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.
We discussed some definitions of "model theory", and then began
discussing the Poincaré model of the hyperbolic plane.
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Jan 15
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We finished
our very brief discussion of the interpretation of the hyperbolic plane
into the Euclidean plane. The discussion informally introduced the
model-theoretic words and notation: structure, (first-order) sentence,
satisfaction, truth, (in)consistency, theory, $T\vdash \varphi$,
interpretation, equiconsistency. We then worked on
this exercise
about modeling mathematical objects as structures.
We ended with quick definitions of ordinals, function sets $A^B$,
and relations.
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Jan 17
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We defined
relations, predicates, functions and constants.
Then we defined "signature", "structure (of a given signature)",
and started on the definition of "language".
We got through the alphabet of symbols and the recursive
definition of "term", but got no further on the definition
of "formula".
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Jan 22
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We stated the
Recursion Theorem. We gave a recursive definition of "term"
and compared it to the book's definition of "term".
We discussed the unique readability of terms.
We defined atomic formulas.
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Jan 24
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We
finished the definition of "formula" and "sentence",
discussed what "language" should mean,
and indicated the recursive definitions of
"subterm", "subformula", "free variable" and
"quantifier rank". We practiced
creating some sentences and formulas.
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Jan 27
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Given a $\sigma$-structure $\mathbf A$,
a $\sigma$-formula $\varphi$ and a valuation
$v\colon \{x_0,\ldots\}\to A$ we defined
the meaning of $\mathbf A\models \varphi[v]$.
We briefly discussed the term algebra
and truth tables.
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Jan 29
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We proved the equivalence of some pairs of formulas;
in particular, enough pairs to enable us to
transform a formula into an equivalent one in prenex form.
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Jan 31
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We discussed the issues that must be considered
when constructing a proof calculus that is sound, complete and effective.
I circulated a document describing one such proof calculus.
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Feb 3
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We discussed: (i) why we define structures with
relations, functions and constants,
(ii) how to deal with multisorted structures,
(iii) why the empty structure is a problem, and
(iv) first-order logic versus second-order logic versus
the infinitary logics $L_{\kappa,\lambda}$.
(We showed that the completeness theorem fails for more
general logics if we require proofs to have finite length.)
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Feb 5
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Proof of the compactness theorem part 1:
extending a finitely satisfiable set $\Sigma$ to a
finitely satisfiable set $\Sigma'\supseteq \Sigma$
that satisfies the completeness condition and has witnesses.
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Feb 7
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Proof of the compactness theorem part 2:
a finitely satisfiable set of sentences
that satisfies the completeness condition and has witnesses
has a model. (We only got to the point where we defined the
universe of the model and the interpretations of the nonlogical
symbols of the language.)
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Feb 10
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Proof of the compactness theorem part 3: almost finished.
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Feb 12
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Finished the proof of the compactness theorem.
Discussed some corollaries.
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Feb 14
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Discussed more corollaries and some "applications"
of the compactness theorem.
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Feb 17
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Discussed ultrafilters.
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Feb 19
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We proved the ultrafilter convergence
theorem, defined ultraproducts
and stated Łos's Theorem.
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Feb 21
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A guest lecturer proved Łos's Theorem.
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Feb 24
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A guest lecturer defined "elementary map"
and proved that the diagonal embedding into an ultrapower
is elementary.
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Feb 26
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A class of structures is axiomatizable iff it is closed
under ultraproducts and elementary equivalence iff it is
closed under ultraproducts and the formation of elementary substructures.
Handout.
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Feb 28
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Comparison of $\mathbf A\leq \mathbf B$ and $\mathbf A\prec \mathbf B$.
Tarski-Vaught Criterion. Relationships between "model complete theory",
"theory with quantifier elimination" and "Skolem theory".
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Mar 3
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We showed that Skolem theories have quantifier elimination
and that every theory can be enlarged to a Skolem theory.
We derived the Downward Lowenheim-Skolem-Tarski Theorem.
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Mar 5
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Upward Lowenheim-Skolem-Tarski Theorem.
We discussed why an elementary substructure of a simple
group must be simple, but an elementary extension
need not be. We introduced the notion of an "elimination set"
and defined "quantifier elimination" (again).
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Mar 7
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We proved that the theory of infinite sets in the language
of equality has q.e.
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Mar 10
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We proved two theorems:
Diagram Lemma.
$\bf B_A$ is a model of the
(elementary) diagram of $\bf A$ iff
the mapping
$h\colon \bf A\to \bf B\colon
a^{\bf A}\mapsto a^{\bf B}$ is an (elementary) embedding.
Theorem.
Let $T$ be an $\mathcal L$-theory and let $\varphi({\bf x})$ be an
$\mathcal L$-formula. TFAE.
(1) $T\models \varphi({\bf x})\leftrightarrow \alpha({\bf x})$ for some
q.f. $\alpha({\bf x})$.
(2) For all $\bf B, \bf C\models T$, $\bf A\leq \bf B, \bf C$ and $\mathbf a\in A$, $\bf B\models \varphi[{\mathbf a}]$ iff
$\bf C\models \varphi[{\mathbf a}]$.
We used the theorem to show that if $T$ is any theory of fields
(in the language of rings) which has quantifier elimination,
then whenever $\bf B \leq \bf C$ and both are models
of $T$, then $\bf B$ is relatively algebraically closed
in $\bf C$. (I.e., any element of $\bf C$
that is algebraic over $\bf B$ lies in $\bf B$.)
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Mar 12
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We explained why ACF has q.e. in the language of rings
and why RCF does not have q.e. in the language of rings.
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Mar 14
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We described all theories extending
$\textrm{ACF}$, in particular the complete extensions are
$\textrm{ACF}_p$ and $\textrm{ACF}_0$.
A complete first-order axiomatization of the field of complex numbers
in the language of rings can be obtained from this.
Namely, a set of axioms for $\textrm{Th}(\bf C)$
consists of the sentences asserting
that it is an algebracally closed field of characteristic zero.
We also showed that
any definable subset of an algebraically closed field is finite or
cofinite.
We explained why $\textrm{RCF}$ is complete, thereby obtaining a first-order
axiomatization of the field of real numbers.
Namely, a basis for its theory consists of the sentences asserting
that it is a field whose nonzero squares form a positive cone for a
strict linear ordering
and that the intermediate value theorem holds.
Along the way we defined "algebraically prime structure",
"algebraically prime model", "having algebraically prime
models", and "prime model". We observed that if $T$ has q.e.
and $T$ has an algebraically prime structure,
then $T$ is complete.
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Mar 17
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We defined types and looked at some examples.
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Mar 19
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Partial types consistent with a theory can be realized in a model.
Supported partial types consistent with a complete theory
must be realized in every model. We started on the proof of the Omitting
Types Theorem.
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Mar 21
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We finished the proof of the Omitting
Types Theorem. I gave an example showing why "countable language"
is necessary. I briefly outlined the difference
needed in statement of the theorem
for uncountable languages. (Our book doesn't have this result,
but see Theorem 2.2.19 of Chang-Keisler.) We explained why
$S_n(T)$ is finite if $T$ is $\aleph_0$-categorical
in a countable language.
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Mar 31
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We explained why (when $T$ is a complete theory in a countable language)
the space $S_n(T)$ has cardinality $2^{\aleph_0}$ or else
cardinality $\leq \aleph_0$. We also explained why, in the first case
(when $|S_n(T)|=2^{\aleph_0}$), the theory $T$ must have
the maximum number of models. The explanations involved
the Urysohn Metrization Theorem, the notions of Polish space
and perfect set,
and the Cantor-Bendixson Theorem.
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Apr 2
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We defined atomic theory and atomic model.
We proved that if $T$ is a complete theory in a countable language,
then $T$ is atomic iff $T$ has a countable atomic model.
We stated a theorem asserting that if ${\bf A}$ and ${\bf B}$
are countable atomic models with the same theory, and they
are back and forth equivalent, then ${\bf A}\cong{\bf B}$.
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Apr 4
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We described Ehrenfeucht's example, back and forth equivalence,
the fact that countable atomic models of a complete theory
are isomorphic and prime.
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Apr 7
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We defined categoricity. We proved that Vaught's completeness
criterion. We proved the Engeler, Ryll-Nardzewski, Svenonius Theorem
characterizing $\aleph_0$-categorical structures.
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Apr 9
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We reviewed some recent topics, then presented Ehrenfeucht's
theories for which $I(T,\aleph_0) = n$ for $n$ equal to
any natural number $\geq 3$. We outlined the proof of Vaught's
Theorem that $I(T,\aleph_0) \neq 2$.
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Apr 11
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We proved that a reduct of an $\aleph_0$-categorical structure
is again an
$\aleph_0$-categorical structure, as is an expansion by finitely
many constants. We defined "$\kappa$-saturated structure"
and gave examples.
We fleshed out our earlier proof-sketch of Vaught's
Theorem that $I(T,\aleph_0) \neq 2$.
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Apr 14
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We proved that two definitions of "$\kappa$-saturated" are equivalent.
(One defn involved the realization of all $n$-types over a set of size
$<\kappa$, while the other involved only the realization of $1$-types.)
We then characterized $\kappa$-saturated models by the
property that an elementary partial map into
a $\kappa$-saturated structure can be properly extended
if its domain has size $<\kappa$.
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Apr 16
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We showed that $\kappa$-saturated = $\kappa$-homogeneous +
$\kappa^+$-universal. We explained why short tuples
of the same type in a $\kappa$-homogeneous model
differ by an automorphism. (Short means: $|{\bf a}|<\kappa$.)
We identified which models of
Ehrenfeucht's 3-model theory are
$\kappa$-saturated, $\kappa$-homogeneous and
$\kappa^+$-universal.
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Apr 18
|
We started the proof that a complete theory $T$ in a countable
language has a countable $\omega$-saturated model
iff $S_n(T)$ is countable for all $n$.
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Apr 21
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We finished the proof that a complete theory $T$ in a countable
language has a countable $\omega$-saturated model
iff $S_n(T)$ is countable for all $n$. In particular,
any complete theory in a countable language that has fewer
than the maximum number of countable models
has a countable $\omega$-saturated model. We noted that
even some theories with the maximum number
of models might have countable type spaces (hence
a countable $\omega$-saturated model),
such as DLO + constants naming an increasing
$\omega^2$-chain.
We also showed that
if ${\bf A}$ is $\kappa$-saturated and
${\bf u}$ is from $A$ and $|{\bf u}|<\kappa$, then
$A_{\bf u}$ is $\kappa$-saturated. We noted
that the corresponding result for $\kappa$-universal
models is false.
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Apr 23
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We discussed (without proving) Morley's Theorem
that a theory in a countable language that is
$\kappa$-categorical for some uncountable $\kappa$ is
$\kappa$-categorical for all uncountable $\kappa$.
We then discussed Vaught's Conjecture, and began
the proof of Morley's theorem that a complete theory
in a countable language has either
$\omega_0, \omega_1$ or $2^{\omega_0}$-many models.
In this lecture we only introduced ${\mathcal L}_{\kappa,\lambda}$ and
mentioned Scott's Isomorphism Theorem.
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Apr 25
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We defined "fragments" of ${\mathcal L}_{\omega_1,\omega}$
and "scattered theory".
To any scattered theory $T$ in a countable language
$\mathcal L$ we defined a sequence of
fragments ${\mathcal L}_{\alpha}$, $\alpha<\omega_1$,
and used them to define $\textrm{tp}^{\mathbf A}_{\alpha}(\bar{a})$.
We proved that every countable model of $T$
has a countable ordinal "height".
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Apr 28
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We proved that if $T$ is a scattered theory in a countable
language, then $T$ has at most $\omega_1$-many countable models.
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Apr 30
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We proved that if $S_n(F,T)$ has size continuum for some
countable fragment $F$, then $T$ has the maximum
number of countable models.
We defined Borel and analytic sets, and started on a proof
that $S_n(F,T)$ is analytic.
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May 2
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We completed the proof that $S_n(F,T)$ is analytic.
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