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\begin{document}
\title{Research statement}
\author{David Pierce}
\date{\today}
\maketitle

\tableofcontents

%\setcounter{section}{-1}
\addsec{Overview}

My specific training, and my publications so far, are in the part of
mathematical logic called \emph{model theory,} and especially in
the model theory of fields and differential fields.  My interests and work also
spread into foundations and history; these subjects are a way to understand mathematics as it is done \emph{today,} and to add to it. 

\section{A primer on model theory}\label{sect:primer}

When model theorists address a general audience, they often feel the
need to explain their subject from the very beginning.  In that
spirit, I compose the present section of this document. 

Model theory is a specific area of mathematics, raising and answering
its own questions.  However, like category theory, it is also a point
of view, a way to think about all of mathematics.  Sometimes this
point of view leads to new insights, proofs, and theorems. 

To my knowledge, the first two textbooks (as opposed to treatises) of model theory are Bell and Slomson's \emph{Models and Ultraproducts} of 1969 \cite{MR0269486} and Chang and Keisler's \emph{Model Theory} of 1973 \cite{Chang--Keisler}.  According to Bell and Slomson,
\begin{quote}
Model theory\dots can be described briefly as the study of the relationship between formal languages and abstract structures.
\end{quote}
Chang and Keisler have a similar description:
\begin{quote}
Model theory is the branch of mathematical logic which deals with the relation between a formal language and its interpretations, or models.
\end{quote}
Chang and Keisler also suggest the equation  
\begin{quote}\centering
universal algebra + logic = model theory;
\end{quote}
but this is perhaps restrictive, simply because universal algebra
is about sets with distinguished \emph{operations,} and not relations
in general; but model theory does study sets with arbitrary
relations. 

The emphasis on a formal language may be relaxed.
According to Wilfrid Hodges in his encyclopedic volume \emph{Model
  Theory} of 1993 \cite[p.~ix]{MR94e:03002},  
\begin{quote}
Model theory is the study of the construction and classification of
structures within specified classes of structures.  A `specified class of structures' is any class of structures that a mathematician might choose to name.  For example, it might be the class of abelian groups, or of Banach algebras, or sets with groups which act on them primitively.  Thirty or forty years ago the founding fathers of model theory were particularly interested in classes specified by some set of axioms in first-order predicate logic---this would include the abelian groups but not the Banach algebras or the primitive groups.  Today we have more catholic tastes, though many of our techniques work best on the first-order axiomatisable classes.
\end{quote}
This definition may misleadingly suppress the \emph{logical}
aspect of model theory.  The classification of the finite simple groups is not really a model-theoretic project, although it has been an inspiration for the project of classifying the infinite simple groups of finite Morley rank,\label{MR} and Morley rank is a logical notion (see page \pageref{MR2}).

I propose the following definition: \textbf{model theory} is the study of structures \emph{qu\^a} models of theories.  For \emph{qu\^a,} one may read \emph{in the capacity of} or \emph{as.}
The three
terms \emph{structure, model,} and \emph{theory} must now be
explained.   

Structures appear in most mathematics.  Groups, rings, ordered fields,
partially ordered sets: all are examples of structures.
A \textbf{structure} then is a set with some extra `structure'.
For us, this extra `structure' consists of:
\begin{compactitem}
\item 
some distinguished relations and
operations on the set,
\item
some specific elements of the set.   
\end{compactitem}
None of the extra structure is actually \emph{required} to be present; a bare set is an example of a structure.

Note that Hodges's example of Banach spaces does not exactly fit this definition of structure.  Model-theorists have indeed developed a Banach-space logic for studying Banach spaces; but this subject is beyond the bounds of the present exposition.

The
ordered field of real numbers is the structure consisting of: 
\begin{compactitem}
\item
the set $\R$ of real numbers;
\item
 the operations of addition, additive inversion, and multiplication on $\R$;
 \item
 the relation `less than' on $\R$;
 \item
the additive identity and the multiplicative identity in $\R$.
\end{compactitem}
In this example, there are standard symbols for the distinguished
operations, relations, and elements, and the structure can be denoted
by 
\begin{equation*}
(\R,+,-,{}\cdot{},<,0,1).
\end{equation*}
We can then speak of \emph{reducts} of this structure, such as
\begin{equation*}
(\R,+,-,{}\cdot{},<).
\end{equation*}
This reduct is not much different from the original structure, since
$0$ and $1$ are \emph{definable} in the reduct, by the formulas 
\begin{align*}
x+x&=x,&x\cdot x&=x,
\end{align*}
respectively;
for example, the solution set of $x+x=x$ in $\R$ is just $\{0\}$.
Also, the relation $<$, considered as the set $\{(x,y)\colon x<y\}$,
is definable in $(\R,+,-,{}\cdot{})$ by the formula 
\begin{equation*}
x\neq y\land\Exists zx+z^2=y
\end{equation*}
(where $z^2$ stands for $z\cdot z$).
Finally, the operation $x\mapsto-x$, considered as the relation
$\{(x,y)\colon y=-x\}$, is definable in $(\R,+,0)$ by 
\begin{equation*}
x+y=0;
\end{equation*}
hence it is definable in $(\R,+)$ by $(x+y)+(x+y)=x+y$.  

In short, everything that can be defined in $(\R,+,-,{}\cdot{},<,0,1)$ can
already be defined in $(\R,+,{}\cdot{})$.  This can be compared to the
situation in propositional logic, where, because of De Morgan's law  
\begin{equation*}
\lnot(P\lor Q)\iff\lnot P\land\lnot Q, 
\end{equation*}
we have $P\lor Q\iff\lnot(\lnot P\land\lnot Q)$,
so that everything that can be said with `or' can be said also with
`and' and `not'. 

However, not every proposition that can be expressed with `and' and
`not' together can be expressed in terms of `and' alone.  Similarly,
multiplication cannot be defined in $(\R,+)$.  To prove this, all we need to
know is that definable operations and relations of a structure are
invariant under automorphisms of the structure. 
Since $(\R,+)$ is a torsion-free divisible abelian group, it can be
understood as a vector-space over $\Q$.  There is an automorphism of
this space that takes the one-dimensional subspace $\Q$ to the subspace $\{x\cdot\surd
2\colon x\in\Q\}$, and such an automorphism does not fix the
subset 
$\{(x,y,z)\colon x\cdot y=z\}$ of $\R^3$.  Therefore multiplication is not
definable in $(\R,+)$. 

The structure $(\R,+,-,{}\cdot{},<,0,1)$ has the \textbf{signature}
$\{+,-,{}\cdot{},<,0,1\}$.  In the signature, the symbols have no
meaning; in the structure, they stand for particular
operations, relations, or elements.  Two structures can have the same
signature.  For example, all abelian groups can be understood to have
the same signature $\{+,-,0\}$; or the signature may be taken to be
simply $\{+\}$, since the operation denoted by $-$ and the element
denoted by $0$ are definable in terms of $+$ by the same formulas in
any abelian group. 

When we are trying to make things precise, and the underlying set of a
structure is $A$, then the structure itself might be denoted by $\str
A$, if we really need to distinguish between the two. (Often we need
not distinguish.)
If $S$ is a symbol in the signature of $\str A$, then the operation or
relation or individual that it denotes in $\str A$ can be denoted also
by $S^{\str A}$.  For example, a function $f$
from a group $G$ to a group $H$ is defined to be a homomorphism if 
\begin{equation*}
f(x\cdot^Gy)=f(x)\cdot^Hf(y);
\end{equation*}
here it is emphasized that the operations of multiplication on the left and right sides are actually distinct.  However, often we do not feel the need to express this distinction notationally.

In the general situation, if $S^{\str A}$ is a relation, then it is considered to be
one of the definable relations of $\str A$.  If $S^{\str A}$ is an $n$-ary
operation for some $n$ in $\N$, then the relation $\{(\vec x,S^{\str
  A}(\vec x))\colon\vec x\in A^n\}$ is a definable $(n+1)$-ary
relation.  If $S^{\str A}$ is an element of $A$, then $\{S^{\str A}\}$ is a
definable singulary relation.  The collection of all definable
relations of $\str A$ is built up by standard set-theoretic operations: binary
intersection, binary union, complementation, and also the coordinate
projections
\begin{equation*}
X\mapsto\{(x_0,\dots,\hat x_i,\dots,x_{n-1})\colon(x_0,\dots,x_{n-1})\in X\}.
\end{equation*}
These are the
operations expressed by the logical symbols
$\land$, $\lor$, $\lnot$, and $\exists x_i$.  Also, the diagonal relation
$\{(x,x)\colon x\in A\}$ is considered definable, and if some relation is definable, then
so is its inverse image under a coordinate projection.  Finally, $\{b\}$ is definable whenever $b\in A$.

Thus a \textbf{definable relation} of a structure is the solution set
of a formula of \emph{first-order logic} in the signature of the
structure, possibly with \emph{parameters} from the structure.  The formula then \textbf{defines} its solution set.  In
high school algebra, when one studies graphs of equations like
$y=mx+k$ or $x^2+y^2=r^2$, one is studying sets definable in
$(\R,+,{}\cdot{})$.  But the formula defining a set may be more than an
equation; it may involve Boolean connectives and quantifiers, as in
the earlier examples.  In $(\R,+,{}\cdot{})$, the interval $[-1,1]$ is
defined by the formula $\Exists yx^2+y^2=1$.   

\emph{First-order} logic is logic in which variables stand for
individuals, not operations or sets.  The Completeness Axiom for
$(\R,<)$ is formulated in \emph{second-order} logic, since the axiom is
that every nonempty \emph{subset} of $\R$ with an upper bound has a
least upper bound.
Similarly, the Induction Axiom\label{IA} for the structure $(\N,1,x\mapsto x+1)$
of the natural numbers is not  first-order, but second-order.

Model theory is generally concerned with first-order logic.
This restriction may seem mathematically unnatural; but it is no more unnatural than restricting one's attention to, say, groups or modules.

Formulas of first-order logic have finite length.  In
particular, the intersection of an infinite collection of definable
sets of $A^n$ is not necessarily definable.  Such intersections are
still of interest though; they are said to be \emph{type-definable}. 

The foregoing argument for why multiplication is not definable in $(\R,+)$
does not actually require that formulas be first-order.  Model theory
normally uses first-order logic because it makes available the
\emph{Compactness Theorem.}  To talk about this, we should first work
out the remaining two undefined terms in the definition of model
theory. 

A formula with $n$ free variables in the signature of $\str A$ defines
a subset of $A^n$.  A formula with \emph{no} free variables is a
\textbf{sentence;} it defines a subset of $A^0$.  It is convenient here to consider the non-negative integers as the set-theorist's natural numbers, as defined by von Neumann~\cite{von-Neumann}: $0=\emptyset$, and $n+1=n\cup\{n\}$.  Then $A^n$ is the set of functions from $\{0,\dots,n-1\}$ to $A$, and in particular $A^0$ is the set of functions with empty domain.  There is only one such function, namely $\emptyset$ or $0$; so $A^0=\{0\}=1$, and its subsets are $0$ and $1$.
In the present context, we can consider these subsets as \textbf{falsehood} and \textbf{truth,} respectively.  To say that a sentence is
\textbf{true} in a structure is just to say that the nullary relation
defined by the sentence is truth. 

Suppose $\Delta$ is a set of first-order sentences in some signature.  A \textbf{model} of $\Delta$ is a structure in which all sentences in $\Delta$ are true.  A sentence $\sigma$ is a \textbf{logical consequence} of $\Delta$, and $\Delta$ \textbf{entails} $\sigma$, if $\sigma$ is true in every model of $\Delta$.  A \textbf{theory} is a set of sentences that contains all of its own logical consequences.  The set of logical consequences of $\Delta$ is then a theory: it is said to be the theory \textbf{axiomatized} by $\Delta$.  Often the distinction between a theory and the sentences that axiomatize it is blurred.

The \textbf{Compactness Theorem} is
that, if every finite subset of some set of sentences has a model, then the whole set has a model.   

For example, suppose $T$ is the theory of finite fields, that is, $T$ consists of all sentences in the signature $\{+,-,{}\cdot{},0,1\}$ that are true in every finite field.  If $n\in\N$, let $\sigma_n$ be the sentence
\begin{equation*}
\Exists{x_0}\cdots\Exists{x_{n-1}}\bigwedge_{i<j<n}x_i\neq x_j,
\end{equation*}
which says that every model has at least $n$ elements.  
Every finite subset of $T\cup\{\sigma_n\colon n\in\N\}$ has a model, namely a finite field that is sufficiently large.
By the Compactness Theorem, the whole set has models, and these are infinite: they are the infinite models of the theory of finite fields. (They are called \emph{pseudofinite fields} and are characterized by Ax \cite{MR0229613}.)

In algebraic geometry, a \textbf{constructible set}\label{constructible} is a set defined
by a \emph{quantifier free} formula (with parameters) in an
algebraically closed field.  A theorem of
Chevalley~\cite[p.~94]{MR0463157} is that a coordinate projection of an
constructible set is a constructible set.  This is equivalent to Tarski's
theorem that the theory of algebraically closed fields admits
\emph{elimination of quantifiers.} 

As suggested in the example of pseudofinite fields, the \textbf{theory
  of} a class $\mathscr K$ of structures in some signature is the set
of sentences in that signature that are true in every structure in
$\mathscr K$.  If $\mathscr K$ has a unique element $\str A$, then the
theory of $\mathscr K$ is the \textbf{theory of} $\str A$.  The theory of $\str A$ is a
\textbf{complete theory:} that is, for every sentence $\sigma$ of the
signature of $\str A$, the theory of $\str A$ contains either $\sigma$ or its
negation $\lnot\sigma$.  For example, the structure $(\N,+,{}\cdot{})$ has
a complete theory.  \textbf{G\"odel's Incompleteness
Theorem}~\cite{Goedel-incompl}\label{Goedel} is that this
theory is not \emph{recursively} axiomatizable: there is no rule for
writing down a set of sentences that axiomatize the theory. 

I shall discuss the proof of G\"odel's theorem in \S\S\ \ref{sect:rec}
and~\ref{sect:sets}.
Meanwhile, there are many common examples of
recursively axiomatizable 
complete theories.  Abraham Robinson's \emph{Complete Theories}~\cite{MR0472504} is all about them.  One of them is $\acf_p$, the
theory of
algebraically closed fields of a given characteristic $p$.  This
completeness can be understood as the logical basis for the
\textbf{Lefshetz Principle,} whereby certain statements proved in $\C$
by analytic methods automatically hold in every algebraically closed
field of characteristic $0$.  Such statements certainly \emph{do} hold
generally, if they are first-order statements.\footnote{Some loosening
  of the first-order requirement is possible; see
  Hodges~\cite[p.~700]{MR94e:03002} for discussion and references.} 

In a lecture at the Mathematical Sciences Research Institute in
Berkeley in 1998, Lou van den Dries~\cite{MR1773701} (citing Ehud
Hrushovski as the source) described model
theory as the `geography of tame mathematics'.  For present purposes,
we can understand a structure to be
\textbf{tame} if its complete theory is recursively axiomatizable.
Model theory provides tools for identifying such structures, which may arise
naturally in the study of \textbf{wild} (non-tame) structures.  For
example, 
\begin{compactitem}
\item
$(\N,+,{}\cdot{})$ is wild, by G\"odel, as noted.
\item
$\N$ is definable in $(\Z,+,{}\cdot{})$, by Lagrange's theorem that every
  positive integer is the sum of four squares; therefore
  $(\Z,+,{}\cdot{})$ is wild.
\item
Therefore $(\Q,+,{}\cdot{})$ is wild, by the theorem of Julia
Robinson~\cite{MR0031446} that $\Z$ is definable in $(\Q,+,{}\cdot{})$. 
\item
However, the Dedekind completion $(\R,+,{}\cdot{})$ and the $p$-adic
completions $(\Q_p,+,{}\cdot{})$ of $(\Q,+,{}\cdot{})$ are tame, by work of
Tarski, Ax, Kochen, and Ershov. 
\end{compactitem}

Making use of the Compactness Theorem and its proof, one shows that a
theory with one infinite model has a model of every infinite
cardinality, at least if that cardinality is not less than the
cardinality of the signature of the theory.  Since $(\R,+,{}\cdot{},<)$ is
the only Dedekind-complete ordered field (up to isomorphism) of any
cardinality, it follows that the Completeness Axiom cannot be recast
in a first-order way.  

The theory of $(\C,+,{}\cdot{})$, which is the recursively axiomatizable
theory $\acf_0$, has just one model
(up to isomorphism) in every uncountable cardinality; in a word, the
theory is \textbf{categorical} in every uncountable cardinality.
Again, as noted, the complete first-order theory of $(\R,+,{}\cdot{},<)$
is also recursively axiomatizable: it is the theory $\rcf$ of
\emph{real-closed ordered fields.}  However, this theory is as far from
categorical as possible: it has
$2^{\kappa}$ nonisomorphic models of cardinality $\kappa$, for every
infinite cardinal $\kappa$.  

A model of $\acf_p$ is determined up to isomorphism by its transcendence
degree.  The study of how theories can have such a property was pursued
by Michael Morley, who showed \cite{MR0175782} that a theory (in a countable signature) that is
categorical in one uncountable cardinality must, like $\acf_p$, be
categorical in every uncountable cardinality.  In the argument, he
used the notion now called \emph{Morley rank}\label{MR2} (see page
\pageref{MR}).  Saharon Shelah
continued this work by identifying those complete first-order theories
whose uncountable models could possibly be \emph{classified:} classified by a
cardinal invariant, as in the case of algebraically closed fields, or
more generally by many cardinals, arranged in a \emph{tree.} The
specific possibilities for this classification were worked out by Hart
and Laskowski \cite{MR1481440,MR1792295}.  (Laskowski was my teacher.)

For an example of how such trees arise, let $T$ be the theory of an
equivalence relation $E$, all of whose classes are infinite.
Then $T$ turns out to be a complete theory, and each model is determined by:
\begin{compactenum}
\item 
the number of $E$-classes;
\item
the size of each $E$-class.
\end{compactenum}
This information can be arranged in a tree.  To be precise, suppose
$\str M$ is a model of $T$ of cardinality $\kappa$.  Then there is an
injective function $f$ or $x\mapsto(x_0,x_1)$ from $M$ into
$\kappa\times\kappa$ such that
\begin{equation*}
x_0=y_0\iff x\mathrel Ey.
\end{equation*}
That is, $x_0$ distinguishes an $E$-class, and $x_1$ distinguishes an element within a given $E$-class.
Then we have a tree whose nodes are all of the form $(\ )$, $(x_0)$, or
$(x_0,x_1)$.  The number of isomorphism-classes of such trees is $2^{\kappa}$.

We have now seen two aspects of model-theoretic practice:
\begin{compactenum}
\item
The study of structures of `ordinary' mathematics, in order to understand their theories.
\item
The creation of new structures whose theories have properties of interest.
\end{compactenum}
Understanding a theory involves understanding the class of all (isomor\-phism classes of) models of the theory; it also involves understanding the definable relations of particular models.

\section{Function fields}\label{sect:ff}

In algebraic geometry, as noted on page \pageref{constructible}, a constructible set is defined by a
quantifier-free formula in the signature of an algebraically closed
field with parameters.  Then an \textbf{algebraic set} is defined by a
\emph{positive} quantifier-free formula, that is, a formula built up
from equations by conjunction and disjunction---by `and' and
`or'---, but not negation.  If this formula cannot be written in a
nontrivial way as a disjunction, then the algebraic set that it
defines is a \textbf{variety} (strictly, an \emph{affine} variety). 
One project of algebraic geometry is to classify varieties up to
\emph{birational equivalence.}  Birational equivalence corresponds to
isomorphism of the \emph{function fields} of the varieties.   

Two structures with the same first-order theory are called \textbf{elementarily equivalent.}
Then isomorphic structures, such as function fields, are elementarily equivalent.
We have already observed that the converse fails, simply because a
theory with infinite models has models of different cardinalities, and
so these models cannot be isomorphic.  Also, two elementarily equivalent structures of
the same cardinality may fail to be isomorphic: an example is algebraically closed fields of the same characteristic, but distinct finite transcendence degrees. 

We may ask whether elementarily equivalent function fields over the
same algebraically closed field are elementarily equivalent.  Then we are in the first aspect of model-theoretic practice mentioned at the end of the last section: studying known structures through consideration of their theories.
Partial
answers to our question are found in work of
Jean-Louis Duret \cite{MR865921,MR1187449} and then in work done by myself
\cite{MR2001a:03080} and independently by Duret's student Xavier Vidaux
\cite{MR1905159}.  

The first thing to note is that function fields of
different dimension are
elementarily inequivalent: this can be established by means of the
Tsen--Lang Theorem.  
Let us then restrict our attention to dimension one.  It turns out that
elementarily equivalent function fields of
curves are isomorphic, \emph{unless} both of the curves are
elliptic curves with complex multiplication. 

Suppose $E_0$ and $E_1$ are elliptic curves with complex multiplication over an algebraically closed field $K$.  This means each endomorphism-ring $\End{E_i}$ is strictly larger than $\Z$.
The condition that the rings $\End{E_i}$ be isomorphic to one another is strictly weaker than the condition that the function fields $K(E_i)$ be isomorphic.
I obtained the result that, in case the characteristic of $K$ is $0$,
the rings $\End{E_i}$ are
isomorphic if and only if the fields $K(E_i)$ agree on all sentences
of the form  
\begin{equation*}
\Forall{x_0}\cdots\Forall{x_{n-1}}\Exists y\phi(x_0,\dots,x_{n-1},y),
\end{equation*}
where $\phi$ is quantifier-free.  Vidaux and I have tried to
determine whether this result can be generalized, even to arbitrary
$\forall\exists$ sentences, that is, sentences of the form 
\begin{equation*}
\Forall{y_0}\cdots\Forall{y_{m-1}}\Exists{x_0}\cdots\Exists{x_{n-1}}\psi(x_0,\dots,x_{n-1},y_0,\dots,y_{m-1}).
\end{equation*}
where $\psi$ is quantifier-free.  If this problem can be solved,
then the following should also be studied:
\begin{compactenum}[1)]
\item 
elliptic curves over finite fields,
\item
arbitrary abelian varieties,
\item
arbitrary varieties.
\end{compactenum}
It may be that the truth lies deep.  A long-standing question of Tarski was whether any two non-abelian free groups of finite rank are isomorphic.  The question has supposedly been answered in the affirmative, by Kharlampovich and Myasnikov, and independently by Sela; but the work is apparently very difficult, and it is not clear (to me at least) whether others have understood it thoroughly.  In fact the Istanbul Model Theory Seminar has spent some time studying this work.  It may possibly illuminate the corresponding question for function fields of elliptic curves, namely, the question of whether two such non-isomorphic fields can be elementarily equivalent.

\section{Differential fields}\label{sect:df}

The field of complex numbers is algebraically closed, and every field
has an algebraic closure.  The field of real numbers is real-closed,
and every ordered field has a real closure.  In model-theoretic terms,
what this means is that, of the \emph{theory} of fields of characteristic $0$ and the \emph{theory} of ordered
fields, each has a \emph{model-companion.}\label{mcomp}  A theory $T^*$
is a \textbf{model companion} of a theory $T$ (in the same signature)
if: 
\begin{compactitem}
\item
$T\included T^*$, and every model of $T$ \emph{embeds} in a model of $T^*$---an \textbf{embedding} is a monomorphism, that is, an injective function that preserves structure;
\item
$T^*$ is \textbf{model complete:}\label{mc} in other words all models of
  $T^*$ are \textbf{existentially closed:} that is, if $\str A$ and $\str B$
  are models of $T^*$, and $\str A\included\str B$ (that is, $A\included B$, and the inclusion of $A$ in $B$ is an embedding of $\str A$ in $\str B$), then every
  quantifier-free formula with parameters from $\str A$ that has a solution in $\str B$ has a solution in $\str A$. 
\end{compactitem}
The question then arises of which other theories have model
companions.  Tame complete theories are often model complete, as for example each theory $\acf_p$ is.  However, the theory of fields of any characteristic also has a model-companion, namely the incomplete theory $\acf$ of algebraically closed fields of any characteristic.

An example developed by Abraham Robinson~\cite{MR0125016} is the
theory $\df$ of \textbf{differential fields,} namely fields with an additional
operation $\delta$ that is an additive endomorphism and obeys the Leibniz
rule 
\begin{equation*}
\delta(x\cdot y)=x\cdot \delta y+y\cdot \delta x.
\end{equation*}
Again a required characteristic on models can be indicated by a subscript.
Robinson showed that $\df_0$ has a model companion; his
student Carol Wood~\cite{MR48:8227} showed the same for $\df_p$ when $p$ is positive. 

But one does not want to know just that there is a model companion;
one wants to understand its models.  Robinson's axioms for $\df_0{}^*$, and then Wood's for $\df_p{}^*$,
were complicated.  In an algebraically closed field, every consistent system of polynomial equations and inequations in any number of variables has a solution; a \textbf{`crude'} axiomatization of $\acf$ says this; but in fact the axioms need only say that every non-constant polynomial in \emph{one variable} has a root.
Lenore Blum~\cite{MR0491149} observed this and found a similarly simple
axiomatization for $\df_0{}^*$; Wood~\cite{MR50:9577} adapted this to $\df_p{}^*$.  An alternative, `geometric' style of
simplification 
was published by Anand Pillay and me~\cite{MR99g:12006} for
$\df_0{}^*$; Piotr Kowalski \cite{MR2119125} carried out this work for $\df_p{}^*$ (he also placed it in a more general setting: that of `derivations of the Frobenius map'). 
I found a slightly simpler form of `geometric' axiomatization and used it in~\cite{MR2114160} to give a model-companion of $\df$ (with no specified characteristic): the models of $\df^*$ are those differential fields $(K,\delta)$ such that
\begin{compactenum}
\item
$K$ is separably closed,
\item
$(K,\delta)$ is \textbf{differentially perfect:} the kernel of $\delta$ is $K^p$, if $K$ has characteristic $p$, which is positive;
\item
for every affine variety $V$ over $K$, if there are $K$-rational maps $\phi$ and $\psi$ from $V$ into affine $n$-space, and $\phi$ is dominant and separable, then $V$ has a $K$-rational point $P$ where $\phi$ and $\psi$ are regular and $\delta(\phi(P))=\psi(P)$.
\end{compactenum}

A model of $\df^*$ is a `universal domain' where all consistent systems of ordinary differential equations and inequations have solutions (although the solutions are algebraic or `formal'; they are not given as functions).  One may ask about partial differential equations: 
does the theory $\df^m$ of fields with $m$ commuting
derivations for some positive integer $m$ have a model-companion?  Wood's student Tracey
McGrail~\cite{MR2001h:03066} showed that it does
in characteristic $0$; but its axioms were `crude' in the sense above.  Similar work was carried out independently by Yaffe~\cite{MR1807840}.

In fact Yaffe's work was in an apparently more general setting: the $m$ derivations need not commute, but the Lie bracket of any two of them is a fixed linear combination of all of them.  I observed in~\cite{MR2000487} that the generalization was only apparent, in the sense that axioms for the model companion of a theory of Yaffe's `Lie differential fields' (of characteristic $0$) could be easily derived from axioms for $(\df_0^m)^*$.  Singer~\cite{MR2286106} made the observation more explicit.

The main point of my paper~\cite{MR2000487} was to give an alternative, `geometric'
axiomatization of $(\df_0^m)^*$; but this supposed axiomatization turned out to be wrong.
Correcting the problem required a whole new approach, presented in~\cite{2007arXiv0708.2769P}.  The
main idea is that, for every insoluble system of differential equations of a
given order, there is a bound on the number of times the equations
must be differentiated to establish the insolubility.  I carried out the work in arbitrary characteristic; in particular, I showed the existence of a model-companion of $\df_p^m$ even when $p>0$: this result is apparently new.

On a field, a derivation is not the only interesting singulary operation: one can consider also an endomorphism $\sigma$, or equivalently the \textbf{difference-operator} $x\mapsto x^{\sigma}-x$.  Derivations and difference-operators are such that their behavior at sums and products is determined by polynomials, and they are $0$ at $0$ and $1$.  Buium~\cite{MR98k:12004} showed that they were the \emph{only} examples of such operations.

Before the paper~\cite{MR99g:12006} of Pillay and me, Hrushovski had shown the the theory of \textbf{difference fields}---fields with an endomorphism---has a model companion (see \cite{MR99c:03046} and \cite{MR2000f:03109}).  Then it is easy to show that the theory of fields with a singulary operation that is either a derivation or a difference-operator has a model-companion; but I did this `geometrically' in \cite{MR2114160} without distinguishing the two cases in the axioms.  More interesting results in that paper occur  when the theory of fields with \emph{both} a derivation and a difference-operator is considered.  In characteristic $0$, there is a model-companion; but there is not, in positive characteristic.  Briefly, the problem is that, in a positive characteristic $p$, if $\delta(x^{\sigma^n})=0$ for \emph{all} nonnegative integers $n$, then $x$ should have a $p$th root in a model of the model-companion; but this condition cannot be made first-order, so there is no model-companion.

\section{Vector spaces}

Structures as defined in the `Primer on model theory'
(\S\ref{sect:primer} above) are more precisely \textbf{one-sorted}
structures, because 
each of them is based on only one set, and we may call that set a
\textbf{sort.}  However, sometimes one wants 
to work with more than one sort.  For example, a vector space has
a sort of scalars and a sort of vectors, so it is a two-sorted structure.

In his \emph{Geometry}~\cite{Descartes-Geometry} of 1637, Ren\'e Descartes
observed that all arithmetic operations on numbers could be
mimicked by manipulations of line segments in a Euclidean plane.  In
fact it is enough for the plane to have the structure of a vector space.
Before Descartes, it had perhaps been felt that the only rigorous form of
mathematics was geometric; presumably it was because of this feeling
that Euclid's \emph{Elements} 
\cite{MR1932864} expressed, in geometric language, theorems of what we
today would call algebra or number theory.  Descartes observed that,
because algebra (or more precisely field theory) \emph{can} be put in
geometric terms, there is no need actually to do so.
In short, algebra can be done with the rigor of geometry.

We may observe conversely that, if field theory can be expressed
geometrically, there is no real need for fields as distinct structures.
More precisely, in a
vector space of 
dimension at least $2$, the sort of scalars is not needed, as long as
the sort of vectors has the relation of parallelism.  

I worked this out in~\cite{MR2505433}.
The result is that there is a certain \emph{equivalence of categories.}  In
one of the categories, the objects are vector spaces of dimension $2$ or more,
considered as quadruples $(V,K,*,\parallel)$,\label{quad} where 
\begin{compactenum}[1)]
\item
$V$ is an abelian group in the signature $\{+,-,\bm0\}$;
\item
$K$ is a field in the signature $\{+,-,{}\cdot{},0,1\}$;
\item
$*$ is the action of $K$ on $V$, that is, a certain function from $K\times V$ to $V$;
\item
$\parallel$ is the binary relation of parallelism on $V$, defined by the formula
\begin{equation*}
\Exists x\Exists y\left(x*\bm u+y*\bm v=\bm0\land(x\neq0\land y\neq0)\right).
\end{equation*}
\end{compactenum}
The arrows in this category are just embeddings (as defined in the last section).  In the other category, the objects are the reducts
$(V,\parallel)$ of the objects $(V,K,*,\parallel)$ in the first
category; the arrows are still embeddings.  Then these two
categories are equivalent.  In fact, if we are given an object $(V,\parallel)$
from the second category, then we can define a set $K$ of certain
equivalence-classes of pairs of parallel vectors, and we can define an
addition and a multiplication on $K$, and an action of $K$ on $V$, so
that $(V,K,*,\parallel)$ is an object of the first category.  In
particular then, the objects of the second category are just the
models of a certain theory. 

There is some subtlety in the choice of arrows for these categories.
For example, even though parallelism is definable in a vector space of
dimension at least two, the categories of vector spaces with and
without a symbol for parallelism are \emph{not} equivalent, if the
arrows are just embeddings.  For example, the identity on the ring
$\Ham$ of quaternions induces an
embedding of the vector space $(\Ham,\R,*)$ in $(\Ham,\C,*)$; but the
same function is
\emph{not} an embedding of $(\Ham,\R,*,\parallel)$ in
$(\Ham,\C,*,\parallel)$.  Indeed, $1$ and $\mi$ (as vectors in $\Ham$)
are not parallel with respect to the scalar field $\R$, but they are
parallel with respect to $\C$. 

The two categories of vector spaces with and without parallelism become equivalent if the
arrows are \emph{elementary} embeddings.  An embedding of structures
is just a function that preserves the truth of \emph{quantifier-free}
sentences with parameters; an \textbf{elementary embedding} preserves
the truth of \emph{all} sentences.  In particular, the inclusion of
$(\Ham,\R,*)$ in $(\Ham,\C,*)$ is not an elementary embedding. 

Suppose a theory $T$ is such that every embedding of its models is elementary.  Then by the definition on page \pageref{mc}, $T$ is model complete.
The converse of this observation is a theorem of Abraham Robinson~\cite[2.3.1]{MR0472504}.

Supposing a theory $T$ has a model companion $T^*$ (as defined on page \pageref{mcomp}), we have that all models of $T^*$ are existentially
closed models of $T^*$; therefore they are also existentially closed models of $T$.  More is true.  First of all,
every existentially closed model of $T$ will be a model of $T^*$, by
work of Eklof and Sabbagh~\cite[Prop.~7.10]{MR0277372}.  Now suppose
$T$ is \textbf{inductive,} that is, it meets either of the following
two conditions, which are equivalent by a theorem of
Chang~\cite{MR0103812} and of \L o\'s and Suszko~\cite{MR0089813}: 
\begin{compactenum}
\item
The union of an ascending chain of models of $T$ is a model of $T$.
\item
$T$ has $\forall\exists$ axioms (in the sense of \S\ref{sect:ff}).
\end{compactenum}
In this case, if there is a theory whose models are precisely the
existentially closed models of $T$, then this theory is a model
companion of $T^*$ \cite[Cor.~7.13]{MR0277372}. 

Loosely, in an existentially closed model of a theory, everything that
\emph{can} happen \emph{does} happen.  When the model is a vector
space, it may seem that two conflicting things can happen:
\begin{compactenum}
\item 
The dimension can always be made higher by addition of new vectors.
\item
Linearly independent vectors can be made dependent by addition of new
scalars. 
\end{compactenum}
Because of the precise definition of existential closedness, it is the
latter tendency that wins out: the new scalars satisfy a quantifier-free formula; the new vectors, only a universal formula.
In an existentially closed
vector space in the usual signature, the scalar field is algebraically
closed; but 
the dimension of the space is simply $1$.  However, in the signature with a
binary symbol for parallelism, the existentially closed vector spaces
have dimension $2$.  More generally, in the signature with an $n$-ary
symbol for linear dependence, the existentially closed vector spaces
have dimension $n$. 
Again, this is worked out in~\cite{MR2505433}.

\section{Interacting rings}

In a vector space, the
vectors may act as derivations of the scalar field.  The theory of
such structures has no model companion, unless one adds some new
symbols to the signature.  These matters were worked out by my student
\"Ozcan Kasal~\cite{Kasal}.

There is a remarkable symmetry in this situation.  We are considering
quadruples $(V,K,*,D)$, where $(V,K,*)$ is a vector space (in the notation on page \pageref{quad}),
and in particular $*$ is the action of $K$ as a scalar field on $V$,
but now also $D$ is an action of $V$ as a space of derivations on $K$.
Also, as $K$ has a multiplication, so we require $V$ to have a
multiplication: the `bracket' operation $(\bm u,\bm v)\mapsto\bm
u\circ\bm v-\bm v\circ\bm u$.  Thus, $V$ is a Lie ring, while $K$ is
an associative, commutative ring.  Then
axioms for these structures $(V,K,*,D)$ come in dual pairs. 

There seems to be some precedent
for referring to $(V,K,*,D)$ as a \textbf{Lie--Rinehart pair.}  
The set $\Der K$ of derivations of a field $K$ can be given the
structure of both a vector-space 
over $K$ and a Lie ring.  Let $V$ be a subspace and sub-ring of $\Der
K$, and let $k$ be the constant field of $V$.  Then $V$ is what is
termed by Rinehart \cite{MR0154906} a \emph{$(k,K)$-Lie
  algebra.}  Other terms include \emph{pseudo-alg\`ebre de
    Lie} \cite{MR0055323} and \emph{Lie $d$-ring} \cite{MR0125867}, as
one may
learn from Stasheff \cite [p.~228]{MR1443334}, in whose own terminology
$(V,K)$ is a \emph{Lie--Rinehart pair over $k$.}  This term applies
more generally to the situation where $k$ is
just a (commutative associative) ring and $K$ is a commutative
algebra over $k$; but I shall require $K$ to be a field.  In any case,
reference to $k$ may distract us from 
seeing the symmetry or dualism present in the pair $(V,K)$, or rather
the quadruple $(V,K,*,D)$ discussed above.  But it is
just this dualism that I want now to emphasize.

Probably the theory of Lie--Rinehart pairs (in the present sense) has
no model companion, 
by a result of Macintyre that was announced~\cite{Macintyre:Lie}, but
not published: namely, the theory of Lie algebras (over a given field)
has no model companion.  In fact, just recovering this result would
be a worthwhile exercise. 

The dualism between associative rings and Lie
rings can be brought out in a way that I presented first at
Logicum Colloquium 2005 in Athens, in a contributed talk.
Suppose $R$ is an abelian group.  Then 
the endomorphisms of $R$ compose an abelian group, $\End R$.  If
$\circ$ is composition in $\End R$, let $\circ'$ be reverse
composition: 
\begin{equation*}
f\circ'g=g\circ f.
\end{equation*}
Then for any pair $(p,q)$ of integers, there is an operation
$p\mathord{\circ}-q\mathord{\circ'}$ on $\End R$, and it is a \textbf{multiplication} on
$\End R$ in
the most general sense: it 
distributes over addition from either side.  Now suppose $\cdot$ is
a multiplication on $R$.  Then for every $x$ in $R$, there is an element
$\lambda_x$ of $\End R$ given by 
\begin{equation*}
\lambda_x(y)=x\cdot y.
\end{equation*}
Moreover, the function $x\mapsto\lambda_x$ is a homomorphism of abelian groups.
Let us say that $(R,{}\cdot{})$ is a \textbf{$(p,q)$-ring} if $x\mapsto\lambda_x$ is a ring-homomorphism from $(R,{}\cdot{})$ to $(\End R,p\mathord{\circ}-q\mathord{\circ'})$.  For example,
\begin{compactitem}
\item
an associative ring is a $(1,0)$-ring;
\item
a Lie ring is a $(1,1)$-ring.
\end{compactitem}
In particular, if
$(p,q)\in\{(1,0),(1,1),(0,0)\}$, then $(\End R,p\mathord{\circ}-q\mathord{\circ'})$ is
itself a $(p,q)$-ring.  The converse is also
true; but I have found no sign
that the result has been published or observed. 

Now go back to the Lie--Rinehart pairs $(V,K,*,D)$ discussed above.  The theory
of these structures is not inductive; but the theory of such
structures in which $\dim_K(V)\leq m$ is inductive, and it has a model
companion, which can be derived from the model companion of $\df^m$
discussed in \S\ref{sect:df}.  Moreover, we can define in $V$ an
isomorphic copy of $K$ as a field, using as a parameter just a single
element $t$ of $K$ with a nonzero derivative.   This $t$ is in
particular an endomorphism of the group-structure of $V$.
Thus we can obtain a model-complete theory of Lie rings in a signature
with a symbol for a singulary function.  (I have written only the draft of an article showing this.)

Again, although it appears that the theory of Lie rings as such has no
model companion, this does not rule out the possibility that there is
a model-complete theory of Lie rings in the usual signature.  This question should be settled.

I return to the work of \"Ozcan Kasal.  He studies Lie--Rinehart pairs, in characteristic $0$, but in a signature \emph{without} a symbol for the bracket operation.  Let $T$ be the theory of such structures.  Kasal characterizes the existentially closed models of $T$ and shows that the class of these models is not \textbf{elementary:} it is not the class of models of a particular theory.  Therefore $T$ has no model companion.

However, Kasal observes that there is a certain relation of dependence among the scalars:  A scalar $x$ depends on a set $Y$ of scalars if $\delta x=0$ for every derivation $\delta$ in $V$ such that $\delta y=0$ for every $y$ in $Y$.  (Here $\delta x$ can also be written as $\delta\mathbin Dx$.)
Kasal then enlarges the signature to contain, for each positive integer $n$, a symbol for the mutual dependence of $n$ scalars.  Then he shows that the theory of \emph{expansions} to this signature of models of $T$ has a model companion.

\section{Recursion and induction}\label{sect:rec}

The remaining sections of this document concern ideas that come out of teaching.  I feel about teaching the way Richard Feynman~\cite[pp.~166 f.]{Feynman-Joking} does:
\begin{quote}
In any thinking process there are moments when everything is going good and you've got wonderful ideas.  Teaching is an interruption, and so it's the greatest pain in the world.  And then there are the \emph{longer} periods of time when not much is coming to you.  You're not getting any ideas, and if you're doing nothing at all, it drives you nuts!  You can't even say `I'm teaching my class.'

If you're teaching a class, you can think about the elementary things
that you know very well.  These things are kind of fun and delightful.
It doesn't do you any harm to think them over again.  Is there a
better way to present them?  Are there any new problems associated
with them?  Are there any new thoughts you can make about them?  The
elementary things are \emph{easy} to think about; if you can't think
of a new thought, no harm done; what you thought about it before is
good enough for the class.  If you \emph{do} think of something new,
you're rather pleased that you have a new way of looking at it. 

The questions of the students are often the source of new research\dots
\end{quote}
In teaching a course called Fundamentals of Mathematics, I observed that Peano~\cite{Peano} had apparently been confused about something that Dedekind~\cite{MR0159773} had got right: the Induction Axiom for $(\N,1,x\mapsto x+1)$ (mentioned in \S\ref{sect:primer} on page \pageref{IA}) does not by itself justify recursive definitions of operations like addition, multiplication, and exponentiation.  In general, one needs the other two of the so-called Peano Axioms:
\begin{compactenum}[1)]
\item
$x\mapsto x+1$ is not surjective;
\item
$x\mapsto x+1$ is injective.
\end{compactenum}
I first discussed these matters publicly at Logicum Colloquium 2008 in Bern in a contributed talk, and Alexandre Borovik has referred to this talk in his own work.  One \emph{can} prove the existence of addition and multiplication by induction alone, and Landau~\cite{MR12:397m} does this, though without dwelling on the logical implications.  However, this fact makes modular arithmetic possible.  The set $\Z/n\Z$ of congruence-classes of integers with respect to a modulus $n$ satisfies the Induction Axiom, and from this alone, it follows that $\Z/n\Z$ is a ring in the usual sense.  Euler's first proof of Fermat's Theorem (reported by Gauss~\cite[\P50, p.~32]{Gauss}) can be understood as a proof by induction in $\Z/p\Z$:  With respect to the modulus $p$, we have $1^p\equiv 1$,
and if $a^p\equiv a\pmod p$, then since $(a+1)^p\equiv a^p+1$, we conclude
$(a+1)^p\equiv a+1$.

The different models of the Induction Axiom, and the operations that can be defined in them, were investigated by Henkin~\cite{MR0120156} (whose proof~\cite{MR0033781} of the \emph{Completeness Theorem}---see \S\ref{sect:sets} below---is the one used today).
However, since Henkin's natural numbers began with $0$ instead of $1$, he missed the following observation: that the recursive definition
\begin{align*}
x^1&=x,&x^{y+1}&=x^y\cdot x
\end{align*}
of multiplication is valid for $\Z/n\Z$ if and only if $n$ is one of
the numbers $1$, $2$, $6$, $42$, and $1806$.  (It turns out that these
numbers were found by Dyer-Bennet~\cite{MR0001234} as the only moduli
with respect to which the congruence of $a_0{}^{b_0}$ and
$a_1{}^{b_1}$ can be inferred from that of the $a_i$ and of the
$b_i$.) 

I wrote an unpublished article~\cite{2011arXiv1104.5311P} on the
relation between induction and recursion, and on confusions about it;
John Baldwin referred to some of this in a talk~\cite{Baldwin-purity}.  We 
should distinguish two kinds of recursive definition.  First, one can
consider the set of natural numbers to be defined recursively by the
rules that $1$ is a natural number, and if $n$ is, then so is $n+1$. 
The set of formulas of a logic is recursively defined in this way.
Recursive definitions of sets justify proofs by induction on those
sets.  However, they do not alone justify recursive definitions of
\emph{functions on} those sets.  This is understood by some writers,
such as Enderton \cite{MR0337470}, but not others. 

In the spirit of Feynman, we might make the presentation of basic
logic more interesting in the following way.  Starting with `atomic'
formulas, we build up other formulas recursively.  Each formula
can then be seen as the root of a tree whose leaves are the atomic
formulas that appear in it.  But then we must prove that the tree is
uniquely determined by its root: this is needed to justify recursive
definitions of functions on the set of formulas.  An example of such a
function is the one
that assigns to each formula the relation that it defines on a given
structure.  We proceed to develop the notion of \textbf{formal proof:}
from a given set of sentences to be considered as axioms, we define
recursively the set of sentences that are to be considered as theorems
provable from those axioms.  Here then a theorem is the root of a tree
whose leaves are axioms.  However, this time the tree is not uniquely
determined by its root.  We have no obvious procedure for finding the
proof of a theorem, other than to enumerate all of the possibilities.
But then we have no obvious procedure for determining whether an
arbitrary sentence is a theorem, unless the set of theorems is a
complete theory.  Indeed, there may be \emph{no} such procedure, and
\emph{G\"odel's Incompleteness Theorem} (see page \pageref{Goedel} in \S\ref{sect:primer} above and
the next section) can be understood to establish this fact for
certain systems of axioms for a theory of $(\N,+,{}\cdot{})$. 

The foregoing paragraph is just an example of how attention paid to
tedious foundational details can give insight into deep results.  The
paper~\cite{2011arXiv1104.5311P} ends with some apparently new
results, although in preparing for a talk in the mathematics
department of Istanbul Bilgi University, October 11, 2011, I found
that revision is needed.  

The Peano Axioms determine the natural
numbers as composing a structure in a signature $\{1,\mathrm S\}$.
This structure turns out
to have a binary relation by which it is well-ordered.  This observation leads
naturally to the von Neumann definition of the natural
numbers~\cite{von-Neumann}, whereby
the least of them is $\emptyset$ (now called $0$), and $n+1$ is
$n\cup\{n\}$; by this definition, the natural numbers compose the set
called $\vnn$.   Then the natural numbers are just the finite examples
of \textbf{ordinal numbers,} in von Neumann's definition.  

This
development of the ordinal numbers has a natural generalization.  The
structure determined by the Peano Axioms is a \emph{free object} in
the category whose objects are structures in $\{1,\mathrm S\}$ and
whose arrows are embeddings.  But for any signature $\mathscr S$ with
no relation symbols, there  
is a free object in the category of structures in $\mathscr S$.
There is then a set-theoretic definition of this free object, resembling
von Neumann's definition of the natural numbers; and then there is a
weakening of the definition that gives us a larger class,
corresponding to the class of ordinals, in which the
free object embeds.  

We can see the elements of this larger class as a
new kind of set.  Each of these new sets has a \textbf{type,} and each
of its elements falls into one or more \textbf{grades.}  Specifically,
there is a type for each symbol of $\mathscr S$, and then there is one
more type for \emph{limits.}  If a symbol is $n$-ary, then the sets of
its type have elements of $n$ different grades.  The constant symbols
are $0$-ary; sets of their type are empty.

Thus there is an analogy between sets in the usual sense, and natural
numbers (in the usual sense):
\begin{compactenum}
\item
The unique least natural number corresponds to the unique empty set $\emptyset$.
\item
There is \emph{one} way of getting new numbers, namely by adding $1$
to old numbers; correspondingly, sets are determined by their
elements.
\item
A new number is obtained by adding $1$ to a \emph{single} natural
number; correspondingly, an element of a set is an element in only one
way.

\end{compactenum}

\section{The Russell Paradox}\label{sect:sets}

This section gives my idea of set theory, as developed while teaching it.  There are some leads that may be pursued further.

Every student of mathematics should be familiar with the Russell
Paradox~\cite{Russell-letter}.  I present it as follows.  Our language
gives us the notion
of a \textbf{collective noun,} which is a singular noun that refers to
many things 
at once.  Suppose we attempt to declare that one
particular collective noun, such as \emph{set,} is going to be the
most general.  Then the sets that do not contain themselves must
compose a set.  This
set contains itself if and only if it does not; and that is absurd.
Therefore there is no most general collective noun.

We can nonetheless choose a collective noun that will be most general
for our purposes.  A similar move is made in model theory, in the
study of a particular theory with infinite models.  There is no
largest model of this theory, but there is a \emph{`monster model',} which is
larger than all of the other models that one wants to study; these
models can be assumed to be elementary substructures of the monster
model.

For the collective noun that is most general for our purposes, the
obvious choice is \emph{collection.}  Such an understanding seems to
be behind accounts of set theory that can be found in standard
textbooks of other areas of mathematics.  Indeed, in two such books
are found
the following statements.
\begin{quote}
  A collection of objects viewed as a single entity will be called a
  \emph{set.}\\ \mbox{}\hfill\cite[p.~32]{MR49:9123}

Intuitively we consider a class to be a collection $A$ of objects
(elements) such that given any object $x$ it is possible to determine
whether or not $x$ is a member (or element) of
$A$.\hfill\cite[p.~2]{MR600654}
\end{quote}
However, such statements suggest questions such as,
\begin{compactitem}
\item 
Is there a collection of things that are not objects?
\item
Is there a collection that is not viewed as a single entity?
\item
Is there a collection in which membership is impossible to determine?
\end{compactitem}
If the answers are no, then the quoted statements can be reduced to
the following, respectively:
\begin{quote}\centering
  A collection will be called a \emph{set.}

Intuitively we consider a class to be a collection.
\end{quote}
These are not very useful.  But if any of the questions above are answered yes, then
examples should be given.

I approach set theory as the study of a particular kind of collection,
to be called a \textbf{set.}  We do not say what a set is, beyond its
being an element of some model of the theory that we develop.  The
models of set theory have one sort, and their signature has just one
symbol, which the binary relation symbol $\in$ for membership.  Thus all
elements of sets must be sets themselves.

We may imagine that set theory has an \emph{intended model,} to be
called $\mathbf V$.  This is a collection of sets, but not necessarily a
set itself.
A singulary formula in the signature of set theory defines a
\emph{collection} of elements of $\mathbf V$; but there is no reason to
assume that this collection is actually \emph{in} $\mathbf V$; that is,
there is no reason to assume that it is a set.  We refer to such a
collection as a \textbf{class.}  Then there is a class of sets that do
not contain
themselves: this is the class defined by the formula $x\notin x$.  Call
this class $\bm R$.  The
Russell Paradox now becomes the theorem that $\bm R$ is not a set.
This theorem can be expressed formally as a sentence of set theory,
namely
\begin{equation*}
\lnot\Exists x\Forall y(y\in x\leftrightarrow y\notin y).
\end{equation*}

The Russell Paradox has an echo in the proof of
G\"odel's Incompleteness Theorem mentioned above in
\S\ref{sect:primer} and the last section.  G\"odel assigns to each
symbol $S$ in the logic of 
$(\N,+,{}\cdot{})$ a distinct element $\gn S$ of $\N$.  Then to a formula
$S_1\cdots S_n$ can be assigned the number
\begin{equation*}
  2^{\gn{S_1}}\cdot 3^{\gn{S_2}}\cdot 5^{\gn{S_3}}\dotsm,
\end{equation*}
that is, $\prod_{i=1}^np_i{}^{\gn{S_i}}$, where $(p_k\colon k\in\N)$
is the sequence of primes.  If the formula is $\phi$, then the
number thus assigned to it can itself be denoted by $\gn{\phi}$.

Given a recursive collection $\Delta$ of axioms for
$(\N,+,{}\cdot{})$, G\"odel has a system of \emph{formal proof} (see the
last section) of deriving their logical
consequences.  Then there is a binary formula $\phi(x,y)$ such that,
for any singulary formula $\psi(x)$ and any $n$ in $\N$, there is a
formal proof of $\psi(n)$ from $\Delta$ if and only
$\phi(\gn{\psi},n)$ is true in $(\N,+,{}\cdot{})$.  Now let $\theta(x)$ be
$\lnot\phi(x,x)$.   Then
\begin{center}
  $\psi(\gn{\psi})$ is provable if and only if
  $\lnot\theta(\gn{\psi})$ is true.
\end{center}
Here the resemblance to the Russell Paradox comes out.  Replace $\psi$
with $\theta$.  Writing $\sigma$ for
$\theta(\gn{\theta})$, we have that $\sigma$ is provable if and only
if $\lnot\sigma$ is true, that is,
$\sigma$ is false.  But every provable sentence is true.
Therefore $\sigma$ is not provable; but it is true.

There now two possibilities.  
\begin{asparaenum}
\item  
One possibility is that our proof system is
\emph{incomplete.}  In particular, although there is no formal proof
of $\sigma$ from $\Delta$, maybe $\sigma$ is still a
\textbf{logical consequence} of $\Delta$: that is, maybe $\sigma$ is
true in every model
of $\Delta$.  In this case, we have not ruled out the possibility that
every sentence in the complete theory of $(\N,+,{}\cdot{})$ is a logical
consequence of $\Delta$. 

Indeed, such a possibility is realized in second-order logic.  The
second-order Peano Axioms for $(\N,1,\mathrm S)$ have only one model,
up to isomorphism; therefore every sentence of the theory of this model
is a logical consequence of the Peano Axioms.  Thus there can be no
complete proof-system for the second-order logic of $(\N,1,\mathrm
S)$, and therefore of $(\N,+,{}\cdot{})$.

However, G\"odel had already proved a \textbf{Completeness Theorem}
for his proof system for 
first-order logic~\cite{Goedel-compl}.  
If there is no first-order proof of $\sigma$ from $\Delta$, then
$\sigma$ is not a logical consequence of $\Delta$.
Therefore the second
possibility must be realized:
\item
Not all
sentences in the first-order theory of $(\N,+,{}\cdot{})$ are logical
consequences of $\Delta$.  The theory axiomatized by $\Delta$ is not
complete. 
\end{asparaenum}
We have not gone through the details of what a recursive collection is, nor
the derivation of $\phi(x,y)$ above.  However, it is easier to carry
out related arguments in set theory.

As an example, in set theory we establish the
\textbf{Undefinability of Truth.}  This is a theorem proved by
Tarski~\cite[p.~247]{MR736686}, who notes his debt to
G\"odel.  We can encode each formula $\phi$ now as a set
$\gn{\phi}$ in $\mathbf V$.  Suppose there were a singulary formula
defining the
collection of codes of all true sentences.  Then there would be a
singulary formula $\phi$ defining the collection of codes of all
singulary formulas $\psi$ such that $\psi(\gn{\psi})$ is
\emph{false}.  In short,
\begin{equation*}
  \phi(\gn{\psi})\iff\lnot\psi(\gn{\psi}).
\end{equation*}
Now we really do have the Russell Paradox.
Putting $\phi$ for $\psi$ gives the contradiction.

\section{Apollonius}

Apollonius of Perga wrote eight books on conic sections.  The last
four were lost in the original Greek, perhaps because they were
too difficult; any case, books V--VII do survive in Arabic
translation.  

I myself have spent much time only with Book I~\cite{MR1660991}.  Here
Apollonius assigns the names \emph{parabola, ellipse,} 
and \emph{hyperbola} to the conic sections: these are names that in Greek allude
to the properties of the curves \emph{as} sections of a cone.
Students today are commonly \emph{told} in textbooks that these curves
can be 
obtained by cutting a cone; but a proof is rarely seen.  A lovely
geometrical account of the sections, with visual proofs, is given by Hilbert and
Cohn-Vossen~\cite{MR0046650}; but this account uses \emph{right}
cones, like most other descriptions of conic sections as such.
However, Apollonius shows that oblique cones can be used.  He in
effect derives our modern equations for the cones; these equations work in
rectangular or oblique coordinate systems.  Then, given a curve
satisfying one of the equations, he shows how to obtain a cone from
which that curve can be cut.

In \emph{Rules for the Direction of the Mind}~\cite{Descartes-Rules},
Descartes suggests that the ancient mathematicians must have had some
kind of algebra for discovering their theorems, but that they felt
obliged to conceal it.  I last worked through Book I of
Apollonius for a course at the Nesin Mathematics Village in the summer
of 2008; then I got the feeling that Descartes was probably not correct,
but that Apollonius's geometric arguments really did show how Apollonius
thought of things.  

However, this is a point worth further
investigation.  Some historians read Apollonius, but probably few
mathematicians: the reading is difficult, and our
Cartesian methods of analysis seem to work more efficiently.  And yet a good
understanding of Apollonius might help prevent inaccurate
generalizations about mathematics.  I said in \S\ref{sect:primer} that
most mathematics involves structures.  This may be true today; but it makes
little sense for ancient mathematics.  And yet ancient mathematics is
unquestionably mathematics, sometimes at the highest level.

\section{Euclid and Archimedes}

Euclid is more commonly read than Apollonius, by mathematicians and others.  The same
may be true of Archimedes.  In the last three summers at the Nesin
Mathematics Village, I have taught a course that I called Non-standard
Analysis.  The title comes from the book of Abraham
Robinson~\cite{MR1373196} that makes rigorous Leibniz's notion of
infinitesimals in calculus.  But I have found it worthwhile to go back
further in history, to study Archimedes's use of infinitesimal
methods, in the quadrature of the parabola for example
\cite{MR2000800,MR13:419b,Heath-Method}, or in showing 
that the surface of a sphere is equal to a circle whose radius is the
diameter of the sphere~\cite{MR2093668}.

I have also worked through the construction of the real numbers from
the natural numbers.  I have aimed to make the construction as
transparent as possible.  Treatments of the construction that I know
of, as in Landau~\cite{MR12:397m} or Spivak~\cite{0458.26001}, have
the aim of satisfying the reader (and the writer) that $\R$ really
exists.  But in the spirit of Feynman (\S\ref{sect:rec} above), one
can get more out of the construction.

For example, one can rediscover H\"older's theorem~\cite{MR1423724}
that there is a unique complete 
densely ordered abelian group, namely $(\R,+,<)$; and every
archimedean ordered abelian group embeds in this.  H\"older derives
his results from considering \emph{magnitudes} as used Book V of
Euclid's \emph{Elements.}

All of this has led to ongoing collaboration with Alexandre Borovik on
constructions of $\R$ and their possible generalizations.  One
observation is the following.  
There seem to be two standard constructions of $\R$.  
\begin{asparaenum}
\item
One is
Dedekind's~\cite{MR0159773}, whereby a real number is a set of
rational numbers with an upper bound, but no maximum element, and
containing every rational that is less than one of its elements.  This
construction uses only the ordering of $\Q$, and so, first of all,
$(\R,\included)$ is obtained as the \textbf{completion} of $(\Q,<)$.
But $\Q$ is also an abelian group, and this structure extends to $\R$;
likewise, the completion of every ordered abelian group has the
structure of an abelian group, \emph{if and only if} the ordered group is
archimedean (this is part of H\"older's theorem).
\item
The other standard construction of $\R$ obtains it as
the quotient of the ring of Cauchy sequences of $\Q$ by the maximal ideal of
sequences convergent to $0$.  But this construction can be applied to
any ordered field, even a non-archimedean one.  In this case,
the Cauchy sequences should have length equal to the
cofinality of the field.  (All Cauchy sequences shorter than this are
eventually constant.)  So this construction should be distinguished
from Dedekind's.
\end{asparaenum}

\setbibpreamble{\smaller}

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\def\cprime{$'$} \def\cprime{$'$} \def\cprime{$'$}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
% \MRhref is called by the amsart/book/proc definition of \MR.
\providecommand{\MRhref}[2]{%
  \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
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\providecommand{\href}[2]{#2}
\begin{thebibliography}{10}

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