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slides-herrsching2023.tex
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\title{Constructive forcing}
\author{Ingo Blechschmidt}
\date{September 20th to September 16th, 2023}
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\begin{document}
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{\usebackgroundtemplate{\begin{minipage}{\paperwidth}\centering\includegraphics[height=\paperheight]{multiverse-faded}\end{minipage}}
\begin{frame}[c]
\jnote{1}{
These are the slides for a three-part lecture series for the autumn school
\fixedhref{https://www.mathematik.uni-muenchen.de/~schwicht/pc23.php}{\emph{Proof and Computation}}
organized by Klaus Mainzer, Peter Schuster and Helmut Schwichtenberg, held in
Herrsching from September 20th to September 16th, 2023.
We explore, in a constructive metatheory, the ``Kripke--Joyal semantics of
the internal language of Grothendieck toposes over sites given by
preorders'' together with applications in constructive algebra and
combinatorics, but without presupposing familiarity with or using language
from topos or category theory.
}
\centering
\color{white}
\bigskip
\bigskip
\bigskip
\bigskip
\scriptsize
\textit{-- an invitation --}
\setbeamercolor{block body}{bg=black!100}
\begin{minipage}{0.35\textwidth}
\begin{block}{}
\centering\normalsize\color{white}
\hil{Constructive forcing}
\end{block}
\end{minipage}
\bigskip
\bigskip
\bigskip
\bigskip
\bigskip
\bigskip
\bigskip
\end{frame}}
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\begin{frame}{}
\jnote{3}{
The slide presents a question on calculus. Constructive forcing
establishes a connection with constructive linear algebra; we will understand
that:
\begin{enumerate}\justifying
\item The eigenvalues (defined as the zeros of the characteristic
polynomial) locally depend continuously on the parameter \hil{because}
it is a theorem of constructive linear algebra that symmetric matrices have
a full list of eigenvalues.
\item The eigenvectors can \hil{not} be expected to locally depend
continuously \hil{because} it is \hil{not} a theorem of constructive
linear algebra that symmetric matrices have an eigenvector basis.
\item There is a dense open subset of the parameter space which restores
continuous dependence \hil{because} it \hil{is} a theorem of constructive
linear algebra that every symmetric matrix does \emph{not~not} have an
eigenvector basis.
\end{enumerate}
A simple example where the eigenvectors cannot be chosen to locally depend
continuously on the parameter is
\fixedhref{https://mathoverflow.net/a/60563/31233}{recorded here}. Removing
the origin there yields a suitable dense open subset.
}
\begin{center}\includegraphics[width=0.2\textwidth]{eigenvector}\end{center}
Let a continuous family of symmetric matrices be given:
\[
\begin{pmatrix}a_{11}(t)&\cdots&a_{1n}(t)\\\vdots&&\vdots\\a_{n1}(t)&\cdots&a_{nn}(t)\end{pmatrix}
\]
Then for every parameter value~$t \in \Omega$, classically there is
\hil{$\blacktriangleright$} a full list of eigenvalues~$\lambda_1(t),\ldots,\lambda_n(t)$ and \\
\hil{$\blacktriangleright$} an eigenvector basis~$(v_1(t),\ldots,v_n(t))$.
\bigskip
\begin{columns}[c]
\begin{column}{0.01\textwidth}
\includegraphics[height=2.4em]{question-mark}
\end{column}
\begin{column}{0.9\textwidth}
\mbox{Can locally the functions~$\lambda_i$ be chosen to be continuous?
\only<2->{\hil{Yes.}}} \\
How about the~$v_i$? \only<2->{\hil{No}\only<3->{, but \hil{yes} on a dense
open subset of~$\Omega$.}}
\end{column}
\end{columns}
\end{frame}
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\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}{
J. 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.
In particular, we will discuss applications of the constructive version of
classical set-theoretic forcing in constructive algebra and combinatorics.
}
\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 parametrized
mathematics.}
\parbox[b]{4.5em}{\hfill\phantom{x}}\makebox[0pt][c]{\phantom{b}}{\color{mypurple}\vrule}\\[-6.5pt]
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\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*{3.35cm}\includegraphics[width=\paperwidth]{topos-horses-lighter}\end{minipage}}
\begin{frame}{Constructive forcing}
\medskip
\begin{columns}[t]
\begin{column}{0.40\textwidth}
\quad\includegraphics[width=0.8\textwidth,valign=t]{branching}
\end{column}
\begin{column}{0.75\textwidth}
\hil{Extending the universe} in various ways, \\[0.7em]
\pause
\quad similarly how we can extend groups or rings, \\[0.7em]
\pause
\quad\quad in and for \hil{constructive mathematics} \\[0.7em]
\pause
\quad\quad\quad \hil{without} presupposing familiarity with \\[0.7em]
\pause
\quad\quad\quad\quad set theory, topos theory, or sheaves.
\pause
\end{column}
\end{columns}
\bigskip
\bigskip
\emph{Outline:}
\vspace*{-0.5em}
\begin{enumerate}
\item What can forcing do for you?
\item Forcing notions and Kripke--Joyal semantics
\item Case studies in constructive algebra and combinatorics
\end{enumerate}
\end{frame}}
\section{What can forcing do for you?}
\begin{frame}{What can forcing do for you?}
\jnote{1}{
Forcing~$\neg\textsc{ch}$ was the historical use case for forcing.
With constructive forcing, where we do not blanketly assume that the base
universe validates~\textsc{lem}, we can inquire the status of~\textsc{lem};
a result in this direction is that~\textsc{lem} is, like~\textsc{ch} in
the context of~\textsc{zfc}, a \emph{switch}:
The base universe can always be extended in such a way as to
force~$\neg\textsc{lem}$ (in fact, most forcing extensions will
falsify~\textsc{lem} even in case the base universe validates it), and also
in such a way to force~\textsc{lem}. The latter provides us with a semantic
view of one of the techniques for extracting constructive content from
classical proofs, namely the
\fixedhref{https://www.speicherleck.de/iblech/stuff/slides-fischbachau2022.pdf}{double-negation
translation combined by the continuation trick}.
}
\jnote{2}{
We will discuss the particular example of the (naive formulation of the)
fundamental theorem of algebra below.
Forcing has been used to construct countermodels to various questions of
constructive (reverse) mathematics, too many to list here. To give just one
pointer, a countermodel for the classical implication ``if there is no
infinite descending chain, then the partial order is inductively
well-founded'' is presented here:
A. Blass, Well-ordering and induction in intuitionistic logic and topoi,
in: \emph{Mathematical Logic and Theoretical Computer Science}. Ed.\@ by
D.\@ Kueker, E.\@ Lopez-Escobar, and C.\@ Smith. Vol.\@ 106. Lect.\@ Notes
Pure Appl. Math. Marcel Dekker, 1987, pp.\@ 29--48.
}
\jnote{3}{
The real numbers don't contain a number~$i$ such that~$i^2 = 0$. Still, for
many results in real analysis, it is convenient to broaden our notion of
existence and pass to the complex plane; the imaginary unit is a
\emph{mathematical phantom} in the
\fixedhref{http://www.wra1th.plus.com/gcw/math/MathPhant.html}{sense of
Gavin Wraith}, a useful tool helping us deduce results about real numbers.
Nowadays there are few ontological concerns about the imaginary unit: We
understand that, in the end, every statement about complex numbers can be
recast as a statement about pairs of real numbers.
In exactly the same fashion, the objects furnished by forcing can be
understood as useful fictions. We will discuss how statements about
the forcing extension can be recast as statements about the base universe.
Given an inhabited set~$X$ in the base universe, most pronouncedly a set
which is uncountable or for which no surjection~$\NN \twoheadrightarrow X$
can be efficiently evaluated, a particularly tantalizing fiction is the
\emph{generic surjection}~$\NN \twoheadrightarrow X$. It exists in a
custom-tailored forcing extension of the base universe and is useful to
apply tools made for the countable setting to the uncountable; we will
discuss an example on the next slide.
}
\jnote{4}{
Typically there are already plenty of maps~$\NN \to X$ in the base
universe; hence constructing a forcing extension which contains a ``fresh''
such map---the so-called \emph{generic sequence}---is not something which
would usually be contemplated in classical set-theoretic forcing.
In the context of constructive mathematics, however, the generic sequence
turns out to be quite useful. It can be used to cast in a familiar naive
language---the language of infinite sequences---definitions, results and
proofs from constructive combinatorics which use constructively more
appropriate inductively defined notions. We will discuss an example below.
}
\jnote{6}{
Slides by Matthias Hutzler:
\fixedhref{https://matthias-hutzler.de/pc23/slides.pdf}{Introduction to
synthetic algebraic geometry}
}
\vspace*{-0.5em}
\begin{columns}
\begin{column}{0.45\textwidth}
\begin{block}{1. Explore foundational possibility}
There are forcing extensions with \\
\textsc{ch}, $\neg$\textsc{ch}, \textsc{lem}, $\neg$\textsc{lem}, \ldots
\end{block}
\end{column}
\pause
\begin{column}{0.45\textwidth}
\begin{block}{2. Demonstrate unprovability}
The fundamental theorem of algebra is \hil{not constructively provable}
as there is a forcing extension where \hil{it is false}.
\end{block}
\end{column}
\end{columns}
\pause
\bigskip
\begin{columns}
\begin{column}{0.45\textwidth}
\begin{block}{3. Harness convenient fictions}
For every set, there is a forcing extension where it is
\hil{countable}.
\end{block}
\end{column}
\pause
\begin{column}{0.45\textwidth}
\begin{block}{4. Constructivize classical theories}
A preorder~$X$ is well iff the \hil{generic sequence} $\NN \to X$ is
good.
\end{block}
\end{column}
\end{columns}
\pause
\bigskip
\begin{columns}
\begin{column}{0.45\textwidth}
\begin{block}{5. Study parametric mathematics}
Eigenvectors depend continuously on the parameter
iff, in a suitable forcing extension, they merely exist.
\end{block}
\end{column}
\pause
\begin{column}{0.45\textwidth}
\begin{block}{6. Develop synthetic accounts}
\emph{As in the lectures by Matthias Hutzler.}
\end{block}
\end{column}
\end{columns}
\end{frame}
% \begin{document}
{\usebackgroundtemplate{\begin{minipage}{\paperwidth}\vspace*{5.95cm}\includegraphics[width=\paperwidth]{fr1}\end{minipage}}
\begin{frame}{Maximal ideals}
\jnote{1-7}{
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{11}{
The iterative construction in the countable case without decidability
assumptions is due to Krivine; it was 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{13-}{
In a suitable forcing extension, 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.
}
\only<1-7>{\textbf{Thm.}
Let~$M$ be a surjective matrix with more rows than columns over a
ring~$A$. Then~$1 = 0$ in~$A$.
\visible<2->{\emph{Proof.} \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<4-7>{\bigskip\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
\medskip
\raggedright
Let~$A$ be a ring. \emph{Does there exist a maximal ideal~$\mmm \subseteq A$?}
\pause
\begin{enumerate}
\item \good{Yes}, if \bad{Zorn's lemma} is available.
\bigskip
\pause
\item \good{Yes}, if~$A$ is countable and membership of finitely generated ideals is decidable:
{\footnotesize
Let~$A = \{ x_0, x_1, \ldots \}$. Then set:
\begin{align*}
\mmm_0 &\defeq \{ 0 \}, &
\mmm_{n+1} &\defeq \begin{cases}
\mmm_n + (x_n), & \text{if $1 \not\in \mmm_n + (x_n)$}, {\qquad\quad\quad\!\!}\\
\mmm_n, & \text{else.}
\end{cases}
\end{align*}}
\pause
\item \good{Yes}, if~$A$ is countable (irrespective of membership decidability):
{\footnotesize\begin{align*}
\mmm_0 &\defeq \{ 0 \}, &
\mmm_{n+1} &\defeq \mmm_n + (\underbrace{\{ x \in A \,|\, x = x_n \wedge
1 \not\in \mmm_n + (x_n) \}}_{\text{a certain subsingleton set}})
\end{align*}}
\vspace*{3.5em}
\visible<11>{{
\centering
\raisebox{0pt}[0pt][0pt]{
\hspace*{2.5em}
\scalebox{0.9}{\begin{tikzpicture}
\node (inner) at (17.3mm,-20mm) {\textit{``a bad joke''}};
\path (0,0) pic{laurel-wreath};
\end{tikzpicture}}
\hspace*{-3em}
\scalebox{0.9}{\begin{tikzpicture}
\node (inner) at (17.5mm,-18mm) {\vbox{\small\centering\textit{``non- \\informative''}}};
\path (0,0) pic{laurel-wreath};
\end{tikzpicture}}
}
\par
}}
\pause
\pause
\vspace*{-3.5em}
\item In the general case: \bad{No}\pause, but \good{yes} in a \emph{suitable forcing extension}\pause, and \\
\emph{bounded first-order consequences} of its existence there \good{do hold} in
the base universe.
\end{enumerate}
\end{frame}
\begin{frame}{Maximal ideals}
\jnote{1}{
Unwinding all the definitions from constructive forcing 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.
}
\textbf{Thm.}
Let~$M$ be a surjective matrix with more rows than columns over a
ring~$A$. Then~$1 = 0$ in~$A$.
\emph{Proof.} (special case) Write~$M = \left(\begin{smallmatrix}x\\y\end{smallmatrix}\right)$. By surjectivity,
we have~$u, v \in A$ 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
$. \qed
\end{frame}}
% \begin{document}
{\usebackgroundtemplate{\begin{minipage}{\paperwidth}\includegraphics[height=\paperheight]{sea-of-clouds-2}\end{minipage}}
\begin{frame}{Infinite data}
\jnote{5-}{
The presented proof rests on the law of excluded middle and hence cannot
immediately be interpreted as a program for finding suitable indices~$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 preorders 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
\textbf{Thm.} Every sequence~$\alpha : \NN \to \NN$ is \hil{good} in that
there exist~$i < j$ with~$\alpha(i) \leq \alpha(j)$.
\pause
\emph{Proof.} \emph{(offensive?)} By~\badbox{\textsc{lem}}, there is a
minimum~$\alpha(i)$.
Set~$j \defeq i + 1$. \qed\par
\pause
\medskip
\textbf{Def.} A preorder~$X$ is \hil{well\only<9->{$^\star$}} iff every sequence~$\NN \to X$ is good.
\textbf{Examples.} $(\NN,{\leq}),\ \
\color{white}\only<7->{\color{red!90}}\astikznode{onlyclass}{$\underbrace{\color{black}X \times Y,\ \ X^*,\ \ \mathrm{Tree}(X)}_{\text{\visible<7->{\bad{only classically}}}}$}$.
\pause
\pause
\medskip
\begin{tikzpicture}[remember picture,overlay]
\node[thick, fill=black, rectangle, inner sep=0.3em, right=2em of onlyclass] (moral) {
\begin{minipage}{6cm}
\begin{columns}
\begin{column}{0.15\textwidth}
\hspace*{1.0em}\color{white}\dbend
\end{column}
\begin{column}{0.95\textwidth}
\color{white}\footnotesize
\it Don't quantify over points of spaces which might not have enough.
\end{column}
\end{columns}