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Mathematical Logic: A Course with Exercises Part II: Recursion Theory, Godel's Theorems, Set Theory, Model Theory

Rene Cori, Daniel Lascar and Donald H. Pelletier

Chapter 8

Some model theory - all with Video Answers

Educators

Chapter Questions

23:32

Problem 1

Consider the first-order language $L$ that has two unary predicate symbols $E$ and $P$ and one binary predicate symbol $A$. Let $T$ be the theory of $L$ consisting of the following formulas:
$$
\begin{aligned}
&H_{0}: \forall v_{0}\left(E v_{0} \Leftrightarrow \neg P v_{0}\right) \\
&H_{1}: \forall v_{0} \forall v_{1}\left(A v_{0} v_{1} \Rightarrow\left(E v_{0} \wedge P v_{1}\right)\right) \\
&H_{2}: \forall v_{1} \forall v_{2}\left(\left(P v_{1} \wedge P v_{2} \wedge \forall v_{0}\left(A v_{0} v_{1} \Leftrightarrow A v_{0} v_{2}\right)\right) \Rightarrow v_{1} \simeq v_{2}\right) \\
&H_{3}: \exists v_{0}\left(P v_{0} \wedge \forall v_{1} \neg A v_{1} v_{0}\right) \\
&H_{4}: \forall v_{1}\left(P v_{1} \Rightarrow \exists v_{2}\left(P v_{2} \wedge \forall v_{0}\left(E v_{0} \Rightarrow\left(A v_{0} v_{1} \Leftrightarrow \neg A v_{0} v_{2}\right)\right)\right)\right) \\
&H_{5}: \forall v_{1} \forall v_{2} \exists v_{3}\left(\left(P v_{1} \wedge P v_{2}\right) \Rightarrow \forall v_{0}\left(A v_{0} v_{3} \Leftrightarrow\left(A v_{0} v_{1} \vee A v_{0} v_{2}\right)\right)\right)
\end{aligned}
$$
and, for every integer $n \geq 1$, the formula
$$
\begin{aligned}
F_{n}=& \forall v_{1} \forall v_{2} \ldots \forall v_{n}\left(\left(\bigwedge_{1 \leq i \leq n} E v_{i}\right)\right.\\
&\left.\Rightarrow \exists w_{1} \forall w_{0}\left(A w_{0} w_{1} \Leftrightarrow\left(\bigvee_{1 \leq i \leq n} w_{0} \simeq v_{i}\right)\right)\right)
\end{aligned}
$$
(a) Let $X$ be a non-empty set and $\wp(X)$ be the set of subsets of $X$, which we assume to be disjoint from $X$. We define an $L$-structure $\mathcal{M}$ as follows:
- the base set is $M_{X}=X \cup \wp(X)$;
- the interpretation of $E$ is $X$;
- the interpretation of $P$ is $\wp(X)$;
- the interpretation of $A$ is the set
$$
\bar{A}=\left\{(x, y) \in M_{X}^{2}: x \in X, y \in \wp(X) \text { and } x \in y\right\} .
$$
Show that $\mathcal{M}$ is a model of $T$.
(b) Does $T$ have a denumerable model?
(c) For which integers $n$, does $T$ have a model whose base set has cardinality $n$ ?
(d) Show that $T$ is equivalent to $\left\{H_{0}, H_{1}, H_{2}, H_{3}, H_{4}, H_{5}, F_{1}\right\}$.
(e) Show that $T$ is not $\aleph_{0}$-categorical.

Chris Trentman
Chris Trentman
Numerade Educator
26:40

Problem 2

This is a follow-up to Exercise 15 from Chapter $3 .$
(a) Let $\mathcal{M}=\langle M, \bar{d}, \bar{g}\rangle$ be an arbitrary model of $T$. Define a binary relation $\approx$ on $M$ by
$$
\begin{array}{r}
a \approx b \quad \text { if and only if } \quad \text { there exist integers } m, n, p, \text { and } q \\
\text { such that } \bar{d}^{m}\left(\bar{g}^{n}(a)\right)=\bar{d}^{p}\left(\bar{g}^{q}(b)\right) .
\end{array}
$$
Show that $\approx$ is an equivalence relation on $M$. An equivalence class under $\approx$ will be called a grill. Show that every grill is stable for $\bar{d}$ and for $\bar{g}$. Show that every grill, together with the restrictions of the mappings $\bar{d}$ and $\bar{g}$, is a substructure of $\mathcal{M}$ that is a model of $T$.
(b) Let $L^{\prime}$ be the language obtained by adding two new constant symbols $\lambda$ and $\mu$ to $L$. For every four-tuple $(m, n, p, q)$ of natural numbers, let $G_{m n p q}$ denote the following closed formula of $L^{\prime}$ :
$$
\neg d^{m} g^{n} \lambda \simeq d^{p} g^{q} \mu .
$$
Using this family of formulas, prove the existence of a non-standard model of $T$ (i.e. a model of $T$ that is not isomorphic to the standard model).
(c) Let $A$ be a non-empty set. Construct a model of $T$ whose set of grills is equipotent with $A$.
(d) Show that two models of $T$ whose sets of grills are equipotent are isomorphic.
(e) Show that $T$ is not $\aleph_{0}$-categorical. Consider a set $\mathcal{X}$ of $L$-structures that has the following properties:
- the elements of $\mathcal{X}$ are denumerable models of $T$;
- if $\mathcal{M} \in \mathcal{X}, \mathcal{N} \in \mathcal{X}$, and $\mathcal{M} \neq \mathcal{N}$, then $\mathcal{M}$ and $\mathcal{N}$ are not isomorphic;
- every denumerable model of $T$ is isomorphic to one of the elements of $\mathcal{X}$. What is the cardinality of $\mathcal{X}$ ?
(f) Let $\kappa$ be an uncountable cardinal. Show that $T$ is $\kappa$-categorical.

Chris Trentman
Chris Trentman
Numerade Educator
23:32

Problem 3

Let $\langle G, \cdot e\rangle$ be a group. With this group, we associate a first-order language $L_{G}$ that has a unary function symbol $f_{\alpha}$ for every element $\alpha \in G$. Let $T$ denote the following theory of $L_{G}$ :
$$
\begin{gathered}
\left\{\forall v_{0} f_{e} v_{0} \simeq v_{0}\right\} \\
\cup\left\{\forall v_{0} f_{\alpha} f_{\beta} v_{0} \simeq f_{\alpha \beta} v_{0}: \alpha \in G \text { and } \beta \in G\right\} \\
\cup\left\{\forall v_{0} \neg f_{\alpha} v_{0} \simeq v_{0}: \alpha \in G \text { and } \alpha \neq e\right\} .
\end{gathered}
$$
(a) Show that for every term $t$ of $L_{G}$, there exists an element $\alpha \in G$ and a symbol $x$ for a variable such that
$$
T \vdash \forall x t \simeq f_{\alpha} x
$$
(b) After observing that any atomic formula of $L_{G}$ can involve at most two variables, show that, for every atomic formula $F=F\left[v_{0}, v_{1}\right]$ of $L_{G}$, one of the following three possibilities holds:
- $T \vdash \forall v_{0} \forall v_{1} F$;
- $T \vdash \forall v_{0} \forall v_{1} \neg F$;
- there exists an element $\alpha \in G$ such that $T \vdash \forall v_{0} \forall v_{1}\left(F \Leftrightarrow v_{0} \simeq f_{\alpha} v_{1}\right)$.
(c) Let $\mathcal{G}$ be the $L$-structure whose base set is $G$ and in which, for every $\alpha \in G$, the symbol $f_{\alpha}$ is interpreted by the map $\beta \mapsto \alpha \cdot \beta$ from $G$ into $G$ (i.e. left multiplication by $\alpha$ ). Show that $\mathcal{G}$ is a model of $T$.
(d) Let $\mathcal{M}=\left\langle M,\left(\phi_{\alpha}\right)_{\alpha \in G}\right\rangle$ be a model of $T$ and let $a$ be an element of $M$. Consider the set
$O(a)=\left\{x \in M:\right.$ there exists $\alpha \in G$ such that $\left.x=\phi_{\alpha}(a)\right\}$
Show that $O(a)$ is a substructure of $\mathcal{M}$ that is isomorphic to $\mathcal{G}$.
Show that $X_{M}=\{O(a): a \in M\}$ is a partition of $M$.
Show that if $\mathcal{M}$ and $\mathcal{N}$ are two models of $T$ and if $X_{M}$ and $X_{N}$ are equipotent, then $\mathcal{M}$ and $\mathcal{N}$ are isomorphic.
(e) Show that if $G$ is infinite, then the theory $T$ is complete.
(f) Suppose that $G$ is finite. Does there exist an infinite cardinal $\lambda$ such that $T$ is $\lambda$-categorical? Is the theory $T$ complete?

Chris Trentman
Chris Trentman
Numerade Educator
26:40

Problem 5

Consider the language $L$ that consists of a single binary relation symbol $R$. Let $L_{\infty}$ denote the language obtained by adding denumerably many new constant symbols $c_{0}, c_{1}, \ldots, c_{n}, \ldots$ to $L$.
For each integer $n$, let $L_{n}$ denote the language $L \cup\left\{c_{0}, c_{1}, \ldots, c_{n}\right\}$.
Given an $L_{\infty}$-structure $\mathcal{M}$ and an integer $n$, let $\mathcal{M}_{n}$ denote the reduct of $\mathcal{M}$ to the language $L_{n}$.

Consider the theory in the language $L$ expressing that the interpretation of $R$ is an equivalence relation that has infinitely many equivalence classes, each of which is infinite.
(a) Write down axioms for the theory $T$ and give an example of a model of $T$.
(b) Show that $T$ is not equivalent to any finite theory in $L$.
(c) For which infinite cardinals $\lambda$ is the theory $T \lambda$-categorical? Find two models $\mathcal{M}_{1}$ and $\mathcal{M}_{2}$ of $T$ such that there is no elementary embedding from $\mathcal{M}_{1}$ into $\mathcal{M}_{2}$ or from $\mathcal{M}_{2}$ into $\mathcal{M}_{1}$.
(d) Is the theory $T$ complete?
(e) Let $T_{+}$be the following theory in the language $L_{\infty}$ :
$$
T_{+}=T \cup\left\{\neg R c_{n} c_{m}: n \in \mathbb{N}, m \in \mathbb{N} \text { and } n \neq m\right\} .
$$
Give an example of a model of $T_{+}$.
Show that $T_{+}$is not equivalent to any finite theory in the language $L_{\infty}$.
(f) For which infinite cardinals $\lambda$ is the theory $T_{+} \lambda$-categorical?
(g) Let $\mathcal{M}_{1}$ and $\mathcal{M}_{2}$ be two denumerable models of $T_{+}$. Show that for eyery integer $n$, the reducts of $\mathcal{M}_{1}$ and of $\mathcal{M}_{2}$ to the language $L_{n}$, which we will denote by $\mathcal{M}_{1}\left\lceil L_{n}\right.$ and $\mathcal{M}_{2} \mid L_{n}$, are isomorphic. Conclude from this that $T_{+}$is a complete theory in $L_{\infty}$.

Chris Trentman
Chris Trentman
Numerade Educator
02:44

Problem 6

Consider the languages $L_{1}=\{f\}$ and $L_{2}=\{f, P\}$, where $f$ is a unary function symbol and $P$ is a unary relation symbol. Let $T_{1}$ denote the following theory in $L_{1}$ :
$$
\begin{aligned}
\{\forall x \forall y(f x \simeq f y \Rightarrow x \simeq y)\} & \cup\{\forall x \exists y f y \simeq x\} \\
& \cup\left\{\forall x \neg f^{n} x \simeq x: n \in \mathbb{N}^{*}\right\}
\end{aligned}
$$
[The term $f^{n} x$ is defined as usual: $f^{0} x=x$ and, for all $n \in \mathbb{N}, f^{n+1} x=$ $\left.f\left(f^{n} x\right) .\right]$
EXERCISES FOR CHAPTER 8
233
Let $T_{2}$ denote the following theory in $L_{2}$ :
$$
T_{1} \cup\{\exists x P x, \quad \exists x \neg P x, \quad \forall x(P x \Leftrightarrow P f x)\}
$$
(a) Show that $T_{1}$ is a complete theory in $L_{1} .$
(b) Show that $T_{2}$ is not categorical in any infinite cardinal.
(c) Use the results from the preceding exercise to show that $T_{2}$ is a complete theory in $L_{2}$.

Christopher Stanley
Christopher Stanley
Numerade Educator
07:09

Problem 7

(a) Let $L_{0}$ be the language consisting of the single binary predicate $R$. Let $T_{0}$ be the theory expressing that the interpretation of $R$ is a total ordering, together with the following two additional formulas:
$$
\begin{aligned}
&\forall v_{1} \exists v_{2}\left(R v_{1} v_{2} \wedge \neg v_{1} \simeq v_{2} \wedge \forall v_{3}\left(\left(R v_{1} v_{3} \wedge \neg v_{1} v_{3}\right) \Rightarrow R v_{2} v_{3}\right)\right) \\
&\forall v_{1} \exists v_{2}\left(R v_{2} v_{1} \wedge \neg v_{1} \simeq v_{2} \wedge\left(\left(R v_{3} v_{1} \wedge \neg v_{1} \simeq v_{3}\right) \Rightarrow R v_{3} v_{2}\right)\right)
\end{aligned}
$$
Show that we can find two models $\mathcal{M}_{0}$ and $\mathcal{M}_{1}$ of $T_{0}$ such that $\mathcal{M}_{0}$ is a substructure of $\mathcal{M}_{1}$, but is not an elementary substructure.
(b) Show that $T_{0}$ is not equivalent to an $\forall \exists$ theory in $L_{0}$.

Chris Trentman
Chris Trentman
Numerade Educator
23:32

Problem 8

Let $L$ be a denumerable language and let $T$ be a consistent theory in $L$ that only has infinite models.
$T$ is called model-complete if and only if for all models $\mathcal{M}$ and $\mathcal{N}$ of $T$, if $\mathcal{N}$ is an extension of $\mathcal{M}$, then it is an elementary extension.

A model $\mathcal{M}$ of $T$ is called a prime model of $T$ if and only if every model of $T$ is isomorphic to a (simple) extension of $\mathcal{M}$.
(a) Show that a model-complete theory that has a prime model is complete.
(b) Show that the following four conditions are equivalent:
(1) $T$ is model-complete;
(2) for every model $\mathcal{M}$ of $T$, every formula of $D(\mathcal{M}$ ) is a consequence of $\Delta(\mathcal{M}) \cup T$ (for the notations, see Section $8.2$ of Chapter 8 );
(3) for every denumerable model $\mathcal{M}$ of $T$, every formula of $D(\mathcal{M})$ is a consequence of $\Delta(\mathcal{M}) \cup T$;
(4) for all denumerable models $\mathcal{M}$ and $\mathcal{M}^{\prime}$ of $T$, if $\mathcal{M} \subseteq \mathcal{M}^{\prime}$, then $\mathcal{M} \prec \mathcal{M}^{\prime} .$
(c) Show that if $T$ is model-complete, $T$ is equivalent to an $\forall \exists$ theory. Is the converse true?
(d) Let $F\left[v_{0}, v_{1}, \ldots, v_{n}\right]$ be a formula of $L$. Consider the following property of $T$ and $F\left[v_{0}, v_{1}, \ldots, v_{n}\right]:$
for all models $\mathcal{M}$ and $\mathcal{M}^{\prime}$ of $T$ such that $\mathcal{M} \subseteq \mathcal{M}^{\prime}$,
for all elements $a_{0}, a_{1}, \ldots, a_{n}$ of $M$,
if $\mathcal{M} \vDash F\left[a_{0}, a_{1}, a_{2}, \ldots, a_{n}\right]$, then $\mathcal{M}^{\prime} \vDash F\left[a_{0}, a_{1}, a_{2}, \ldots, a_{n}\right]$.
(*)
Prove that $(*)$ holds if and only if there is an existential formula $G\left[v_{0}, v_{1}, v_{2}, \ldots, v_{n}\right]$ of $L$ such that
$$
T \vdash \forall v_{0} \forall v_{1} \ldots \forall v_{n}\left(F\left[v_{0}, v_{1}, \ldots, v_{n}\right] \Leftrightarrow G\left[v_{0}, v_{1}, \ldots, v_{n}\right]\right) .
$$
(Hint: Adjoin constant symbols $c_{0}, c_{1}, \ldots, c_{n}$ and reread the proof of Theorem $8.37$ for inspiration.)
(e) Show that $T$ is model-complete if and only if for every universal formula $F\left[v_{0}, v_{1}, \ldots, v_{n}\right]$ of $L$, there exists an existential formula $G\left[v_{0}, v_{1}, \ldots, v_{n}\right]$ of $L$ such that
$$
T \vdash \forall v_{0} \forall v_{1} \ldots \forall v_{n}\left(F\left[v_{0}, v_{1}, \ldots, v_{n}\right] \Leftrightarrow G\left[v_{0}, v_{1}, \ldots, v_{n}\right]\right) .
$$

Chris Trentman
Chris Trentman
Numerade Educator
26:40

Problem 9

The purpose of this exercise is to prove Lindstrom's theorem: let $T$ be an $\forall \exists$ theory in a denumerable language; if $T$ has no finite models and is categorical in some infinite cardinal, then $T$ is model-complete.
We resume with the notations and results from the preceding exercise.
(a) Let $\lambda$ be an infinite cardinal and let $F\left[v_{0}, v_{1}, \ldots, v_{n}\right]$ be a formula of $L$. Show that the following two conditions are equivalent:
(1) For all models $\mathcal{M}$ and $\mathcal{M}^{\prime}$ of $T$ of cardinality $\lambda$ such that $\mathcal{M} \subseteq \mathcal{M}^{\prime}$ and for all elements $a_{0}, a_{1}, \ldots, a_{n}$ of $M$, if $\mathcal{M} \vDash F\left[a_{0}, a_{1}, a_{2}, \ldots, a_{n}\right]$, then $\mathcal{M}^{\prime} \vDash F\left[a_{0}, a_{1}, a_{2}, \ldots, a_{n}\right]$.
(2) There is an existential formula $G\left[v_{0}, v_{1}, \ldots, v_{n}\right]$ of $L$ such that
$$
T \vdash \forall v_{0} \forall v_{1} \ldots \forall v_{n}\left(F\left[v_{0}, v_{1}, \ldots, v_{n}\right] \Leftrightarrow G\left[v_{0}, v_{1}, \ldots, v_{n}\right]\right) .
$$
(b) Assume that $T$ is an $\forall \exists$ theory. Let $F\left[v_{0}, v_{1}, \ldots, v_{n}\right]$ be a universal formula and let $\lambda$ be an infinite cardinal. Prove that $T$ has a model $\mathcal{M}$ of cardinality $\lambda$ which has the following property:
for every model $\mathcal{M}^{\prime}$ of $T$ such that $\mathcal{M} \subseteq \mathcal{M}^{\prime}$, for all elements $a_{0}, a_{1}, \ldots, a_{n}$ of $M$, if $\mathcal{M} \vDash F\left[a_{0}, a_{1}, a_{2}, \ldots, a_{n}\right]$, then $\mathcal{M}^{\prime} \vDash F\left[a_{0}, a_{1}, a_{2}, \ldots, a_{n}\right] .(* *)$
(c) Prove Lindstrom's theorem.
(d) Assume that the language consists of a single unary function symbol $f$ and set
$$
T_{0}=\left\{\forall v_{0} \forall v_{1}\left(f v_{0} \simeq f v_{1} \Rightarrow v_{0} \simeq v_{1}\right)\right\} \cup\left\{\forall v_{0} \neg f^{n} v_{0} \simeq v_{0}: n \in \mathbb{N}\right\} .
$$
Show that $T_{0}$ is an $\forall \exists$ theory that is neither complete nor model-complete.

Chris Trentman
Chris Trentman
Numerade Educator
23:32

Problem 10

The language $L$ consists of a binary predicate symbol $R$ and a denumerably infinite set of constant symbols $\left\{c_{0}, c_{1}, \ldots, c_{n}, \ldots\right\}$.

Let $A$ be a closed formula of $L$ expressing that the interpretation of $R$ is a strict, dense, total ordering without endpoints. For every $n \in \mathbb{N}, F_{n}$ is the formula $R c_{n} c_{n+1}$.
Consider the theory
$\{A\} \cup\left\{F_{n}: n \in \mathbb{N}\right\} .$
We let $\mathcal{A}, \mathcal{B}$, and $\mathcal{C}$ denote the three $L$-structures whose base set is $\mathbb{Q}$, in which the interpretation of $R$ is the usual strict ordering, and in which the sequence of constant symbols $\left(c_{n}\right)_{n \in \mathbb{N}}$ is interpreted, respectively, by the following sequences of rationals:
$$
\alpha=\left(\alpha_{n}\right)_{n \in \mathbb{N}}, \quad \beta=\left(\beta_{n}\right)_{n \in \mathbb{N}}, \quad \gamma=\left(\gamma_{n}\right)_{n \in \mathbb{N}}
$$
where
$$
\alpha_{n}=n, \quad \beta_{n}=-\frac{1}{n+1}, \quad \gamma_{n}=\sum_{k=0}^{n} \frac{1}{k !} .
$$
(a) Show that $T$ is complete in $L$.
(b) Show that every denumerable model of $T$ is isomorphic to one of the three structures $\mathcal{A}, \mathcal{B}$, or $\mathcal{C}$.
(c) Show that the theory $T$ is model-complete (see Exercise 8).
(d) Show that every denumerable model of $T$ has an elementary extension that is isomorphic to $\mathcal{B}$ and an elementary extension that 'is isomorphic to $\mathcal{C}$.

Chris Trentman
Chris Trentman
Numerade Educator
23:32

Problem 11

Let $L$ be the first-order language consisting of a unary function symbol $f$ and a binary relation symbol $R$. Let $A$ denote the conjunction of the following seven formulas:
$$
\begin{aligned}
&\forall v_{0} R v_{0} v_{0} \\
&\forall v_{0} \forall v_{1}\left(\left(R v_{0} v_{1} \Leftrightarrow R v_{1} v_{0}\right) \Rightarrow v_{0} \simeq v_{1}\right) \\
&\forall v_{0} \forall v_{1} \forall v_{2}\left(\left(R v_{0} v_{1} \wedge R v_{1} v_{2}\right) \Rightarrow R v_{0} v_{2}\right) \\
&\forall v_{0} \exists v_{1}\left(f v_{1} \simeq v_{0} \wedge \forall v_{2}\left(f v_{2} \simeq v_{0} \Rightarrow v_{2} \simeq v_{1}\right)\right) \\
&\forall v_{0} \forall v_{\mathrm{I}}\left(R v_{0} v_{1} \Leftrightarrow R f v_{0} f v_{1}\right) ; \\
&\forall v_{0}\left(R v_{0} f v_{0} \wedge \neg v_{0} \simeq f v_{0}\right) ; \\
&\forall v_{0} \forall v_{1}\left(\left(\neg v_{0} \simeq v_{1} \wedge R v_{0} v_{1}\right) \Rightarrow R f v_{0} v_{1}\right)
\end{aligned}
$$
(a) Show that in every model of the formula $A$, the interpretation of the symbol $R$ is a total ordering of the base set of the model, with no least or greatest element, such that every element has a successor, i.e. a strict least upper bound.
(b) Show that $\mathbb{Z}$ with its usual ordering and the successor function is a model of $A$.
Let $X=\langle B, \leq\rangle$ be an arbitrary totally ordered set. Consider the following $L$-structure $\mathcal{M}_{X}$ :
- the base set of $\mathcal{M}_{X}$ is the set $B \times \mathbb{Z}$;
- the interpretation of $R$ in $\mathcal{M}_{X}$ is the set
$$
\left\{((x, n),(y, m)) \in(B \times \mathbb{Z})^{2}: x<y \text { or }(x=y \text { and } n \leq m)\right\}
$$
- the interpretation of $f$ in $\mathcal{M}_{X}$ is the mapping that, with $(x, n) \in(B \times \mathbb{Z})$, associates $(x, n+1)$.
Show that $\mathcal{M}_{X}$ is a model of $A$.
(c) Let $\mathcal{M}=\langle M, \bar{f}, \bar{R}\rangle$ be a model of $A$. We wish to prove that there exists a totally ordered set $X$ such that $\mathcal{M}$ is isomorphic to $\mathcal{M}_{X}$.

On the base set $M$ of $\mathcal{M}$, we define two binary relations $\ll$ and $\approx$ as follows: for all $a$ and $b$ in $M$,
$a \ll b \quad$ if and only if $\quad$ for all $n \in \mathbb{N}, \mathcal{M} \vDash R f^{n} a b$
and
$a \approx b$ if and only if there exist integers $n$ and $p$
such that $\mathcal{M} \vDash f^{n} a \simeq f^{p} b$.
Show that $\ll$ is irreflexive and transitive, that $\approx$ is an equivalence relation, and that
$a \approx b$ if and only if $a \ll b$ and $b \ll a$ are both false.
Show that each equivalence class modulo $\approx$ is a substructure of $\mathcal{M}$ that is isomorphic to $\mathbb{Z}$.

Show that the relation $\ll$ allows us to define a total ordering on the set $M / \approx$ of equivalence classes.

Show that if $X=\langle C, \Delta\rangle$ is the ordered set obtained in this way, then $\mathcal{M}$ is isomorphic to $\mathcal{M}_{X}$.
(d) Show that if $X$ and $Y$ are two totally ordered sets, then $\mathcal{M}_{X}$ and $\mathcal{M}_{Y}$ are isomorphic if and only if $X$ and $Y$ are isomorphic.

Show that $A$ only has infinite models and is not categorical in any infinite cardinal.
(e) We wish to show that $\{A\}$ is a complete theory.
(1) Show that if $a$ and $b$ are two points in a model $\mathcal{M}$ of $A$ which satisfy $a \ll b$, then there exists an elementary extension $\mathcal{M}_{1}$ of $\mathcal{M}$ and a point $c$ in $\mathcal{M}_{1}$ such that
(2) Show also that if $a$ is a point of $M$, then there exists an elementary extension $\mathcal{M}_{1}$ of $\mathcal{M}$ and points $b$ and $c$ of $M_{1}$ such that
$$
b \ll a \text { and } a \ll c .
$$
(3) Let $\mathcal{M}$ and $\mathcal{N}$ be two models of $A$ and let $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ and $\left(b_{1}, b_{2}, \ldots, b_{n}\right)$ be two finite sequences of the same length from $\mathcal{M}$ and $\mathcal{N}$, respectively.
Consider the following condition
$$
P\left(\left(\mathcal{M}, a_{1}, a_{2}, \ldots, a_{n}\right),\left(\mathcal{N}, b_{1}, b_{2}, \ldots, b_{n}\right)\right):
$$
for every atomic formula $F\left[v_{1}, v_{2}, \ldots, v_{n}\right]$ of $L$,
$\mathcal{M} \vDash F\left[a_{1}, a_{2}, \ldots, a_{n}\right] \quad$ if and only if $\mathcal{N} \vDash F\left[b_{1}, b_{2}, \ldots, b_{n}\right]$
Show that this condition is equivalent to
for all integers $i$ and $j$ such that $1 \leq i, j \leq n$ and for all $k \in \mathbb{N}$, $\mathcal{M} \vDash a_{i} \simeq f^{k} a_{j} \quad$ if and only if $\quad \mathcal{N} \vDash b_{i} \simeq f^{k} b_{j}$, and $\mathcal{M} \vDash R a_{i} a_{j} \quad$ if and only if $\quad \mathcal{N} \vDash R b_{i} b_{j}$
(4) Assume that the condition
$$
P\left(\left(\mathcal{M}, a_{1}, a_{2}, \ldots, a_{n}\right),\left(\mathcal{N}, b_{1}, b_{2}, \ldots, b_{n}\right)\right)
$$
is satisfied.
Show that if $c$ is an element of $M$, then
- if $c \approx a_{i}$ for some index $i$ between 1 and $n$ inclusive, then there exists a point $d \in N$ such that
$$
P\left(\left(\mathcal{M}, a_{1}, a_{2}, \ldots, a_{n}, c\right),\left(\mathcal{N}, b_{1}, b_{2}, \ldots, b_{n}, d\right)\right)
$$
- if not, there exists an elementary extension $\mathcal{N}^{\prime}$ of $\mathcal{N}$ and a point $d$ of $N^{\prime}$ such that $P\left(\left(\mathcal{M}, a_{1}, a_{2}, \ldots, a_{n}, c\right),\left(\mathcal{N}^{\prime}, b_{1}, b_{2}, \ldots, b_{n}, d\right)\right)$.
(5) Use induction on the height of the formula $G\left[v_{1}, v_{2}, \ldots, v_{n}\right]$ to prove the following assertion:
If $\mathcal{M}$ and $\mathcal{N}$ are two models of $A$ and if $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ and $\left(b_{1}, b_{2}, \ldots, b_{n}\right)$ are two sequences from $\mathcal{M}$ and $\mathcal{N}$, respectively, then $P\left(\left(\mathcal{M}, a_{1}, a_{2}, \ldots, a_{n}\right),\left(\mathcal{N}, b_{1}, b_{2}, \ldots, b_{n}\right)\right)$ implies $\mathcal{M} \vDash G\left[a_{1}, a_{2}, \ldots, a_{n}\right]$ if and only if $\mathcal{N} \vDash G\left[b_{1}, b_{2}, \ldots, b_{n}\right] .$
(6) Conclude from this that $\{A\}$ is a complete theory.

Chris Trentman
Chris Trentman
Numerade Educator
07:27

Problem 12

Let $L$ be the language consisting of a single binary predicate symbol $\leq$ and let $T$ be the theory of dense linear orderings with no first or last element. Show that for every formula $F\left[v_{0}, v_{1}^{\prime}, v_{2}, \ldots, v_{n}\right]$, there exists a quantifier-free formula $H\left[v_{0}, v_{1}, v_{2}, \ldots, v_{n}\right]$ such that
$$
T \vdash \forall v_{0} \forall v_{1} \ldots \forall v_{n}\left(F\left[v_{0}, v_{1}, v_{2}, \ldots, v_{n}\right] \Leftrightarrow H\left[v_{0}, v_{1}, v_{2}, \ldots, v_{n}\right]\right)
$$

Chris Trentman
Chris Trentman
Numerade Educator
07:48

Problem 13

Let $L$ be a language. A class $\mathcal{C}$ of $L$-structures is closed under ultraproducts if for every set $I$, for every ultrafilter $\mathcal{U}$ on $I$, and for every sequence $\left(M_{i}: i \in I\right)$ of structures from $\mathcal{C}, \prod_{i \in I} \mathcal{M}_{i} / \mathcal{U}$ belongs to $\mathcal{C}$.

Let $T$ be a theory in $L$. Show that the class of $L$-structures that are not models of $T$ is closed under ultraproducts if and only if $T$ is finitely axiomatizable.

Chris Trentman
Chris Trentman
Numerade Educator
03:13

Problem 14

Let $K$ be a field (the language is $\{0,1,+, \times\}$ ). Let $I$ be a set and let $\mathcal{F}$ be a filter on $I$.
(a) Show that $K^{I} / \mathcal{F}$ is a ring and that it is a field if and only if $\mathcal{F}$ is an ultrafilter on $I$.
(b) Let $\mathcal{J}$ be the subset of $K^{l}$ defined by
$$
\mathcal{J}=\left\{\left(k_{i}: i \in I\right) \in K^{i}:\left\{i \in I: k_{i}=0\right\} \in \mathcal{F}\right\} .
$$
Show that $\mathcal{J}$ is an ideal in the ring $K^{I}$ and that the quotient ring $K^{I} / \mathcal{J}$ is equal to $K^{I} / \mathcal{F}$.

Gideon Idumah
Gideon Idumah
Numerade Educator
01:30

Problem 15

The language is that of ordered sets: $L=\{\leq\}$.
(a) Let $\alpha$ be an infinite ordinal. Show that there exists an ordered set that is elementarily equivalent to $\langle\alpha, \leq\rangle$ but is not a well-ordering.
(b) Show that there exists a denumerable ordinal $\alpha$ such that
$$
\langle\alpha, \leq\rangle \prec\left\langle\aleph^{2}, \leq\right\rangle .
$$
(\alephs denotes the least uncountable ordinal.)
(c) Show that there exist two distinct denumerable ordinals $\alpha$ and $\beta$ such that
$$
\langle\alpha, \leq\rangle \prec\langle\beta, \leq\rangle .
$$

Nick Johnson
Nick Johnson
Numerade Educator
26:40

Problem 16

Consider the uncountable language $L$ that contains:
- for every integer $n$ a constant symbol $\underline{n}$;
- for every subset $A$ of $\mathbb{N}$, a unary predicate symbol $\underline{A}$;
- for every mapping $f$ from $\mathbb{N}$ into $\mathbb{N}$, a unary function symbol $f$.
Let $\mathcal{N}$ be the $L$-structure whose base set is $\mathbb{N}$ and in which each symbol $\underline{X}$ of $L$ is interpreted by $X$. Let $T$ be the theory of $\mathcal{N}$.
(a) Show that every model of $T$ is isomorphic to an elementary extension of $\mathcal{N}$.
(b) Let $\mathcal{M}$ be a proper elementary extension of $\mathcal{N}$ and let $a$ be a point of $\mathcal{M}$ that does not belong to $\mathbb{N}$. Show that the set
$$
\mathcal{F}_{a}=\{A \subseteq \mathbb{N}: \mathcal{M} \vDash \underline{A} a\}
$$
is a non-trivial ultrafilter on $\mathbb{N}$.
(c) Let $\alpha$ be a bijection from $\mathbb{N}^{2}$ onto $\mathbb{N}$. For every positive real number $r$, choose two sequences of natural numbers $\left(p_{r}(n): n \in \mathbb{N}\right)$ and $\left(q_{r}(n): n \in \mathbb{N}\right)$
such that the sequence $\left(p_{r}(n) / q_{r}(n): n \in \mathbb{N}\right)$ converges to the limit $r$. Define the mapping $f_{r}$ from $\mathbb{N}$ into $\mathbb{N}^{2}$ by setting
$$
f_{r}(n)=\alpha\left(p_{r}(n), q_{r}(n)\right)
$$
for all $n \in \mathbb{N}$.
Show that if $r$ and $s$ are distinct positive reals, then the set
$$
\left\{n \in \mathbb{N}: f_{r}(n)=f_{s}(n)\right\}
$$
is finite.
(d) Show that every model of $T$ that is not isomorphic to $\mathcal{N}$ must have cardinality greater than or equal to $2^{\aleph_{0}}$.
Show that $T$ is $\aleph_{0}$-categorical.
(e) Let $L^{\prime}$ be the language obtained by adding a new unary predicate symbol $X$ to $L$ and let $T^{\prime}$ be the theory $T \cup\{X \underline{n}: n \in \mathbb{N}\}$. Show that $T^{\prime}$ has no finite model, is $\aleph_{0}$-categorical, and is not complete.

Chris Trentman
Chris Trentman
Numerade Educator