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slides-hottest2023.lagda.tex
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slides-hottest2023.lagda.tex
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\title{Towards multiversal modal operators for homotopy type theory}
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\input{images/primes.tex}
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\begin{document}
\addtocounter{framenumber}{-1}
{\usebackgroundtemplate{\begin{minipage}{\paperwidth}\centering\includegraphics[height=\paperheight]{multiverse-faded}\end{minipage}}
\begin{frame}[c]
\centering
\color{white}
\bigskip
\bigskip
\bigskip
\bigskip
\scriptsize
\textit{-- an invitation --}
\setbeamercolor{block body}{bg=black!100}
\begin{minipage}{0.49\textwidth}
\begin{block}{}
\centering\normalsize\color{white}
\hil{Towards multiversal modal operators for homotopy type theory}
\end{block}
\end{minipage}
\bigskip
\bigskip
\bigskip
\bigskip
\bigskip
\bigskip
\bigskip
\end{frame}}
\definecolor{mypurple}{RGB}{150,0,255}
\setbeamercolor{structure}{fg=mypurple}
{\usebackgroundtemplate{\begin{minipage}{\paperwidth}\includegraphics[height=\paperheight]{sea-of-clouds-2}\end{minipage}}
\begin{frame}{How should this notion be formalized in HoTT?}
\jnote{5-}{
The presented proof rests on the law of excluded middle and hence cannot
immediately be interpreted as a program for finding suitable indices~$i < j$.
However, constructive proofs are also possible (for instance by
induction on the value of a given term of the sequence), and furthermore
constructive proofs can be extracted from the presented classical proof.
}
\jnote{7-}{
The class of well quasiorders is stable under cartesian products, lists and
trees, by Dickson's Lemma, Higson's Lemma and Kruskal's Theorem,
respectively. However, in their naive formulations, these are merely
theorems of classical mathematics. For general constructive results, the
definition of ``well'' needs to be improved.
Classical texts often employ needless negations in their definitions; this
is an XXX
}
\vspace*{-1em}
\[ \astikznodetransparentlycircled{xm}{7}\!,
\quad \astikznodetransparentlycircled{x0}{4}\!,
\quad \only<1-2>{\astikznodetransparentlycircled{t1}{3}}\only<3->{\astikznodecircled{t1}{mypurple}{3}}\!,
\quad \only<1>{\ldots}\pause \astikznodetransparentlycircled{x1}{1}\!,
\quad \only<2>{\ldots}\pause \astikznodecircled{t2}{mypurple}{8}\!,
\quad \only<3>{\ldots} \visible<4->{\astikznodetransparentlycircled{x2}{2}\!,}
\quad \only<4->{\ldots} \]
{\centering\begin{tikzpicture}[remember picture,overlay]
\node[draw=mypurple, circle, thick, inner sep=0.1em] (t3) {\scriptsize$\leq$};
\path[draw=mypurple,thick]
(t1)
to [out=-90, in=180] (t3)
to [out=0, in=-90] (t2);
\end{tikzpicture}\par}
\medskip
\pause
\begin{block}{}
\justifying
\textbf{Thm.} Every sequence~$\alpha : \NN \to \NN$ is \hil{good} in that
there exist~$i < j$ with~$\alpha\,i \leq \alpha\,j$.
\end{block}
\pause
\vspace*{-0.6em}
\emph{Proof.} \emph{(offensive?)} By~\badbox{\textsc{lem}}, there is a
minimum~$\alpha\,i$.
Set~$j \defeq i + 1$. \qed\par
\pause
\textbf{Def.} (classically) A quasiorder~$X$ is \hil{well} iff every sequence~$\NN \to X$ is good.
\textbf{Examples.} (classically)\ \ $(\NN,{\leq}),\ \ X \times Y,\ \ X^*,\ \ \mathrm{Tree}(X).$
\bigskip
\pause
\emph{A naive formalization attempt:}
\textbf{Def.} $\displaystyle\mathsf{Well}_\infty(X,{\leq}) \defeq
\prod\limits_{\alpha \? \NN \to X} \, \Bigl\|\sum\limits_{i \? \NN} \sum\limits_{j \? \NN}
i < j \times \alpha\,i \leq \alpha\,j\Bigr\|_{-1}$
\pause
\qquad\scalebox{1.5}{\dbend}
\medskip
\begin{itemize}
\item[\bad{\xmark}] philosophically~strenuous %: needless reference of the transfinite
\item[\bad{\xmark}] not practical %: cannot verify closure properties
\item[\bad{\xmark}] not faithful(?) %: missing out on some sequences
\end{itemize}
\end{frame}}
\begin{frame}{Sequences provided by \textsc{lem}}
\justifying
\begin{block}{}
\justifying
\textbf{Lemma.} Let~$X$ be a well quasiorder. Let~$\alpha : \NN \to X$ be a
sequence. Assuming~\bad{\textsc{lem}}, there merely is a monotonic
subsequence~$\alpha\,i_0 \leq \alpha\,i_1 \leq \cdots$.
\end{block}
\vspace*{-0.6em}
\emph{Proof.} The type
\[ I \defeq \sum_{i \? \NN} \neg\sum_{j \? \NN} i < j \times \alpha\,i \leq \alpha\,j \]
cannot be in bijection with~$\NN$, as else the~$I$-extracted subsequence
of~$\alpha$ would not be good. By~\bad{\textsc{lem}}, the type~$I$ is finite.
Any index~$i_0$ larger than all the indices in~$I$ is a suitable starting
point for a monotonic subsequence. \qed
\medskip
\pause
\begin{block}{}
\justifying
\textbf{Prop.} Let~$X$ and~$Y$ be well quasiorders.
Assuming~\bad{\textsc{lem}},~$X \times Y$ is well.
\end{block}
\vspace*{-0.6em}
\emph{Proof.} Let an infinite sequence~$\gamma : \NN \to X \times Y$ be
given. Write~$\gamma\,k = (\alpha\,k,\beta\,k)$. By the lemma, there is a
monotonic subsequence~$\alpha\,i_0 \leq \alpha\,i_1 \leq \cdots$. Because~$Y$
is well, there are indices~$n < m$ such that~$\beta\,i_n \leq \beta\,i_m$. As
also~$\alpha\,i_n \leq \alpha\,i_m$, the sequence~$\gamma$ is good. \qed
\medskip
\pause
\dnote{\emph{We cannot trust~\bad{\textsc{lem}}-provided sequences to be
available in the type~$\NN \to X$. Similarly with~\bad{\textsc{dc}}.}}
\end{frame}
\begin{frame}{Sequences depending on an environment}
Let~$X$ be an hset such that there is no surjection~$\NN \twoheadrightarrow X$.
Then the type~$\NN \to X$ misses the \hil{generic enumeration}~$\genalpha$ of~$X$.
\medskip
\pause
\emph{Idea:} Approximate (fictitious) surjections~$\NN \twoheadrightarrow X$
by finite sequences~$\sigma = x_0 \ldots x_{n-1} \? X^*$. \\
Starting with the empty sequence~$\varepsilon$, over time, such an
approximation can grow to
\begin{enumerate}
\item one of $\sigma y$, where~$y \? X$, so it becomes \emph{more defined}; or
\item one of~$\sigma \tau$, where~$\tau \? X^*$ such that~$a \in
\sigma\tau$, so it becomes \emph{more surjective}.
\end{enumerate}
\medskip
\pause
For~$k : \NN$ and~$a : X$, the expression~``$\genalpha\,k = a$'' does not
denote a proposition but rather a \hil{stage-dependent proposition}~$X^* \to
\mathsf{Prop}$, namely $\lambda x_0 \ldots x_{n-1}.\ k < n \times x_k = a$.
\medskip
\pause
Given a stage-dependent proposition~$P$, $\hil{$\nabla P$}\,\sigma$ expresses that no
matter how~$\sigma$ evolves to a better approximation~$\tau$,
eventually~$P\,\tau$ will hold:
\AgdaHide{
\begin{code}
{-# OPTIONS --no-prop --no-import-sorts --allow-unsolved-metas #-}
open import Agda.Primitive using (Set)
Prop = Set
infixl 5 _∈_
postulate
X : Set
X* : Set
[] : X*
_∷_ : X* → X → X*
_⧺_ : X* → X* → X*
_∈_ : X → X* → Prop
module Foo where
\end{code}
}
{\small\begin{code}
data ∇ (P : X* → Prop) (σ : X*) : Prop where
now : P σ → ∇ P σ
later₁ : ((y : X) → ∇ P (σ ∷ y)) → ∇ P σ
later₂ : (a : X) → ((τ : X*) → a ∈ σ ⧺ τ → ∇ P (σ ⧺ τ)) → ∇ P σ
\end{code}}
\vspace*{-0.8em}
For instance, for~$P_a\,\sigma \defeq (a \in \sigma)$,
we have~$\nabla P_a \, \varepsilon$. \quad ``$\genalpha$ is surjective'':
\AgdaSymbol{(}\AgdaBound{a}\AgdaSpace{}%
\AgdaSymbol{:}\AgdaSpace{}%
\AgdaPostulate{X}\AgdaSymbol{)}\AgdaSpace{}%
\AgdaSymbol{→}\AgdaSpace{}%
\AgdaDatatype{∇}\AgdaSpace{}%
\AgdaBound{P$_a$}\AgdaSpace{}%
\AgdaBound{$\varepsilon$}
\end{frame}
\begin{frame}{Well quasiorders revisited}
\emph{A naive formalization attempt:}
\textbf{Def.} $\displaystyle\mathsf{Well}_\infty(X,{\leq}) \defeq
\prod\limits_{\alpha \? \NN \to X} \, \Bigl\|\sum\limits_{i \? \NN} \sum\limits_{j \? \NN}
i < j \times \alpha\,i \leq \alpha\,j\Bigr\|_{-1}$
\pause
\qquad\scalebox{1.0}{\dbend}
\bigskip
\emph{An inductive rephrasing:}
\textbf{Def.} $\displaystyle\mathsf{Well}(X,{\leq}) \defeq
\nabla \mathsf{Good} \, \varepsilon$, where~$\mathsf{Good}\,x_0 \ldots
x_{n-1} \defeq \mathop{\exists}\limits_{i:\NN}\mathop{\exists}\limits_{j:\NN} (i < j \times x_i \leq
x_j)$ and
{\vspace*{-0.7em}\small\begin{code}
data ∇ (P : X* → Prop) (σ : X*) : Prop where
now : P σ → ∇ P σ
later : ((y : X) → ∇ P (σ ∷ y)) → ∇ P σ
\end{code}}
\vspace*{-0.8em}
In other words: \emph{A quasiorder~$X$ is well iff the generic sequence~$\NN \to X$
is good.}
\begin{block}{}
\justifying
\textbf{Prop.} Let~$X$ and~$Y$ be well quasiorders.
\good{Without} assuming~\bad{\textsc{lem}},~$X \times Y$ is well.
\end{block}
\pause
\dnote{We have~$\mathsf{Well}(X,{\leq}) \to
\mathsf{Well}_\infty(X,{\leq})$, but the converse only holds in presence of \hil{bar
induction}. \emph{How much stronger exactly is the inductive rephrasing?}}
\end{frame}
{\usebackgroundtemplate{\begin{minipage}{\paperwidth}\vspace*{4.95cm}\includegraphics[width=\paperwidth]{topos-horses-lighter}\end{minipage}}
\begin{frame}{Topos for reifying generic models}
Grothendieck toposes (= categories of sheaves over sites) are mathematical universes:
\begin{itemize}
\item The \hil{generic sequence} is an honest function~$\NN \to X$
\hil{in} a certain topos~$\mathcal{E}$.
\item The \hil{generic surjection} is an honest surjection \hil{in} an
appropriate subtopos of~$\mathcal{E}$.
\item In the \emph{double-negation subtopos} of the base, a predicate~$Q : \NN
\to \mathsf{Prop}$ such that
\[ Q\,x \times Q\,y \,\longrightarrow\, x = y \quad\text{and}\quad
\neg\neg \sum_{n\?\NN} Q\,n \]
looks like an ordinary natural number.
\end{itemize}
A statement holds in a topos iff a certain topos-directed translation holds
in the base. For instance, for the double-negation subtopos the translation
substitutes
\[\begin{array}{rcl@{\quad\text{where}\quad}rcl}
\exists & \leadsto & \exists^\mathrm{cl},
& \exists^\mathrm{cl} &\defeqv& \neg\neg\exists, \\
\vee & \leadsto & \vee^\mathrm{cl},
& \alpha \vee^\mathrm{cl} \beta &\defeqv& \neg\neg(\alpha \vee \beta), \\
= & \leadsto & =^\mathrm{cl},
& s =^\mathrm{cl} t &\defeqv& \neg\neg(s = t).
\end{array} \]
In general, stage-dependent~$\nabla$ instead of~$\neg\neg$.
\end{frame}}
{\usebackgroundtemplate{\begin{minipage}{\paperwidth}\vspace*{5.95cm}\includegraphics[width=\paperwidth]{fr1}\end{minipage}}
\begin{frame}{Maximal ideals as convenient fictions}
\jnote{1}{
The theorem on the slide is a generalization of a fact from undergraduate
linear algebra: Over a field, no surjective matrix can have more rows than
columns. (``Surjective'' here means that the induced linear map is
surjective.)
}
\jnote{6-7}{
The slide presents a standard proof as offered by most textbooks on
commutative algebra. The proof is quite efficient from a viewpoint of
mathematical organization, as it quickly succeeds in reducing to the field
situation. As such, it is short and memorable.
However, the proof can also be critized for appealing to the transfinite
two times; the methods of the proof are at odds with the concreteness of
the statement of the theorem---from given equations witnessing
surjectivity,~$Mv_i = e_i$, we are asked to deduce the equation~$1 = 0$.
For this reason, the theorem and its classical proof are often used as case
studies for tools and techniques aiming to extract constructive content
from classical proofs. One such technique employs constructive forcing.
The first(?)\@ constructive proof, found directly without using extraction
techniques, is laid out in a
\fixedhref{https://www.ams.org/journals/proc/1988-103-04/S0002-9939-1988-0954974-5/S0002-9939-1988-0954974-5.pdf}{beautiful short note by Richman}.
}
\jnote{8-9}{
For the countable case, there is an iterative construction of a maximal
ideal, making do without any decidability assumptions (due to Krivine,
later later clarified by Berardi and Valentini). It is a parlor trick,
resulting in a subset which formally verifies the axioms for a maximal
ideal but without carrying out any actual work. Indeed, the resulting ideal
will in general not be a detachable subset of the ring.
Surprisingly, there is still computational content in this construction, as
explored in \fixedhref{https://arxiv.org/abs/2207.03873}{this joint paper
with Peter Schuster}; one interpretation of our observation is that classical
proofs don't ``really'' require a maximal ideal; they just use that notion
for structuring hidden computations.
}
\jnote{9}{
In a suitable topos, the ring appears countable. Hence we can
carry out the iterative maximal ideal construction there. The resulting ideal
will not be part of the base universe (instead, from the point of view of
the base universe we will just have constructed a certain sheaf of ideals on a
certain pointfree space), but bounded first-order consequences of its
existence still pass down to the base.
}
\jnote{10}{
Unwinding all the definitions from toposes and from the
iterative maximal ideal construction, and eliminating the application
of~\textsc{lem} from the classical proof presented before, we mechanically
arrive at the constructive direct proof presented on the slide.
}
\begin{block}{}
\textbf{Thm.}
Let~$M$ be a surjective matrix with more rows than columns over a
commutative ring~$A$. Then~$1 = 0$ in~$A$.
\end{block}
\only<1-9>{
\visible<2->{\emph{Proof.} (classical) \bad{Assume not.}}
\visible<3->{Then there is~a \bad{maximal ideal} $\mmm$.}
\visible<5->{The matrix is surjective over~$A/\mmm$.}
\visible<6->{Since~$A/\mmm$ is a field, this is a contradiction to basic linear algebra.\qed}
}
\only<4-9>{\medskip\par\centering\scalebox{0.9}{\centering\begin{tikzpicture}
\node (0) at (0,1) {$(0) = \{0\}$};
\node (1) at (0,5) {$(1) = \ZZ$};
\node (2) at (-2,4) {$(2)$};
\node [right of=2] (3) {$(3)$};
\node [below of=2] (4) {$(4)$};
\node [below of=2, xshift=0.7cm] (6) {$(6)$};
\node [right of=3] (5) {$(5)$};
\node [right of=5] (7) {$(7)$};
\node [right of=7] (7d) {$\ldots$\phantom{(}};
\node [right of=7d, xshift=3cm, yshift=-2cm] (max)
{\vbox{\small{\it maximal among the proper ideals} \\ \medskip \hspace*{-6.75em}\textbullet \quad $\neg(1 \in
\mmm)$ \\ \medskip \textbullet \quad $\neg\bigl(1 \in \mmm + (x)\bigr) \Rightarrow x \in \mmm$}};
\node [below of=4] (8) {$(8)$};
\node [right of=8, xshift=3cm] (8d) {$\ldots$};
\draw (0) -- (8);
\draw (0) -- (8d);
\draw (0) -- (6);
\draw (2) -- (1);
\draw (3) -- (1);
\draw (5) -- (1);
\draw (7) -- (1);
\draw (7d) -- (1);
\draw (4) -- (2);
\draw (8) -- (4);
\draw (6) -- (2);
\draw (6) -- (3);
\draw [mypurple!30, thick, shorten <=-2pt, shorten >=-2pt, ->] (max) to [out=120, in=-30] (7d);
\begin{pgfonlayer}{background}
\draw[decorate, very thick, draw=mypurple!30]
($(2.south west) + (8pt, 0)$) arc(270:180:8pt) --
($(2.north west) + (0, -8pt)$) arc(180:90:8pt) --
($(7d.north east) + (-8pt, 0)$) arc(90:0:8pt) --
($(7d.south east) + (0, 8pt)$) arc(0:-90:8pt) --
cycle;
\end{pgfonlayer}
\end{tikzpicture}\par}\par}
\pause
\pause
\pause
\pause
\pause
\pause
\raggedright
\only<7-9>{
\emph{Does there exist a maximal ideal?}
\pause
\good{Yes}, if~$A$ is countable.
In the general case: \bad{No}\pause, but \good{yes} in a \emph{suitable topos}, and
\emph{bounded first-order consequences} of its existence there \good{pass down} to
the base.
\pause
}
\only<10>{
\emph{Proof.} (constructive, special case) Write~$M =
\left(\begin{smallmatrix}x\\y\end{smallmatrix}\right)$. By surjectivity,
we have~$u, v \? A$ with
\[
u \left(\begin{smallmatrix}x\\y\end{smallmatrix}\right) = \left(\begin{smallmatrix}1\\0\end{smallmatrix}\right)
\quad\text{and}\quad
v \left(\begin{smallmatrix}x\\y\end{smallmatrix}\right) = \left(\begin{smallmatrix}0\\1\end{smallmatrix}\right).
\]
Hence
$
1 = (vy) (ux) = (uy) (vx) = 0
$. \qed
}
\end{frame}}
\newcommand\ytl[2]{%
\parbox[b]{4.5em}{\hfill{#1}~$\cdots\cdots$~}%
\makebox[0pt][c]{\color{mypurple}$\bullet$}{\color{mypurple}\vrule}\quad%
\parbox[c]{9.3cm}{\vspace{7pt}\raggedright#2\\[7pt]}%
\\[-2.0pt]}
\begin{frame}{A brief timeline}
\jnote{4}{
Gödel's proof is by the \hil{$L$-translation}, where $L$ is the
``constructible universe''. This translation ``relativizes quantification
to~$L$'', for instance the~$L$-translation of
\[ \varphi \defeqv \bigl(\forall x\_ \exists y\_ \ldots\bigr) \qquad\text{is}\qquad
\varphi^L \equiv \bigl(\forall(x \in L)\_ \exists(y \in L)\_ (\ldots)^L\bigr). \]
We can then verify, in a weak metatheory such as~\textsc{pra}, that for
every formula~$\varphi$ in the language of set theory:
If $\textsc{zfc}\text{+}\textsc{ch} \vdash \varphi$, then $\textsc{zf} \vdash
\varphi^L$.
Specializing to~$\varphi \defeqv \bot$ we obtain in particular: If~\textsc{zfc} is
inconsistent, then so is~\textsc{zf}. The axiom of choice does not
introduce new inconsistencies.
In modern semantic language: While the axiom of choice and~\textsc{ch} might fail in the
base universe~$V$ (= the class of all sets), they always hold in~$L$.
Gödel's~$L$ was the first \emph{inner model} (= class-sized model of set
theory) explicitly studied, nowadays we know many.
}
\jnote{5}{
For his proof, Cohen invented the technique of \emph{forcing}, situated in
classical mathematics where the base universe~$V$ is assumed to validate
the axioms of~\textsc{zfc}.
Recall that a given ring~$R$ or group can be extended in various ways, to include
``generic elements'' as in~$R[X]$ or elements with prescribed relations as
in~$R[X]/(X^2+1) =\vcentcolon R[i]$. The idea of forcing is to construct
similar such extensions, but not of rings but of universes (traditionally set-sized models
of~\textsc{zf} or~\textsc{zfc}, but also class-sized models, or models
of intuitionistic set theories, or models of type theories, or even models
of arithmetic).
In semantic language, from a high level the idea of Cohen's independency proof is the following: Whether
the base universe~$V$ contains a cardinal number intermediate
between~$\aleph_0$ and~$2^{\aleph_0}$ is uncertain. But there is a certain
extension of the base universe---constructed by forcing---which does
contain such a number. Like the base~$V$, this forcing extension still
validates the axioms of~\textsc{zfc}. Hence there cannot be a
\textsc{zfc}-proof of~\textsc{ch}, as in Cohen's extension~$\neg\textsc{ch}$
holds.
Syntactically, Cohen's forcing provides us with an explicit formula
translation~$\varphi \mapsto \varphi^C$ such that~\textsc{pra} proves:
For every formula~$\varphi$, if~$\textsc{zfc}{\text{+}}\neg\textsc{ch} \vdash \varphi$,
then~$\textsc{zfc} \vdash \varphi^C$.
\bigskip
}
\jnote{6}{
Joel David Hamkins argues: In view of our rich experience with worlds which
validate~\textsc{ch} and worlds which don't, we shouldn't be surprised that
no proposed new axiom for settling~\textsc{ch} is ultimately convincing.
Instead, we should embrace the multiverse of all models of set theory and
explore how the truth values of statements of interest change when we
travel the multiverse (for instance, by passing from a universe to one
of its forcing extensions).
In this generalized sense, the continuum hypothesis is settled: We have a
good understanding of the stability properties of~\textsc{ch} under
important constructions. In particular, for a certain precise meaning of
``universe'' and ``extension'', we know that~\textsc{ch} is a
\emph{switch}: $\necessary(\possible\textsc{ch} \wedge
\possible\neg\textsc{ch})$; in words: Every universe can be extended both to a
universe in which~\textsc{ch} holds and to a universe in
which~$\neg\textsc{ch}$ holds.
An exposition and references for further reading about the multiverse
position can be found
\fixedhref{https://www.speicherleck.de/iblech/stuff/multiverse.pdf}{here}.
}
\jnote{8}{
Judith Roitman. ``The uses of set theory''. \emph{Math.\@ Intelligencer}
\textbf{14}(1) (1992), 63--69.
Forcing is useful not only to explore the range of foundational
possibility; it has many more applications across several subjects of
mathematics.
}
\begin{columns}[t]
\begin{column}{0.80\textwidth}
\ytl{1878}{Cantor advances the \hil{continuum hypothesis}, the claim
that~$2^{\aleph_0} = \aleph_1$.}
\pause
\ytl{1910s}{Zermelo--Fraenkel set theory emerges.}
\pause
\ytl{1920s}{Set theorists pursue additional axioms to
settle~\textsc{ch} \\ (one way or another).}
\pause
\ytl{1938}{Gödel proves: If~\textsc{zfc} is consistent, so
is~\textsc{zfc}+\textsc{ch}.}
\pause
\ytl{1963}{Cohen proves: If~\textsc{zfc} is consistent, so
is~\textsc{zfc}+$\neg$\textsc{ch}.}
\pause
\ytl{2011}{Hamkins offers his paper on the \hil{multiverse position} in
the philosophy of set theory.}
\pause
%arguing that the program of pursuing
%additional axioms (while successful in many ways) is doomed to fail
%to settle~\textsc{ch}.}
\ytl{2016}{Oldenziel proposes to study the modal multiverse of toposes.}
\parbox[b]{4.5em}{\hfill\phantom{x}}\makebox[0pt][c]{\phantom{b}}{\color{mypurple}\vrule}\\[-6.5pt]
\parbox[b]{4.5em}{\hfill\phantom{x}}\makebox[0pt][c]{\phantom{b}}{\color{mypurple}\vrule}\\[-6.5pt]
\parbox[b]{4.5em}{\hfill\phantom{x}}\makebox[0pt][c]{\phantom{b}}{\color{mypurple}\vrule}\\[-6.5pt]
\parbox[b]{4.5em}{\hfill\phantom{x}}\makebox[0pt][c]{\phantom{b}}{\color{mypurple}\vrule}\\[-6.5pt]
\parbox[b]{4.5em}{\hfill\phantom{x}}\makebox[0pt][c]{\phantom{b}}{\color{mypurple}\vrule}\\[-6.5pt]
\end{column}
\begin{column}{0.20\textwidth}
\pause
\centering\includegraphics[width=\textwidth,valign=t]{roitman}
\scriptsize Judith Roitman %(* 1945)
\medskip
\emph{Mainstream mathematics is beginning to see results using
modern set theoretic techniques.}
\end{column}
\end{columns}
\end{frame}
{\usebackgroundtemplate{\begin{minipage}{\paperwidth}\vspace*{3.59cm}\includegraphics[width=\paperwidth]{staircase}\end{minipage}}
\begin{frame}{The modal set-theoretic multiverse}
\begin{tikzpicture}[remember picture,overlay]
\node[xshift=-2.5cm,yshift=-4cm] at (current page.north east)
{\includegraphics[width=3cm]{branching}};
\end{tikzpicture}
\justifying
\vspace*{-2em}
\textbf{Def.} A \hil{model of set theory} is a (perhaps class-sized) structure $(M,{\in})$
satisfying axioms such as those of~\textsc{zfc}.
{\small\emph{Examples.}\vspace*{-0.8em}
\begin{itemize}
\item $V$, the class of all sets \vspace*{-0.6em}
\item $L$, Gödel's constructible universe \vspace*{-0.6em}
\item $V[G]$, a forcing extension containing a generic filter~$G$ of \\ some
poset of forcing conditions \vspace*{-0.6em}
\item Henkin/term models from consistency of (extensions of)~\textsc{zfc}
\end{itemize}}
\pause
\emph{We embrace all models of set theory:}
\textbf{Def.} $\possible\varphi$ iff~$\varphi$ holds in \hil{some extension} of
the current universe. \\
\phantom{\textbf{Def.}} $\necessary\varphi$ iff~$\varphi$ holds in \hil{all extensions}
of the current universe.
\begin{itemize}
\item $\necessary(\possible\textsc{CH} \wedge \possible\neg\text{CH})$,
the continuum hypothesis is a \hil{switch}.
\item $\necessary\possible\necessary(\text{$X$ is countable})$,
existence of an enumeration is a \hil{button}.
\end{itemize}
\end{frame}}
\begin{frame}{The modal topos-theoretic multiverse}
\textbf{Def.} A statement~$\varphi$ holds \ldots
\begin{enumerate}
\small
\item \hil{everywhere} ($\necessary\varphi$) iff it holds in every topos
(over the current base).
\item \hil{somewhere} ($\possible\varphi$) iff it holds in some positive topos.
\item \hil{proximally} ($\xpossible\varphi$) iff it holds in some positive overt topos.
\end{enumerate}
\bigskip
\begin{tikzpicture}[overlay]
\node[anchor=south east,inner sep=0] (image) at (14.8,0.2) {
\includegraphics[width=0.2\textwidth]{multiverse}
};
\end{tikzpicture}\vspace*{-2.3em}
\pause
\emph{Traveling the multiverse:}
\begin{itemize}
\item For every inhabited set~$X$, \emph{proximally} there is an enumeration~$\NN \twoheadrightarrow X$.
\item A quasiorder is well iff \emph{everywhere}, every sequence is good.
\item A ring element is nilpotent iff all prime ideals \emph{everywhere} contain it.
\item For every ring, \emph{proximally} there is a maximal ideal.
\item A relation is well-founded iff \emph{everywhere}, there is no descending chain.
\item \emph{Somewhere}, the law of excluded middle holds.
\end{itemize}
\pause
\only<1-3>{
\small
\begin{block}{}
\justifying
\textbf{Prop.} Let~$(X,{\leq})$ be a well quasiorder.
Then~$({<})$, where~$x < y \equiv (x \leq y \wedge \neg(y \leq x))$,
is well-founded.
\end{block}
\vspace*{-0.5em}
\emph{Proof.} Everywhere, there can be no infinite descending chain, as any
such would also be good. \qed
Unrolling this proof gives a program~$\nabla\mathsf{Good}\,\varepsilon \to
\prod_{x:X} \mathsf{Acc}\,x$.
}
\pause
\scriptsize
\emph{Foreshadowed by:}
\begin{itemize}
\item[1984\phantom{s}] André Joyal, Miles Tierney. ``An extension of the Galois theory of Grothendieck''.
\item[1987\phantom{s}] Andreas Blass. ``Well-ordering and induction in intuitionistic logic and topoi''.
\item[2010s] Milly Maietti, Steve Vickers. Ongoing work on arithmetic universes.
\item[2011\phantom{s}] Joel David Hamkins. ``The set-theoretic multiverse''.
\item[2013\phantom{s}] Shawn Henry. ``Classifying topoi and preservation of higher order logic by geometric morphisms''.
\end{itemize}
\end{frame}