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\begin{document}

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\section*{forcing}

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

\hypertarget{topos_theory}{}\paragraph*{{Topos Theory}}\label{topos_theory}

[[!include topos theory - contents]]

\hypertarget{category_theory}{}\paragraph*{{Category Theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{forcing}{}\section*{{Forcing}}\label{forcing}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{an_analogy_to_polynomial_rings}{An analogy to polynomial rings}\dotfill \pageref*{an_analogy_to_polynomial_rings} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{ReferencesInTermsOfClassifyingToposes}{In terms of sheaves and classifying toposes}\dotfill \pageref*{ReferencesInTermsOfClassifyingToposes} \linebreak


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

In [[set theory]], \textbf{forcing} is a way of ``adjoining indeterminate objects'' to a [[model]] in order to make certain [[axioms]] [[true]] or [[false]] in a resulting new model.

The language of forcing is generally used in [[material set theory]]. From the point of view of [[structural set theory]]/[[categorical logic]] it is more or less equivalent to the construction of [[categories of sheaves]] in [[topos theory]] and [[structural set theory]]: here a [[classifying topos]] of a [[theory]] is a ``category of sets'' that is ``forced'' to contain a generic model of the theory.

\hypertarget{an_analogy_to_polynomial_rings}{}\subsection*{{An analogy to polynomial rings}}\label{an_analogy_to_polynomial_rings}

There are three ways in which to describe the construction of forcing, which can be explained by analogy to a situation in algebra. Suppose that we have a [[ring]] $R$ and we want to know what $R$ would be like if it had a nilsquare element, i.e. an $x$ such that $x^2=0$, although we don't know whether $R$ \emph{actually} has any nilsquare elements. There are several things we could do.

\begin{enumerate}%
\item We could consider some larger ring $S$ which contains $R$ as a subring, and also contains some nilsquare element $x$, and then study the subring of $S$ generated by $R$ and $x$, using only our knowledge that it contains $R$ and $x$ and that $x$ is nilsquare.


\item We could work formally in the [[theory]] of $R$ with an extra variable $x$ and the axiom that $x^2=0$, avoiding consideration of any new rings at all.


\item We could construct the polynomial ring $R[x]$ and quotient by the ideal $(x^2)$, resulting in the \emph{universal} ring $R[x]/(x^2)$ containing $R$ and also a nilsquare element.



\end{enumerate}
Obviously these are more or less equivalent ways of doing the same thing. Arguably, most modern mathematicians would find the third the most natural. It is certainly the most [[category theory|category-theoretic]] and the most in line with the [[nPOV]].

Now suppose instead that we have a model $V$ of [[ZF]] set theory (for instance), and we want to know what $V$ would be like if it contained a set $G$ satisfying some particular properties. We can likewise take three approaches.

\begin{enumerate}%
\item We can assume that $V$ sits inside some larger model of ZF, in such a way that there is a set $G$ outside of $V$ and the ``model generated by $V$ and $G$'' called $V[G]$ satisfies the desired properties. The $G$ is called a \emph{generic} set for the desired ``notion of forcing.'' There is an extra wrinkle in this approach in that such a $G$ will not generally exist unless we assume that $V$ is [[countable set|countable]]. But for purposes of independence proofs in [[classical mathematics]], this is no problem, since the downward [[Löwenheim-Skolem theorem]] guarantees that if ZF is consistent, then it has a countable model.


\item We can define a new notion of ``truth'' for statements about $V$, which is not the same as the old one, and which will usually take values not in ordinary [[truth values]] but in some more complicated [[Boolean algebra]] (or [[Heyting algebra]]), in such a way so that the statement ``there exists a $G$ with the desired properties'' is ``true.'' This is called \emph{forcing semantics}.


\item We can construct a new model of set theory out of whole cloth from $V$, rather than trying to find it as a submodel of some assumed larger model. In material set theory, this construction is usually called a \emph{Boolean valued model} (or a \emph{Heyting valued model} if it is [[constructive mathematics|intuitionistic]]). In structural set theory, this construction is simply that of the [[topos of sheaves]] on a suitable [[site]], in the model $V$. The topos of sheaves is moreover \emph{universal} over $V$ such that it contains an [[object]] $G$ satisfying the desired properties in its [[internal logic]]---in other words, it is a [[classifying topos]] of the [[theory]] of such a $G$. See at \emph{\hyperlink{ReferencesInTermsOfClassifyingToposes}{References -- In terms of classifying toposes}} for more.



\end{enumerate}
Although by analogy, it would seem that a modern perspective should prefer the third approach, most material set theorists still seem to prefer one of the first two.

Before [[Paul Cohen|Cohen]] showed how forcing could give rise to models of ZF(C), [[Abraham Fraenkel|Fraenkel]] introduced the method of [[permutation model]]s (later refined by Mostowski and Specker), which gave models of [[ZFA]], a version of ZF with [[urelements|atoms]]. See for example the [[basic Fraenkel model]].

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

\begin{itemize}%
\item [[double negation topology]]


\item [[homotopy type theory FAQ]] -- \href{http://ncatlab.org/nlab/show/homotopy+type+theory+FAQ}{What is homotopy type theory? -- For set theorists}



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

Exposition is in

\begin{itemize}%
\item [[Timothy Chow]], \emph{A beginner's guide to forcing}, Contemp. Math (\href{https://arxiv.org/abs/0712.1320}{arXiv:0712.1320})

\end{itemize}
Another good introduction is in \hyperlink{Schoenfield71}{Schoenfield 71}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

\begin{itemize}%
\item [[Paul Cohen]], \emph{Set theory and the [[continuum hypothesis]]}, Benjamin, New York 1966


\item [[Gonzalo Reyes]], \emph{Typical and generic in a Baire space for relations}, thesis 1967


\item [[Abraham Robinson]], \emph{Infinite forcing in model theory}, in Proc. 2nd Scand. Logic Synp. pp. 317-340, ed. J. E. Fenstad


\item Jon Barwise ed. \emph{Handbook of mathematical logic}, 1977, in chapters by Macintyre, Burgess and Keisler


\item J. R. Schoenfield, \emph{Unramified forcing}, pp. 383--395 in: Axiomatic set theory, vol. 1, ed. D. S. Scott, Proc. Symp. Pure Math. \textbf{13} (1971) ((\href{https://www.math.ucsd.edu/~sbuss/CourseWeb/Math260_2012F2013W/Shoenfield_UnramifiedForcing.pdf}{pdf}))


\item S. Shelah, \emph{Proper and Improper Forcing} , Perspectives in Math. Logic vol. 5 Springer Heidelberg 1998. (\href{http://projecteuclid.org/euclid.pl/1235419814#toc}{toc})


\item [[Paul Cohen]], \emph{The Discovery of Forcing}, Rocky Mountain J. Math. Volume 32, Number 4 (2002), 1071-1100. (\href{http://projecteuclid.org/euclid.rmjm/1181070010}{Euclid})


\item math overflow, what is the generic poset used in forcing really?, \href{http://mathoverflow.net/questions/51187/what-is-the-generic-poset-used-in-forcing-really}{web}



\end{itemize}
\hypertarget{ReferencesInTermsOfClassifyingToposes}{}\subsubsection*{{In terms of sheaves and classifying toposes}}\label{ReferencesInTermsOfClassifyingToposes}

Discussion of the interpretation of forcing as the passage to [[classifying toposes]] includes

\begin{itemize}%
\item [[Michael Makkai]], [[Gonzalo Reyes]], \emph{First Order Categorical Logic}, Lecture Notes in Mathematics Volume 611, Springer 1977


\item Andrej Ščedrov, \emph{Forcing and classifying topoi}, Memoirs of the American Mathematical Society 1984; 93 pp


\item [[Andreas Blass]], Andrej Ščedrov, \emph{Classifying topoi and finite forcing} (\href{http://deepblue.lib.umich.edu/bitstream/handle/2027.42/25225/0000666.pdf}{pdf})


\item [[David Roberts]], \emph{Class forcing and topos theory}, talk at \emph{\href{https://indico.math.cnrs.fr/event/747/}{Topos at l'IHES}} 2015 (\href{https://doi.org/10.4225/55/5b2252e3092af}{talk notes}, \href{https://youtu.be/4AaSySq8-GQ}{video recording})



\end{itemize}
On \hyperlink{Scedrov84}{Scedrov 84}, [[Peter Johnstone]] writes in his review (\href{http://www.jstor.org/stable/2274338}{The Journal of Symbolic Logic Vol. 50, No. 3 (Sep., 1985), pp. 852-85}):

\begin{quote}%
Until recently, logicians could have been forgiven for treating topos theory as a closed book, whose applications to logic (however interesting they might appear from the outside) were accessible only to insiders well schooled in the mysteries of category theory $[...]$ $[$ Ščedrov's book makes $]$ a very clear case for the advantages of the topos-theoretic approach to forcing namely, that it provides a clear conceptual view of what generic structures really are, as well as a workable mean $[...]$ Although categorical ideas and terminology are inevitably present, they are handled so skilfully that the categoriphobic reader need have few qualms about them $[...]$ Ščedrov's work ought to go a long way towards demolishing the wall of incomprehension that has hitherto prevented many logicians from appreciating what topos theory has to offer their subject.


\end{quote}
[[!redirects forcing]]



\end{document}