Date
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What we discussed/How we spent our time
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Jan 17
|
Syllabus. Text. HW.
We introduced the concept
of an abstract logic $(\mathcal{K},\Sigma,\models)$
and indicated that a first-order logic
can be generated from an appropriate
alphabet of nonlogical symbols.
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Jan 19
|
Room change! Chem 145 flooded. We will meet in
Math 350 when we return to in-person meetings.
We followed these slides,
which identify the main ingredients of Model Theory.
Then we started discussing this handout
concerning the construction of first-order formulas.
(We only covered up to the definition of atomic formulas.)
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Jan 22
|
We will meet in Math 350 starting
Wednesday January 24.
We finished these slides.
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Jan 24
|
We defined (algebraic) closure operator
and Galois connection. The main result
on Galois connections is Theorem 5 of
this handout.
The Galois connection that is of importance for us is the
GC between syntax and semantics for a first-order language $L$.
Given a set $U$ of $L$-sentences, $U^{\perp}$
is the class of all $L$-structures that satisfy $U$.
Given a class $X$ of $L$-structures, $X^{\perp}$
is the set of all $L$-sentences true throughout the class
$X$.
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Jan 26
|
We briefly described the Galois Connection
of Galois Theory and then the GC of
Algebraic Geometry. I identified three 'projects'
one has once an GC has been identified:
Give an internal description of Galois Closure.
(E.g., Hilbert's Nullstellensatz, Gödel's
Completeness Theorem)
Describe the lattice of Galois-closed subsets.
Exploit the Galois Correspondence.
We discussed the first four frames of
these slides.
We circulated (but did not yet discuss)
this handout.
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Jan 29
|
We got to the middle of slide 9 of
these slides.
This involved discussing the Deduction Theorem,
Lindenbaum's Theorem, and Henkin theories.
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Jan 31
|
We finished the proof of the
Completeness
Theorem. The proof showed that if $\Sigma$
is a consistent set of sentences
in a signature of size $\kappa$,
then $\Sigma$ has a model of size at most $\kappa+\omega$.
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Feb 2
|
We discussed the first HW assignment briefly,
then had a vocabulary review.
Finally, we discussed the Compactness Theorem
and some of its applications.
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Feb 5
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We discussed the 4th application
of the Compactness Theorem.
Then we discussed
the space of complete theories
in a given signature.
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Feb 7
|
As an exercise, we reconstructed a finite topological space from its
frame of open sets.
Then we completed the slides on
the space of complete theories
in a given signature.
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Feb 9
|
We started discussing homomorphisms
and constructions of new models
following
these slides.
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Feb 12
|
We discussed embeddings,
congruences, quotient structures and product
structures following
these slides.
At one point I posed the problem of finding
all the quotients (up to isomorphism) of the symmetric graph
$\langle V ; E(x, y)\rangle$ that is a 4-element path.
We calculated in class
that there are 15 equivalence relations on a 4-element set
and computed some of the quotients with respect to these
equivalence relations. The complete
list of isomorphism types of quotients should contain
1 graph of size 4, 4 graphs of size 3, 3 graphs of size 2, and 1
graph of size 1, for a total of 9 isomorphism types of quotients.
At another point I asked for a description of
the product of two symmetric, loopless graphs, which are
both 3-vertex paths. The answer is a graph with 2 connected components
where one is a loopless 4-cycle and the other is a loopless
graph of size 5 with one vertex of degree
4 and four vertices of degree 1.
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Feb 14
|
Today we discussed ultrafilter convergence.
We defined the following:
(principal, nonprincipal) filter, ultrafilter
filter convergence (a filter $\mathcal{F}$ converges
to a point $p$ of a space $X$ if $\mathcal{F}$
contains the neighborhood filter of $p$)
reduced product, ultraproduct.
We discussed relevant examples, too, including
(i) a space $X$ of size $\kappa$ whose open sets
are the subsets $U\subseteq X$ with $|X\setminus U| <\lambda$,
and (ii) a successor ordinal in the order topology.
Both examples show that $\omega$-indexed sequences
are insufficient to define convergence.
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Feb 16
|
Today we discussed
ultraproducts and
Łos’s Theorem.
We discussed the possible isomorphism types of an ultraproduct
of the family $\{\mathbb Z_n\;|\;n\in \mathbb N^+\}$.
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Feb 19
|
Today we discussed
why a collection
of subsets of a set $I$ which has the finite
intersection property generates a proper filter on $I$,
the fact that a proper filter on $I$ can be extended
to an ultrafilter on $I$,
the relationship (= definitional equivalence)
between the Boolean algebra of truth values,
$\mathbb B_2 = \langle \{0,1\}; \wedge,\vee,\neg,0,1\rangle$
and the $2$-element ring/field
$\mathbb F_2 = \langle \{0,1\}; \cdot,+,\neg,0,1\rangle$.
Under this equivalence, $a\wedge b = ab$, $0=0$, and $1=1$, while
$a\vee b = a+b+ab, \neg a = 1-a$, and
$a+b = (a\wedge \neg b)\vee (b\wedge \neg a), -a = a$.
This yields a correspondence between
the Boolean algebra ${\mathcal P}(I)\cong 2^{|I|}$
and the commutative ring $\mathbb F_2^{|I|}$.
Under this correspondence, ideas from commutative ring
theory can be imported into Boolean algebra.
For example, one may derive the Ultrafilter Lemma
from the theorem of ring theory that asserts that every ideal
can be extended to a maximal ideal.
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Feb 21
|
We proved that
A filter is an ultrafilter iff it is maximal
among proper filters.
Any proper filter can be extended to an ultrafilter.
A nonprincipal ultraproduct of the form
$\prod_{\mathcal{U}} \mathbb Z_p$,
where $\mathcal U$ is an ultrafilter on the set
of prime numbers,
will be an infinite, torsion-free, divisible, abelian group.
I mentioned that the theory of
infinite, torsion-free, divisible, abelian groups
is $\kappa$-categorical for uncountable $\kappa$,
but we haven’t proved that statement yet.
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Feb 23
|
We proved that the group
$\prod_{\mathcal U} \mathbb Z_p$
is isomorphic to $\bigoplus^{2^{\aleph_0}} \mathbb Q$
when $U$ is a nonprincipal ultrafilter
on the set of primes.
This involved the observation that $\prod_{\mathcal U} \mathbb Z_p$ is a
torsion-free, divisible, abelian group
of cardinality $2^{\aleph_0}$ when $\mathcal U$
is nonprincipal, and that
every torsion-free, divisible, abelian group
is a reduct of a uniquely determined
rational vector space.
We then discussed why the group
$\prod_{\mathcal U} \mathbb Z_n$
will also be
isomorphic to $\bigoplus^{2^{\aleph_0}} \mathbb Q$
for some choices of $\mathcal U$, but
for other choices of $\mathcal U$ it is not hard
to see that the group
$\prod_{\mathcal U} \mathbb Z_n$ will have
a nontrivial torsion subgroup.
During the lecture we made reference to the main result of
Saharon Shelah,
On the cardinality of ultraproducts of finite sets
Journal of Symbolic Logic, 1970.
One consequence of this paper is the fact that
the class of ultraproducts of finite sets is not
elementary. This implies that the process of generating an elementary
class is more complicated than simply closing under ultraproducts.
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Feb 26
|
We reviewed past material (definition of a logic,
the associated Galois connection, definition of a frame,
the space of complete $L$-theories is a compact
Hausdorff zero-dimensional space,
ultrafilter convergence in the space of complete $L$-theories).
Then we discussed the first three
slides
about the topological structure of the space
of complete $L$-theories.
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Feb 28
|
We proved the Cantor-Bendixson Theorem
(existence of decomposition only)
following
these slides.
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March 1
|
Ultraproducts roundup!.
During this discussion, we referred to
$\kappa$-complete ultrafilters
and to regular ultrafilters.
|
March 4
|
We discussed elementary diagrams,
the upward Lowenheim-Skolem Theorem,
and the Los-Vaught Test for completeness.
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March 6
|
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, $|\overline{\mathbb Q(X)}|=|X|+\aleph_0$.
This shows that the theory of algebraically
closed fields of characteristic zero is
$\kappa$-categorical for all uncountable
$\kappa$, but is not $\aleph_0$-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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March 8
|
During the meeting we reviewed the argument
that the theory $\textrm{ACF}_0$ of algebraically closed
fields of characteristic $0$ is $\kappa$-categorical
for uncountable $\kappa$. By the Los-Vaught Test,
$\textrm{ACF}_0$ is complete, and hence
$\textrm{Th}(\mathbb{C})=\text{ACF}_0$.
Then we introduced the theory $\text{RCF}$ of
real closed fields and outlined the main ideas
needed to prove that this theory has an algebraically prime model
$\mathbf{P}=\overline{\mathbb{Q}}\cap\mathbb R$.
This algebraically prime model
is actually a prime model for the theory, because
$\text{RCF}$ has quantifier elimination.
The existence of a prime model is sufficient to establish
the completeness of $\text{RCF}$, so $\textrm{Th}(\mathbb{R})=\text{RCF}$.
Finally, we introduced the theory $\textrm{DLO}$
of dense linear orders without endpoints,
and used the Back and Forth method
to prove that this theory is $\aleph_0$-categorical
and complete. Hence
$\textrm{Th}(\langle \mathbb{Q}; <\rangle)=\text{DLO}$.
We pointed out that
$\textrm{DLO}$ is not $\kappa$-categorical
for $\kappa=|\mathbb R|$.
|
March 11
|
Let $L$ be the language of one unary predicate $E(x,y)$.
A graph is a structure $\langle V; E(x,y)\rangle$
in this language satisfying axioms that say that
$E(x,y)$ is an irreflexive symmetric relation.
For a graph $G=\langle V; E(x,y)\rangle$,
if $U, W\subseteq V$ are finite sets satisfying
$|U|=m, |W|=n, U\cap W=\emptyset$, then we say that $z\in V$
is correctly joined to $(U,W)$ if
$z$ is adjacent to every element of $U$ and to no element
of $W$. Let $\varphi_{m,n}$ be the sentence
expressing that $\forall (U,V)$, if
$|U|=m, |W|=n, U\cap W=\emptyset$, then $\exists z\notin U\cup W$
that is correctly joined to $(U,W)$. Let $T$ be the theory
axiomatized by all $\varphi_{m,n}$, $m,n\geq 0$.
We explained why $T$ is $\aleph_0$-categorical
and has no finite models, hence $T$ is complete.
We described a construction of a countably infinite
model $R$ of $T$, which we called the Random Graph.
We observed that every countable graph is embeddable
in the Random Graph, and that any isomorphism
between finite subgraphs of $R$ extends to an automorphism
of $R$.
We made references to related results that we did not explore.
Let me list some of these here:
- The Random Graph is called random because
it can be constructed by a random process:
Start with vertex set $V=\mathbb N$ and choose
edges randomly by flipping a fair coin for each $(i,j)\in \mathbb N^2$
and defining $E(i,j)=\top$ if the result is heads and
$E(i,j)=\bot$ if the result is tails. The resulting graph
satisfies any given sentence $\varphi_{m,n}$ with probability $1$.
- The symmetrization of a
countable directed graph model of ZFC
is random. (Here, if $V$ is a countable model of ZFC,
then say that $a, b\in V$ are adjacent
in the symmetrization of the associated directed graph model
provided $a\in b$ or $b\in a$.)
- Let $V$ be the set of primes that are congruent to $1$ modulo $4$.
Say that $p$ is adjacent to $q$ if $p$ is a nonzero square modulo
$q$ (i.e., if $p\neq q$ and
the Legendre symbol $\left(\frac{p}{q}\right)$
evaluates to $1$). The resulting graph
is random. [In this example, I am restricting to the set of primes
that are $1$ modulo $4$ to ensure that the edge relation
is symmetric.]
- The theory of the Random Graphs satisfies a
zero-one law:
If $\sigma$ is a sentence in the language of graphs
and $P_n(\sigma)$ is the proportion of graphs with vertex set
$V=\{0,1,\ldots,n-1\}$
that satisfy $\sigma$, then $\lim_{n\to \infty} P_n(\sigma)$
exists and it is either $0$ or $1$. This limit is $1$
if $\sigma\in \textrm{Th}(R)$ and is $0$
if $\neg \sigma\in \textrm{Th}(R)$. For this reason,
we call $\textrm{Th}(R)$ the almost sure theory
of the class of graphs.
- All the results mentioned above about the Random Graph
can be generalized to the class of all structures
in a given finite relational signature. The results do not hold
for signatures with operations. In a finite signature with at least
operation of arity at least $2$, there will be a sentence $\sigma$
such that the limit $\lim_{n\to \infty} P_n(\sigma)$
does not exist.
In a finite signature where all operations have arity $1$,
the limit $\lim_{n\to \infty} P_n(\sigma)$ will always exist,
but might be strictly between $0$ and $1$.
|
March 13
|
We reviewed the Back and Forth method
which we applied to show that the theories
of (i) the ordered set
of rational numbers and (ii) the random graph
are $\aleph_0$-categorical. We isolated the
key element that allowed us to derive our conclusions, namely that
the structures (i) and (ii) are $\aleph_0$-saturated.
This led to a discussion of types
following these slides.
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March 15
|
Snow Day!
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March 18
|
We continued discussing these slides
(up to, but not including, Vaught’s Corollary).
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March 20
|
We discussed Vaught’s Corollary, the Downward LS Theorem,
and Skolem’s Paradox to complete these slides.
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March 22
|
Read Section 3.1.
We introduced quantifier elimination
following these slides.
Some key points were:
- In theories $T$ with q.e., every embedding between
models of $T$ is elementary.
- q.e. helps to establish completeness of $T$.
- q.e. implies $\aleph_0$-categoricity for the
complete theory of an infinite structure in a finite relational language.
- To establish q.e. for $T$ by brute force, it suffices
to show how to eliminate a single $\exists y$ from formulas of the
form $(\exists y)\varphi(\mathbf{x},y)$ where
$\varphi(\mathbf{x},y)$ is a primitive q.f. formula.
|
April 1
|
Read Section 3.1.
I announced that I will be at a 4-day conference,
and plan to post videos for
the three lectures: April 5, April 8, April 10.
We reviewed the brute force method of showing that
a theory has q.e. Then we began
these slides
which begin to develop alternative
methods for establishing that theory has q.e.
Our discussion included a discussion of atomic
diagrams and a proof of the Diagram Lemma.
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April 3
|
I announced that I will be at a 4-day conference,
and plan to meet by Zoom on April 5 and post videos for
the lectures on April 8, April 10.
Read Sections 3.1 and 3.2.
We completed the proof of
the local characterization
of q.e., and also explained why
an $L$-theory $T$ has q.e. if and only if, whenever
$\mathbf{A}$ is a substructure of a model
of $T$, the $L_A$-theory
$T\cup \textrm{Diag}(\mathbf{A})$ is complete.
We introduced algebraically prime extensions,
and explained why the theory of fields and the
theory $\textrm{ACF}_0$ have algebraically prime extensions.
(E.g., if $\mathbb D$ is a substructure of a model of
$\textrm{ACF}_0$, then $\mathbb D$ is an integral domain
of characteristic zero. The algebraic closure $\mathbb D^*$
of the field of fractions of $\mathbb D$ is
algebraically prime over $\mathbb D$ with respect
to $\textrm{ACF}_0$.)
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April 5
|
Meeting ID: 946 8655 4534
Read Section 3.2.
We completed
these q.e. slides. For applications,
we established that the theory of $\mathbb C$ has q.e.,
while the theories of $\mathbb R$ and $\mathbb Q$ do not.
|
April 8
|
Meeting ID: 946 8655 4534
Read Sections 4.1 and 4.2.
We reviewed the concept of a type following
these slides.
This review included showing that any
$n$-type of $\mathbf{A}$ can be
realized in an elementary extension of $\mathbf{A}$.
We started discussing the Omitting Types Theorem following
these slides.
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April 10
|
Meeting ID: 946 8655 4534
Read Section 4.2.
We finished these slides
which discuss the proof of the Omitting Types Theorem.
|
April 12
|
We discussed the topological space $S_1(T)$
where $T$ is the elementary diagram of
$\langle \mathbb Q; < \rangle$.
|
April 15
|
Read Section 4.2.
We proved the Atomic Model Theorem.
|
April 17
|
Read Sections 4.2 and 4.3.
We finished the slides on atomic models
(handout form)
and started the slides on
saturated models
(handout version).
|
April 19
|
Read Section 4.3.
We got to slide 10 of
saturated models
(handout version).
|
April 22
|
Read Section 4.3.
We continued the discussion of
saturation
(handout version),
introducing the concepts of
- $\omega$-saturation,
- (strong and ordinary) $\omega$-homogeneity,
- $\omega^+$-universality.
|
April 24
|
Read Section 4.3.
We discussed the existence and uniqueness
of $\omega$-saturated models,
and explained why an algebraically closed
field of characteristic $p$ is saturated
iff it has an infinite transcendence base.
|
April 26
|
Read Section 4.4.
We proved the theorem of
Engeler, Ryll-Nardzewski, and Engeler
(handout version)
providing many characterizations of
$\omega$-categorical theories and models.
|
April 29
|
Read Section 4.4.
We proved
Vaught‘s Theorem
(handout version)
that proves that there is no complete theory
in a countable language with exactly
two isomorphism types of countable models.
We then discussed Vaught‘s Conjecture,
Morley‘s Theorems and the Baldwin-Lachlan Theorem.
|
May 1
|
We spent the hour finalizing HW solutions.
|