Date
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What we discussed/How we spent our time
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Jan 11
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Syllabus. Text.
We discussed some definitions of "model theory", and then
informally discussed structure,
sentence, $\vdash$, $\models$, and independence.
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Jan 13
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We discussed the alphabet of symbols
of a language, the definition of structures,
and started talking about formulas.
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Jan 15
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We
discussed terms, atomic formulas, formulas,
and the satisfaction of a formula in a structure
at a value ($\mathbf A\models \varphi[v]$).
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Jan 20
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We discussed
Galois connections induced by a relation.
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Jan 22
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We discussed the Galois connection between $\mathcal L$-structures and
$\mathcal L$-sentences induced by $\models$.
We noticed that the collection of $\mathcal L$-theories
forms a lattice under intersection and join (= theory generated by the union).
Each element of the lattice is extendible to a maximal proper element
(a complete theory). We noted that the complete
theories are exactly the theories of single structures.
We introduced the notion of elementary equivalence.
[Some of our observations relied on Gödel's Completeness Theorem.]
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Jan 25
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We discussed the space of complete
$\mathcal L$-theories
(or $\mathcal L$-structures modulo elementary equivalence),
and argued that it is compact, Hausdorff and
zero-dimensional. [The proof relied on Gödel's Completeness Theorem.]
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Jan 27
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We discussed theories of fields. In particular, we showed that every
field of positive characteristic satisfies
$\exists x\exists y(x^2+y^2=-1)$, but no totally ordered field
satisfies this sentence.
We used this to show that
$\mathbb Q$ and $\mathbb R$
do not belong to the elementary class generated by
fields of positive characteristic.
We showed how to express ``There is an irreducible monic cubic
in $\mathbb F[x]$'' in a first-order way, and used this to show that
$\mathbb R$ and $\mathbb C$
do not belong to the elementary class generated by finite fields.
We started
talking about topological convergence via
limits over
(ultra)filters.
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Jan 29
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We proved the results of the first of Wednesday's handouts,
and discussed the results of the second.
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Feb 1
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We proved that $x\in \overline{Y}$ iff $\lim_{\mathcal U} f = x$
for some $I, f, {\mathcal U}$ such that $f(I)\subseteq Y$.
We then proved that ${\mathcal X}$ is compact iff
for any $I, f, {\mathcal U}$
there is at least one $x\in {\mathcal X}$ such that
$\lim_{\mathcal U} f=x$.
I asked you to read the proof that
${\mathcal X}$ is Hausdorff iff
for any $I, f, {\mathcal U}$
there is at most one $x\in {\mathcal X}$
such that
$\lim_{\mathcal U} f=x$.
We then discussed compactness of the space of complete $\mathcal L$-theories.
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Feb 3
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We started by discussing the difference between ``consistency''
and ``satisfiability''. We stated the Compactness Theorem.
we reviewed basic terminology associated to maps of sets/spaces/structures,
e.g.: domain, codomain, image, fiber, coimage, inclusion map,
natural map, induced map, $\varphi = \iota\circ \overline{\varphi}\circ \nu$,
bijective morphism versus isomorphism,
weak topology for a family of outgoing maps, strong topology
for a family of incoming maps, subgroup or subspace, quotient
group or quotient space. We then defined homomorphism
of first-order structures.
weak interpretation (of relation symbols)
for a family of incoming maps, strong interpretation
for a family of outgoing maps, substructure and quotient structure.
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Feb 5
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We defined substructure, quotient structure,
product, and
ultraproduct.
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Feb 8
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We proved Łos's Theorem
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Feb 10
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We derived some corollaries from Łos's Theorem:
Cor. A sentence holds in an ultraproduct iff it holds in almost
every coordinate modulo the ultrafilter.
Cor. A class of $\mathcal L$-structures is 1rst-order axiomatizable iff
it is closed under elementary equivalence and ultraproducts.
Thm (Keisler-Shelah). $\mathbb A \equiv \mathbb B$ iff $\mathbb A$
and $\mathbb B$ have isomorphic ultrapowers. (Not proved.)
Cor. A class of $\mathcal L$-structures is 1rst-order axiomatizable iff
it is closed under elementary equivalence and ultraproducts.
Compactness Theorem, version 1. A set $\Sigma$ of $\mathcal L$-structures
is satisfiable iff it is finitely satisfiable.
Compactness Theorem, version 2. A set $\Sigma$ of $\mathcal L$-structures
is not satisfiable iff there is some finite subset
$\Sigma_0\subseteq \Sigma$ that is not satisfiable.
Compactness Theorem, version 3. $\Sigma\models \sigma$ iff
$\Sigma_0\models \sigma$ for some finite $\Sigma_0\subseteq \Sigma$.
Cor. A class $\mathcal K$ is finitely axiomatizable iff
$\mathcal K$ and $\overline{\mathcal K}$ are axiomatizable.
(A topological proof, and a proof via the Compactness Theorem.)
Cor. If $\Sigma$ and $\Sigma'$ axiomatiza the same class,
and $\Sigma$ is finite, then there is a finite subset
$\Sigma_0\subseteq \Sigma$ that axiomatizes the same class.
Example. We stated Birkoff's Theorem, that a class of structures
in an algebraic lannguage is axiomatizable by identities
iff it is closed under the formation of homomorphic images,
substructures and products. Then we claimed that it is not
hard to show that the class of 2-step nilpotent groups
is closed under the formation of homomorphic images,
substructures and products, hence this class is
axiomatizable by identities. It is also finitely axiomatizable
by the group laws and the 1rst-order sentence
$\forall x\forall y\exists z\forall w((xy=yxz)\wedge (zw=wz))$,
which is not an identity.
It follows that the class of 2-step nilpotent
groups is axiomatizable by the group
laws plus finitely many other identities.
(With some work, the desired identity can be found:
$\forall x\forall y\forall z(zx^{-1}y^{-1}xy=x^{-1}y^{-1}xyz)$.)
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Feb 15
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We discussed the use of language expansion to understand the structure
of ultraproducts. We used the method to show that
(i) the ultraproduct
of the sets $\{0,1\}^n$, $n\in \omega$, over a nonprincipal ultrafilter
on $\omega$ has size $2^{\omega}$, (ii) any ultraproduct of
nonsimple groups is nonsimple, (iii) an ultrapower of
the ordered field $\mathbb R$
over a nonprincipal ultrafilter on $\omega$ is nonarchimedian.
Then we began a discussion of regular
ultrafilters.
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Feb 17
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We finished discussing
regular ultrafilters, and then turned to discussing
when the diagonal embedding into an ultrapower,
$\Delta\colon \mathbb A\to \prod_{\mathcal U} \mathbb A$, is
surjective. This led to a discussion of
$\kappa$-complete
ultrafilters.
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Feb 19
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We definitely defined definable. We also defined
$\textrm{tp}^{\mathbb A}(\bar{a}/X)$.
Among our examples, we noted that $\mathbb C$ with conjugation
(in the language of fields with an automorphism)
is definable in $\mathbb R$ (in the language of fields),
and conversely $\mathbb R$ is definable in
$\mathbb C$ with conjugation.
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Feb 22
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We defined interpretable, and showed that the projective
plane over a finite field $\mathbb F$
is interpretable in affine 3-space over $\mathbb F$.
We turned back to types and made a number of remarks,
including the definitions of partial type
versus complete type, quantifier-free type, and
elementary embedding.
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Feb 24
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We defined elementary substructure and
elementary extension, and
noted that if [$\mathbb C\prec \mathbb B$] then
[$\mathbb C\leq \mathbb B$ and $\mathbb C\equiv \mathbb B$]
but not conversely. I raised the question of
when a field extension is elementary. We defined the
diagram language (${\mathcal L}_A$),
diagram expansion ($\mathbb A_A$),
atomic diagram ($\textrm{Diag}(\mathbb A)$), and
elementary diagram ($\textrm{Th}(\mathbb A_A)$)
of a structure $\mathbb A$. We talked about the Diagram
Lemma and some corollaries, namely the fact that the elementary
extensions of $\mathbb A$ are the $\mathcal L$-reducts
of the models of the elementary diagram of $\mathbb A$
and the fact that every infinite structure has arbitrarily
large elementary extensions.
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Feb 26
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We discussed an upcoming sequence of theorems aimed
at understanding the structure the category of
models of a complete theory under elementary embeddings.
The theorems were: the Tarski-Vaught Criterion (for
testing if a substructure is elementary),
the upward and downward Löwenheim-Skolem Theorems (for size control
of elementary extensions), the Robinson Consistency Theorem
(for finitary type unification), Tarski's Theorem on Elementary Chains
(for infinitary type unification), and the idea of a saturated
elementary extension.
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Feb 29
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We established the validity of the Tarski-Vaught Test,
the Löwenheim-Skolem Theorems, and the
Łos-Vaught Test.
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Mar 2
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We expanded on the previous lecture by discussing
Skolemization, categoricity, and the back and forth method.
We sketched a proof that the theory of dense linear orders
without endpoints is $\omega$-categorical.
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Mar 4
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We discussed the uncountable categoricity of
the theories $\textrm{ACF}_0$ and $\textrm{ACF}_p$.
We proved the elementary joint embedding property.
We derived the Robinson Joint Consistency Theorem from the
Craig Interpolation Theorem.
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Mar 7
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We proved the Elementary Chain Principle.
We discussed several preservation theorems, including the
preservation theorems of Łos and Tarski (about $\Pi_1$ axiomatizations
and $\Sigma_1$ axiomatizations),
Lyndon's Preservation Theorem (about positive axiomatizations),
and the Chang-Łos-Suszko Theorem (about $\Pi_2$ axiomatizations).
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Mar 9
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We proved the Łos-Tarski about $\Pi_1$ axiomatizations.
We defined
type, what it means to
realize or omit a type, and
weakly saturated model.
We discussed what is known about the number of countably infinite
models of a complete theory in a countable language.
(It is known that this number lies in the interval
$[0,2^{\omega}]$, that it cannot be $2$, that it cannot
be any uncountable cardinal other than $\omega_1$ or
$2^{\omega}$, but it can be any finite
cardinal other than $2$, and it can be $\omega$ or $2^{\omega}$.
It is open whether it can be $\omega_1$.) Our discussion referred
to Vaught's ``Never Two'' Theorem, Morley's Theorem and Vaught's Conjecture.
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Mar 11
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We discussed Ehrenfeucht's models
and defined what it means for a theory
to have quantifier elimination.
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Mar 14
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We discussed q.e. for the theory of Ehrenfeucht's example.
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Mar 16
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We finished discussing q.e. for the theory of Ehrenfeucht's example,
and explained why this theory is complete.
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Mar 18
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We discussed a model-theoretic method for establishing q.e., namely
Theorem 3.1.4.
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Mar 28
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We defined type spaces and some continuous maps between them.
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Mar 30
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We discussed realizing and omitting types,
focusing today on realizing types.
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Apr 1
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We defined supported/isolated (partial) types, and showed that
a supported type compatible with complete theory
$T$ is realized in every model of $T$.
We gave an example of a theory $T$ and
an unsupported type $p\in S_n(T)$ that is realized
in every model.
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Apr 4
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We started the proof that a Henkin Theory has a Henkin model.
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Apr 6
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We finished the proof that a Henkin Theory has a Henkin model
and started the proof of the omitting types theorem.
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Apr 8
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We finished the proof of the omitting types theorem, stated in
the following form: let $\mathcal L$ be a countable language
and let $T$ be a satisfiable $\mathcal L$-theory.
If $M_n\subseteq S_n(T)$ is a meager set for each $n\lt\omega$,
then $T$ has a model omitting all complete types in each $M_n$ for all $n$.
We closed the lecture by defining atomic models and theories.
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Apr 11
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Call a theory an X-theory if it is a complete satisfiable
theory in a countable language.
We proved that an X-theory is atomic iff it has an atomic model.
Then we proved that a model of an X-theory
is prime iff it is countable and atomic. We then explained why
prime models of X-theories are unique up to isomorphism,
and why two $n$-tuples in such a model have the same type
iff they differ by an automorphism.
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Apr 13
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We discussed perfect and scattered (sub)spaces
and the Cantor-Bendixson Theorem.
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Apr 15
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We proved that if $X$ is a second countable Stone space
with perfect kernel $P$
and $I$ is the subspace of isolated points, then
$X = \overline{I}\cup P$. In particular, if $X$ is scattered,
then its isolated points are dense. We then observed that:
$|X|\lt 2^{\aleph_0}$ implies $X$ is scattered. Turning to
type spaces we observed that $I(T,\aleph_0)\lt 2^{\aleph_0}$
implies $|S_n(T)|\lt 2^{\aleph_0}$ for each $n$, which
implies isolated types in $S_n(T)$ are dense for every $n$.
We concluded by characterizing the $\aleph_0$-categorical
complete theories in a countable language as those
where $|S_n(T)|<\aleph_0$ for all $n$.
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Apr 18
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We explained why $\textrm{Th}(\mathbb A)$ is $\aleph_0$-categorical
iff $\textrm{Aut}(\mathbb A)$ is oligomorphic.
We gave some examples of countably infinite structures with
$\aleph_0$-categorical theories ($\langle \omega; \emptyset\rangle$,
$\langle {\mathbb Q}; \lt\rangle$, a countably infinite vector
space over a finite field, a countably infinite $G$-set
where $|G|<\omega$). We began discussing the random graph.
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Apr 20
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We defined the ``extension axioms'' for graph theory.
We proved that the theory axiomatized by these axioms
is complete, $\aleph_0$-categorical,
and has quantifier elimination. We call the countably infinite
model the random graph. We showed that the random graph
is ultrahomogeneous and universal. Part of our arguments established
the general fact that the theory of
an ultrahomogeneous $\aleph_0$-categorical structure
has quantifier elimination.
We ended the lecture by noting
that the expansion of an $\aleph_0$-categorical structure by
naming finitely many elements with new constants is again
$\aleph_0$-categorical. The reverse is also true: any reduct of an
$\aleph_0$-categorical structure is again $\aleph_0$-categorical.
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Apr 22
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We defined $\kappa$-saturated structures and proved
the equivalence of two definitions (one using
$S^{\mathbb A}_n(U)$
and the other using $S^{\mathbb A}_1(U)$).
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Apr 25
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We explained why any infinite structure in a language
of cardinality $\leq\kappa$ has a $\kappa^+$-saturated
elementary extension of cardinality at most $|A|^{\kappa}$.
We proved that two elementarily equivalent saturated structures
of the same cardinality are isomorphic.
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Apr 27
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We reviewed saturation and $\kappa$-saturation,
and noted that it is possible for
two elementarily equivalent $\kappa$-saturated structures
of the same cardinality to be nonisomorphic.
We then proved that if $T$ is a complete satisfiable
theory in a countable language, then $T$ has a
countable saturated model iff $|S_n(T)|\leq \aleph_0$
for all $n$. ($T$ is small.) Hence
$I(T,\aleph_0)\leq\aleph_0$ $\Longrightarrow$
$T$ has a countable saturated model
$\Longleftrightarrow$
$|S_n(T)|\leq \aleph_0$ for all $n$ $\Longrightarrow$
$T$ is atomic
$\Longleftrightarrow$
$T$ has a countable atomic model.
We argued that $T$ is $\aleph_0$-categorical iff $T$
has a countable model that is atomic and saturated.
We started on the proof of Vaught's ``Never Two''
Theorem, which asserts that $I(T,\aleph_0)\neq 2$
whene $T$ is a complete theory in a countable language.
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Apr 29
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We finished the proof of Vaught's ``Never Two''
Theorem. We discussed what to read next.
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