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\begin{document}

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\section*{universe}

\begin{quote}%
This entry is about the notion in [[mathematics]]/[[logic]]/[[type theory]]. For the notion of the same name in [[physics]] see at \emph{[[observable universe]]}.


\end{quote}

\vspace{.5em} \hrule \vspace{.5em}
\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{categories_as_universes}{Categories as universes}\dotfill \pageref*{categories_as_universes} \linebreak
\noindent\hyperlink{universes_of_pure_sets}{Universes of pure sets}\dotfill \pageref*{universes_of_pure_sets} \linebreak
\noindent\hyperlink{universes_inside_}{Universes inside $SET$}\dotfill \pageref*{universes_inside_} \linebreak
\noindent\hyperlink{reflection}{Reflection}\dotfill \pageref*{reflection} \linebreak
\noindent\hyperlink{universes_in_type_theory}{Universes in type theory}\dotfill \pageref*{universes_in_type_theory} \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{in_type_theory}{In type theory}\dotfill \pageref*{in_type_theory} \linebreak


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

A universe is a realm within which (some conceived part, naively and virtually all, of) [[mathematics]] may be thought of as taking place. Universes can be purely metamathematical, but we can also reflect upon them and bring them into mathematics. There are several different kinds of `universes'. For a physical notion of universe see [[observable universe]].

\hypertarget{categories_as_universes}{}\subsection*{{Categories as universes}}\label{categories_as_universes}

Much of [[ordinary mathematics]] can be thought of as taking place [[internalization|inside]] ``the archetypical [[category]] [[Set|SET]] of sets''. Typically, the properties of $SET$ are formulated in [[first-order logic]] using a [[set theory]] such as [[ZFC]] or (more directly) [[ETCS]].

We can generalise this from $SET$ to any other [[category]] $C$. Without further assumptions on the category, there is in general very little mathematics that can be formulated inside it, but a few extra properties and structures are usually sufficient to provide something interesting. This is the general topic of \emph{[[internalisation]]}. The attitude to take is that any \emph{specific} category is merely one [[model]], while a general \emph{class} of categories is a [[theory]] (really a [[doctrine]], or [[2-theory]]).

We can also use [[higher category theory|higher categories]] instead of mere categories here. Even for ordinary mathematics, this means starting with $\infty$-[[infinity-Grpd|GRPD]] instead of $SET$.

\hypertarget{universes_of_pure_sets}{}\subsection*{{Universes of pure sets}}\label{universes_of_pure_sets}

The idea of the [[large category]] $SET$ as the universe of mathematics has an analogue in pre-category-theoretic [[material set theory]]. The \textbf{von Neumann universe} $V$ is the [[proper class]] of all [[axiom of foundation|well-founded]] [[pure sets]].

More explicitly: for every [[ordinal number]] $\alpha$, we have a [[set]] $V_\alpha$ (the von Neumann universe of \textbf{rank} $\alpha$), defined [[recursion|recursively]] using the operations of [[power set]] and (material) [[union]] as

\begin{displaymath}
V_\alpha \coloneqq \bigcup_{\beta \lt \alpha} \mathcal{P}V_\beta .
\end{displaymath}
Then $V$ itself is the union of all of the $V_\alpha$.

A similar but more complicated definition allows us to define the universe $L$ of [[constructible set]]s, called the [[constructible universe]].

See also a \href{https://secure.wikimedia.org/wikipedia/en/wiki/Universe_%28mathematics%29}{Wikipedia article} written largely by [[Toby Bartels]] in another lifetime.

\hypertarget{universes_inside_}{}\subsection*{{Universes inside $SET$}}\label{universes_inside_}

If set theory is the [[foundation of mathematics]], then the universes above are part of metamathematics rather than mathematics itself. However, we can also look for [[small categories]] or sets that exist within set theory and have the properties of a universe.

There is already a hint of this in the hierarchy $V_\alpha$ out of which the von Neumann universe is built. For some values of $\alpha$, $V_\alpha$ might be a sufficiently large and complete collection of [[sets]] in which to do ordinary mathematics. From the [[nPOV]], we can instead look at the category $Set_\alpha$ of these sets and the [[functions]] between them, although it's more common to think about $Set_\kappa$ for some [[cardinal number]] $\kappa$.

An [[infinite set]] is a simple example, as [[finite mathematics]] can be done inside it. Going beyond this, a [[Grothendieck universe]] is a set within which other infinite sets exist but which is complete under the operations of [[material set theory]] that are needed for ordinary mathematics. The [[structural set theory|structural]] analogue is a [[universe in the topos]] $SET$.

In general, we need some axioms to state the existence of such universes; we can think of these as [[large cardinal]] axioms. (The existence of an infinite set is the [[axiom of infinity]]; the existence of a Grothendieck universe is the existence of an [[inaccessible cardinal]].)

The use of such universes is convenient when we want to work with [[large categories]] ``as if they were small.'' That is, if we redefine ``small'' to mean ``element of some fixed universe,'' then the categories of all small structures of some sort will still be sets (rather than [[proper classes]]) in the bigger ambient universe, and thus we can for instance form [[functor categories]] between them freely. We do sometimes then need a way to ``move a category'' from one universe to another; see [[universe enlargement]].

\hypertarget{reflection}{}\subsection*{{Reflection}}\label{reflection}

In [[logic]], we use [[reflection principle]]s (see \href{https://secure.wikimedia.org/wikipedia/en/wiki/Reflection_principle}{Wikipedia}) to systematically identify features of our meta-universe and see what is needed to prove the existence of an internal universe with these features.

This follows the following outline:

\begin{enumerate}%
\item We assume that we understand \emph{a priori} what it means to use [[first-order logic]] (or some other finitary base logic).


\item Using this base logic, we can formulate a foundational [[set theory]] that describes a meta-universe such as $SET$.


\item Of course, $SET$ cannot be described from inside itself. But there may be objects in $SET$ ([[internal categories]]) that \emph{look like} $SET$ itself.


\item We add an additional axiom to our set theory stating that such objects, the \emph{internal universes}, exist. These are then [[models]] of our original set theory.


\item Now we are using a new, stronger set theory; repeat.



\end{enumerate}
\hypertarget{universes_in_type_theory}{}\subsection*{{Universes in type theory}}\label{universes_in_type_theory}

Set theory is not the only [[foundation of mathematics]]. For example, there are also various foundational [[type theories]], closely related to [[structural set theory]]. Then we have a meta-universe of all types, and we can also add axioms for the existence of internal [[type universes]].

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

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


\item [[universe polymorphism]]


\item [[type of types]]

\begin{itemize}%
\item [[subobject classifier]], [[object classifier]], [[partial map classifier]]

\end{itemize}

\item [[reflective subuniverse]]


\item [[G-universe]]



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

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

(\ldots{})

\hypertarget{in_type_theory}{}\subsubsection*{{In type theory}}\label{in_type_theory}

\begin{itemize}%
\item [[Martin Hofmann]], section 2.1.6 of \emph{Syntax and semantics of dependent types} (\href{http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.36.8985}{web})

\end{itemize}
[[!redirects universe]] [[!redirects universes]] [[!redirects universe {\tt \symbol{62}} history]]

[[!redirects von Neumann universe]] [[!redirects von Neumann universes]]



\end{document}