Date
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What we discussed/How we spent our time
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Jan 13
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Syllabus. Text. HW.
We discussed some definitions of "model theory",
and introduced the symbols $\vdash$ and $\models$.
I circulated this handout
and
this handout.
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Jan 15
|
No meeting, reading assignment only:
Please read Section 1.1.
Try the following exercises.
(No need to turn these in.)
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Jan 17
|
We discussed exapmples of structure
and described the alphabet for a
language for structures of a given signature.
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Jan 20
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MLK, Jr Day.
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Jan 22
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We discussed predicates, terms, and first-order formulas.
This handout includes
some of the definitions and also some relevant exercises.
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Jan 24
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We defined $\mathbb A\models \varphi$.
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Jan 27
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We discussed the
Galois connection between
syntax and semantics, and defined the
relation $\Sigma\models \sigma$.
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Jan 29
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We began a discussion
of the completeness theorem for
first-order logic.
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Jan 31
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We discussed the current HW assignment,
including the definitions of
``elementary equivalence'' and ``definable relation''.
Then we discussed the construction of ``expansion by constants''.
Finally, explained why each consistent theory
can be extended to a complete theory with witnesses.
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Feb 3
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We explained why any Henkin theory has a model.
We derived some corollaries from the completeness theorem:
A theory is consistent iff it has a model.
A theory is complete iff it is the theory of a single model
(not uniquely determined, usually).
A theory is complete and has witnesses
iff it is the theory of a single model
in which all elements are interpretations of constants.
(Compactness) A theory is satisfiable iff each finite subset
is satisfiable.
We showed how to use the
compactness theorem to
establish the existence of an elementary extension
of the natural numbers that has an infinitely large element.
We also showed how to use the compactness theorem to
establish the existence of an elementary extension
of the real numbers that has a positive infinitesimal.
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Feb 5
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We discussed ordinals and cardinals.
Then we discussed how to estimate the number of different
$L$-structures defined on a domain that is a given cardinal $\kappa$.
We estimated this number up to equality and also up to isomorphism.
We defined a language $L$ to be the set of formulas
in some signature $S$; we discussed how to determine
the cardinality of $L$;
we denoted this cardinality by $\|L\|$.
We outlined the argument that any consistent
$L$-theory $T$ can be enlarged to a complete, consistent,
theory $T'$, with witnesses, in a language $L'$ satisfying
$\|L\| = \|L'\|$. Finally, we explained why a consistent
$L$-theory has a model of cardinality $\leq \|L\|$.
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Feb 7
|
Chapter 1 Roundup!
(We rounded up some of the ideas that were stragglers.)
First we explained why ``finite''
is not elementary. Specifically, we discussed
There is a single first-order sentence
$\sigma_{\textrm{card}= n}$
(or $\tau_{\textrm{card}\geq n}$)
which is satisfied by an $L$-structure $\mathbb A$
iff $|A|=n$ (or $|A|\geq n$).
We can express
``$|A|\in \{m_1,\ldots,m_k\}$'' or
``$|A|\notin \{n_1,\ldots,n_{\ell}\}$'' with a single sentence.
We can express ``$|A|$ is infinite'' with a set of sentences
(e.g., with $\Sigma_{\infty}$, taken to be either
$\{\neg \sigma_{\textrm{card}= 0}, \neg \sigma_{\textrm{card}= 1},\ldots\}$ or with
$\{\tau_{\textrm{card}\geq 0}, \tau_{\textrm{card}\geq 1},\ldots\}$).
There does not exist a set of sentences expressing
which holds for an $L$-structure iff the structure is finite.
${\bf Thm.}$ If $\{{\mathcal K}, {\mathcal K}'\}$ is a partition
of the class of all $L$-structures into two, complementary,
axiomatizable classes, then each of ${\mathcal K}$ and
${\mathcal K}'$ are finitely axiomatizable. (Equivalently,
a theory has a complement in the lattice of $L$-theories
iff it is finitely axiomatizable.)
${\bf Thm.}$ Any theory with arbitrarily large finite models
has an infinite model.
${\bf Cor.}$ If the class of finite models of $T$ is elementary,
then there is a finite $N$ such that every finite model
of $T$ has size at most $N$.
Second, we explained why relations definable (without
parameters) are invariant under automorphisms.
Specifically, we discussed
The definition of isomorphism and automorphism.
The fact that if $\alpha: \mathbb A\to \mathbb B$ is an isomorphism,
then $\alpha$ induces a bijection $v\mapsto \alpha\circ v$
between the set of all valuations
in $\mathbb A$ and the set of all valuations in $\mathbb B$.
If $\alpha: \mathbb A\to \mathbb B$ is an isomorphism,
$t$ is a term, and $v: \textrm{Variables}\to A$ is a valuation in $\mathbb A$,
then $\alpha(t^{\mathbb A}[v])=t^{\mathbb B}[\alpha\circ v]$.
If $\alpha: \mathbb A\to \mathbb B$ is an isomorphism,
$\varphi$ is a formula,
and $v: \textrm{Variables}\to A$ is a valuation in $\mathbb A$,
then $\mathbb A\models \varphi[v]$ iff
$\mathbb B\models \varphi[\alpha\circ v]$.
${\bf Cor.}$ $\mathbb A\cong \mathbb B$ implies
$\mathbb A\equiv \mathbb B$.
If $\alpha: \mathbb A\to \mathbb B$ is an isomorphism,
$\varphi$ is a formula, $R:=\varphi[A]$ is the relation
defined in $\mathbb A$ by $\varphi$, and
$S:=\varphi[B]$ is the relation
defined in $\mathbb B$ by $\varphi$, then $\alpha(R)=S$.
In particular, if $\mathbb A=\mathbb B$ and $\alpha$ is
an automorphism of $\mathbb A$, then the relation
defined by $\varphi$ in $\mathbb A$ is invariant
under $\alpha$.
In particular, if $\mathbb A$ is an infinite set in the
language of equality, then the only definable unary relations
are $\emptyset$ and $A$. They are definable since it is easy
to define them, while no other subset of $A$ is definable
since no other subset is invariant under all permutations
(=automorphisms) of $\mathbb A$.
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Feb 10
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We discussed the spectrum of complete $L$-theories,
following this handout.
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Feb 12
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We proved that the space of complete $L$-theories is
compact, Hausdorff and has a basis of clopen sets.
We examined the space of theories of the language of equality.
Our argument for showing that we had a complete description of this space
involved the concept of $\kappa$-categorical theory.
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Feb 14
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We calculated the spectrum of the language with
one unary relation. It is a countable space with
Cantor-Bendixson rank 2.
We discussed some background topology (separation
axioms $T_i$, isolated and nonisolated points,
Urysohn's Metrization Theorem, compact metric spaces are complete,
Cantor derivative, the Cantor-Bendixson
Theorem, Cantor-Bendixson rank, scattered spaces,
perfect kernel, Brouwer's Theorem characterizing the Cantor space
as the unique perfect Stone space with countable basis).
We pointed
out that the class of nonfinitely axiomatizable $L$-structures
is an elementary class, but the class of finitely axiomatizable
$L$-structures is not an elementary class.
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Feb 17
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We began to talk about convergence in
topological spaces in terms of filters.
Some concepts that were defined:
neighborhood, filter, nhood filter, cofinite filter,
finer/coarser filters, filter $\mathcal F$ converges to $p$
($\mathcal F\to p$).
We explained why the push-forward of a filter is a filter:
If $a: I\to X$ is a function and $\mathcal F$
is a filter on $I$, then
$a_*(\mathcal F) = \{U\subseteq A\;|\;a^{-1}(U)\in\mathcal F\}$
is a filter on $X$.
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Feb 19
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We discussed ultrafilters following these handouts:
I and
II.
The most important elements of these handouts are
(i) the definition of an ultrafilter, (ii)
the fact that any collections of sets satisfying the FIP
can be extended to an ultrafilter, and
(iii) the topology of any compact Hausdorff space
is completely determined by the ultrafilter limit
operations. (In this way, the category of compact Hausdorff spaces
may be thought of as a category of infinitary algebraic structures
where the operations are the ultrafilter limit operations.)
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Feb 21
|
We discussed this handout,
which begins with the canonical factorization of a function
$f=\iota\circ \overline{f}\circ n$, and continues through
a sequence of basic definitions. We discussed
morphisms of structures,
the universal properties of images and coimages,
substructures and quotient structures,
kernels and congruences.
We were interrupted by the bell while discussing
the universal property of products.
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Feb 24
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We discussed the construction of products
and ultraproducts.
We started discussing Los's Theorem.
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Feb 26
|
We sampled the proof of Los's Theorem and discussed
these exercises.
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Feb 28
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We proved the compactness theorem with ultraproducts.
Using ultrapowers we found a structure elementarily
equivalent to the ordered field $\mathbb R$ which has
a positive infinitesimal.
Using ultrapowers we found a structure elementarily
equivalent to the well-ordered set $\omega$ which has
a decreasing $\omega$-chain.
We defined the diagonal embedding of $\mathbb A$
into any ultrapower, and asserted that the embedding
is elementary.
We introduced regular ultrafilters,
explained why any infinite set supports a regular ultrafilter,
and asserted that it is easy to control the size
of an ultrapower over a regular ultrafilter.
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Mar 2
|
We stated the Keisler-Shelah Isomorphism Theorem.
We proved the Frayne-Morel-Scott
Theorem about sizes of ultrapowers over regular ultrafilters.
We pointed out why
Principal ultrafilters on infinite sets cannot be regular.
Nonprincipal ultrafilters on $\omega$ must be regular.
There exist regular ultrafilters on any infinite set.
There exist irregular ultrafilters on any uncountable set.
The diagonal map $\Delta: A\to \prod_{\mathcal U} A$
is always an injection. When $A$ is finite,
it is also a bijection. In fact, it is a bijection exactly when
$I$ cannot be partitioned into $\leq |A|$ measure zero sets.
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Mar 4
|
We discussed
The process of thinning the coordinate set
of an ultraproduct (if $J\subseteq I$ and $J\in{\mathcal U}$,
then $\prod_{\mathcal U} \mathbb A_i\cong
\prod_{{\mathcal U}|_J} \mathbb A_i$).
The fact that the diagonal map
$\Delta: \mathbb A\to \prod_{\mathcal U} \mathbb A$
is surjective iff ${\mathcal U}$ is closed under
$\leq |A|$-fold intersections. We discussed
the connection with measurable cardinals.
We defined the type of a tuple.
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Mar 6
|
We continued our discussion of types.
At one point we mentioned that asserting that a structure realizes
a type is the same as asserting that the structure satisfies
a certain infinitary sentence.
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Mar 9
|
We proved the Tarski-Vaught Test
and the Downward Lowenheim-Skolem Theorem.
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Mar 11
|
We proved the Upward L-S Theorem.
We discussed Skolem's Paradox and its resolution.
We proved the Los-Vaught Test for completeness.
We also mentioned the ``iff version''
of the Los-Vaught Test: If $\kappa\geq \|L\|$,
then $T$ is complete iff all models of $T$ of size $\kappa$
are elementarily equivalent.
Finally, we mentioned that the Los-Vaught Test can be used
to establish that $\textrm{ACF}_p$, $\textrm{DAG}$,
and $\textrm{DLO}$ are complete.
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Mar 13
|
We held our first remote meeting!
During the meeting we outlined the proof that every
algebraically closed field $\mathbb F$ of characteristic zero
has the
form $\overline{\mathbb Q(X)}$ where
$X$ is a transcendence base for $\mathbb F$
over the prime subfield $\mathbb Q$.
Moreover, $|\overline{\mathbb Q(X)}|=|X|+\omega$.
This shows that the theory of algebraically
closed fields of characteristic zero is
$\kappa$-categorical for all uncountable
$\kappa$, but is not $\omega$-categorical.
It also shows that a complete first-order
axiomatization of the field $\mathbb C$ of complex numbers is
given by:
axioms defining fields,
axioms saying the characteristic is zero, and
axioms saying that the field is algebraically closed.
Key points.
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Mar 16
|
Chapter 2 Roundup!
We explained why DAG is uncountably categorical
and why DLO is countably categorical.
(The $\omega$-categoricity of the theory of dense
linear orders without endpoints is called Cantor's Theorem.)
The proof of $\omega$-categoricity of DLO was
through a back and forth argument.
We derived some corollaries from the result:
The automorphism group of $\langle \mathbb Q; \leq \rangle$
acts transitively on $n$-tuples of the same order type,
hence the order type of any $n$-tuple determines
its elementary type. This also establishes that
$\langle \mathbb Q; \leq \rangle$ has only finitely
many $n$-types for every $n$.
Key points.
Make sure to enumerate the domain and codomain
before starting a back and forth argument.
The back and forth strategy for proving isomorphism
of countable structures $A$ and $B$ relies on the idea that
$A$ and $B$ should have $n$-tuples of the same type, and
whenever $\bar{a}\in A^n, \bar{b}\in B^n$ have the same type
we are able to extend these tuples to longer tuples of the same type:
for all $c\in A$ there should exist $d\in B$ such that $\bar{a}c$
has the same type as $\bar{b}d$, and
for all $d\in B$ there should exist $c\in A$ such that $\bar{a}c$
has the same type as $\bar{b}d$.
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Mar 18
|
We discussed quantifier
elimination.
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Mar 20
|
We finished the quantifier elimination handout.
We defined model completeness.
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Mar 30
|
We explained why the theory of $\mathbb R$
in the field language does not have
quantifier elimination (since the subset
of positive reals is definable in the field
language, but not q.f.-definable since
it is infinite and co-infinite).
Nevertheless this theory is model complete
(since $\mathbb R_{<}$ has q.e. and the
relation $<$ has both a $\Sigma_1$
and a $\Pi_1$ definition in the field language).
We ended by discussing a theorem which can be viewed
as saying ``A theory has q.e. iff the type
of a tuple in a model is determined by its q.f. part''.
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Apr 1
|
We discussed this.
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Apr 3
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We discussed realizing types.
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Apr 6
|
We discussed omitting types.
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Apr 8
|
We finished the proof of the Omitting
Types Theorem, and
outlined the proof of
the Craig Interpolation Theorem.
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Apr 10
|
We derived the Robinson Joint Consistency Theorem
from the Basic Interpolation Theorem, then discussed
the Beth Definability Theorem.
We then turned our attention to the project of
understanding models of complete
theories in a countable language
in terms of the spaces $S_n(T)$.
We proved the
the Atomic Model Theorem.
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Apr 13
|
We proved the uniqueness of atomic models
and then began discussing
saturated models.
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Apr 15
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We continued discussing
saturated models.
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Apr 17
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We continued discussing
saturated models.
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Apr 20
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We continued discussing
saturated models.
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Apr 22
|
We discussed
$\omega$-categoricity.
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Apr 24
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We discussed the situation when
$I(T,\aleph_0)$ is finite.
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Apr 27
|
We discussed
universal classes.
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Apr 27
|
We discussed
preservation theorems.
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