Akihiro Kanamori (2012). \emph{In Praise of Replacement}. The Bulletin of Symbolic Logic 18:1 (2012 March). \href{http://math.bu.edu/people/aki/20.pdf}{pdf}.



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

\begin{itemize}%
\item [[Zermelo-Fraenkel set theory]]


\item [[reflection principle]]


\item [[axiom of choice]]


\item [[ETCS]]



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

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


\item [[Georg Cantor]], \emph{Brief an Dedekind vom 22. Juli 1899}, pp.443-447 in Cantor, \emph{Gesammelte Abhandlungen}, Springer Berlin 1932. English transl. pp.113-117 of van Heijenoort (ed.), \emph{From Frege to G\"o{}del} , Harvard UP 1967.


\item A. Fraenkel, \emph{Zu den Grundlagen der Cantor-Zermeloschen Mengenlehre}, Math. Ann. \textbf{86} (1922) pp.230-237. (\href{http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002268760}{gdz})


\item [[Harvey Friedman|H. J. Friedman]], \emph{Higher Set Theory and Mathematical Practice}, Ann. Math. Logic \textbf{2} (1971) pp.326-357.


\item [[André Joyal]], [[Ieke Moerdijk]], \emph{A categorical theory of cumulative hierarchies of sets}, C. R. Math. Rep. Acad. Sci. Canada \textbf{13} (1991) pp.55-58.


\item [[F. William Lawvere]], \emph{Variable Quantities and Variable Structures in Topoi}, pp.101-131 in Heller, Tierney (eds.), \emph{Algebra, Topology and Category Theory: a Collection of Papers in Honor of Samuel Eilenberg} , Academic Press New York 1976.


\item [[F. William Lawvere]], \emph{Comments on the development of topos theory}, pp.715-734 in Pier (ed.), \emph{Development of Mathematics 1950 - 2000} , Birkh\"a{}user Basel 2000. (\href{http://www.tac.mta.ca/tac/reprints/articles/24/tr24abs.html}{tac reprint})


\item [[Colin McLarty]], \emph{Exploring Categorical Structuralism}, Phil. Math. \textbf{12} no.3 (2004) pp.37-53. (\href{http://www.cwru.edu/artsci/phil/PMExploring.pdf}{pdf})


\item D. A. Martin, \emph{Borel determinacy}, Ann. Math. \textbf{102} (1975) pp.363-371.


\item [[Gerhard Osius]], \emph{Categorical Set Theory: A Characterization of the Category of Sets}, JPAA \textbf{4} (1974) pp.79-119.


\item [[Thoralf Skolem]], \emph{[[Some remarks on axiomatized set theory|Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre]]}, Mathematikerkongressen i Helsingfor 4-7 Juli 1922. English transl. pp.290-301 of van Heijenoort (ed.), \emph{From Frege to G\"o{}del} , Harvard UP 1967.


\item [[Thomas Streicher]], \emph{Universes in Toposes}, pp.78-90 in Crosilla, Schuster (eds.), \emph{From Sets and Types to Topology and Analysis} , Oxford UP 2005. (\href{http://www.mathematik.tu-darmstadt.de/~streicher/NOTES/UniTop.pdf}{preprint})


\item [[Paul Taylor]], \emph{[[Practical Foundations of Mathematics]]}, Cambridge UP 1999. (ch. 9)


\item George Tourlakis, \emph{Lectures in Logic and Set Theory}, Volume 2: \emph{Set Theory}, Cambridge University Press (2003). (section III.8)



\end{itemize}
[[!redirects axiom of replacement]] [[!redirects axioms of replacement]] [[!redirects axiom scheme of replacement]] [[!redirects axiom schemes of replacement]] [[!redirects axiom schema of replacement]] [[!redirects axiom schemas of replacement]] [[!redirects axiom schemata of replacement]] [[!redirects replacement axiom]] [[!redirects replacement axioms]] [[!redirects replacement axiom scheme]] [[!redirects replacement axiom schemes]] [[!redirects replacement axiom schema]] [[!redirects replacement axiom schemas]] [[!redirects replacement axiom schemata]]

[[!redirects axiom of collection]] [[!redirects axioms of collection]] [[!redirects axiom scheme of collection]] [[!redirects axiom schemes of collection]] [[!redirects axiom schema of collection]] [[!redirects axiom schemas of collection]] [[!redirects axiom schemata of collection]] [[!redirects collection axiom]] [[!redirects collection axioms]] [[!redirects collection axiom scheme]] [[!redirects collection axiom schemes]] [[!redirects collection axiom schema]] [[!redirects collection axiom schemas]] [[!redirects collection axiom schemata]]

category: foundational axiom



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