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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 6

Formalization of arithmetic, Gödel's theorems - all with Video Answers

Educators

Chapter Questions

14:09

Problem 1

Let $X$ be a non-empty set and let $f$ be a function from $X \times X$ into $X$. Consider the $\mathcal{L}_{0}$-structure $\mathcal{M}$ whose base set is $M=\mathbb{N} \cup(X \times \mathbb{Z})$ and in which the symbols $\underline{S}, \underline{+}$, and $\underline{\times}$ are interpreted by the functions $S,+$, and $\times$ that are defined by the following conditions:
- $M$ is an extension of $\mathbb{N}$;
- if $a=(x, n) \in M-\mathbb{N}$, then $S(a)=(x, n+1)$;
- if $a=(x, n) \in M-\mathbb{N}$ and $m \in \mathbb{N}$, then $a+m=m+a=(x, n+m) ;$
- if $a=(x, n)$ and $b=(y, m)$ are elements of $M-\mathbb{N}$, then $(x, n)+(y, m)=$ $(x, n+m)$
- if $a=(x, n) \in M-\mathbb{N}$ and $m \in \mathbb{N}$, then $(x, n) \times m=(x, n \times m)$ if $m \neq 0$ and $(x, n) \times 0=0$;
- if $a=(x, n) \in M-\mathbb{N}$ and $m \in \mathbb{N}$, then $m \times(x, n)=(x, m \times n)$;
- if $a=(x, n)$ and $b=(y, m)$ are elements of $M-\mathbb{N}$, then $(x, n) \times(y, m)=$ $(f(x, y), n \times m)$.
(a) Show that $\mathcal{M}$ is a model of $\mathcal{P}_{0}$.
(b) Show that none of the following formulas is a consequence of $\mathcal{P}_{0}$ :
(i) $\forall v_{0} \forall v_{1} v_{0} \pm v_{1} \simeq v_{1} \pm v_{0}$;
(ii) $\forall v_{0} \forall v_{1} \forall v_{2} v_{0} \times\left(v_{1} \times v_{2}\right) \simeq\left(v_{0} \times v_{1}\right) \times v_{2} ;$
(iii) $\forall v_{0} \forall v_{1}\left(\left(v_{0} \leq v_{1} \wedge v_{1} \leq v_{0}\right) \Rightarrow v_{0} \simeq v_{1}\right)$;
(iv) $\forall v_{0} \underline{0} \times v_{0} \simeq \underline{0}$.
(c) Construct a model of $\mathcal{P}_{0}$ in which addition is not associative.

Anthony Ramos
Anthony Ramos
Numerade Educator
14:09

Problem 2

Let $X$ be a non-empty set and let $f$ be a function from $X \times X$ into $X$. Consider the $\mathcal{L}_{0}$-structure $\mathcal{M}$ whose base set is $M=\mathbb{N} \cup(X \times \mathbb{Z})$ and in which the symbols $\underline{S}, \underline{+}$, and $\underline{\times}$ are interpreted by the functions $S,+$, and $\times$ that are defined by the following conditions:
- $M$ is an extension of $\mathbb{N}$;
- if $a=(x, n) \in M-\mathbb{N}$, then $S(a)=(x, n+1)$;
- if $a=(x, n) \in M-\mathbb{N}$ and $m \in \mathbb{N}$, then $a+m=m+a=(x, n+m) ;$
- if $a=(x, n)$ and $b=(y, m)$ are elements of $M-\mathbb{N}$, then $(x, n)+(y, m)=$ $(x, n+m)$
- if $a=(x, n) \in M-\mathbb{N}$ and $m \in \mathbb{N}$, then $(x, n) \times m=(x, n \times m)$ if $m \neq 0$ and $(x, n) \times 0=0$;
- if $a=(x, n) \in M-\mathbb{N}$ and $m \in \mathbb{N}$, then $m \times(x, n)=(x, m \times n)$;
- if $a=(x, n)$ and $b=(y, m)$ are elements of $M-\mathbb{N}$, then $(x, n) \times(y, m)=$ $(f(x, y), n \times m)$.
(a) Show that $\mathcal{M}$ is a model of $\mathcal{P}_{0}$.
(b) Show that none of the following formulas is a consequence of $\mathcal{P}_{0}$ :
(i) $\forall v_{0} \forall v_{1} v_{0} \pm v_{1} \simeq v_{1} \pm v_{0}$;
(ii) $\forall v_{0} \forall v_{1} \forall v_{2} v_{0} \times\left(v_{1} \times v_{2}\right) \simeq\left(v_{0} \times v_{1}\right) \times v_{2} ;$
(iii) $\forall v_{0} \forall v_{1}\left(\left(v_{0} \leq v_{1} \wedge v_{1} \leq v_{0}\right) \Rightarrow v_{0} \simeq v_{1}\right)$;
(iv) $\forall v_{0} \underline{0} \times v_{0} \simeq \underline{0}$.
(c) Construct a model of $\mathcal{P}_{0}$ in which addition is not associative.
Show that the relation $R$ is a total ordering. Show that $E$, with respect to this ordering, has a least element but does not have a greatest element. Show that $R$ is a dense ordering of $E$.

Anthony Ramos
Anthony Ramos
Numerade Educator
00:29

Problem 3

Prove the Chinese remainder theorem (Theorem $6.13$ ).

James Kiss
James Kiss
Numerade Educator
04:55

Problem 4

Prove the converse of the representation theorem (Theorem $6.8$ ): if a function from $\mathbb{N}^{p}$ into $\mathbb{N}$ is representable, then it is recursive.

Muhammad Saleem
Muhammad Saleem
Numerade Educator
07:27

Problem 5

Let $T$ be a theory in a finite language. Assume that $T$ is recursively enumerable,
i.e. that the set
$$
\{\# F: F \in T\}
$$

Chris Trentman
Chris Trentman
Numerade Educator
01:11

Problem 6

Show that if Fermat's last theorem,
$$
\neg(\exists x>0)(\exists y>0)(\exists z>0)(\exists t>2)\left(x^{t}+y^{t} \simeq z^{t}\right),
$$
is not refutable in $\mathcal{P}_{0}$, then it is true in $\mathbb{N}$.
(At the time this text was first written, Fermat's last theorem was still a conjecture; it has since been proved by Andrew Wiles.)

Carson Merrill
Carson Merrill
Numerade Educator
01:09

Problem 7

In this exercise, $\operatorname{Drv}\left[v_{0}, v_{1}\right]$ is a formula that represents the set
Drv $=\{(a, b): b$ is the Gödel number of a derivation in $\mathcal{P}$ of the formula whose Gödel number is $a\}$.
Among the following assertions, which are true for any closed formula $F ?$
(a) $\mathbb{N} \vDash \exists v_{1} \operatorname{Dr} v\left[\# F, v_{1}\right] \Rightarrow F$;
(b) $\mathcal{P} \vdash \exists v_{1} \operatorname{Drv}\left[\# F, v_{1}\right] \Rightarrow F$;
(c) $\mathbb{N} \vDash F \Rightarrow \exists v_{1} \operatorname{Drv}\left[\# F, v_{1}\right]$;
(d) $\mathcal{P} \vdash F \Rightarrow \exists v_{1} \operatorname{Drv}\left[\# F, v_{1}\right]$.

Carson Merrill
Carson Merrill
Numerade Educator
01:42

Problem 8

Show that there exists a formula $F\left[v_{0}\right]$ of $\mathcal{L}_{0}$ such that $\mathbb{N} \vDash \neg \exists v_{0} F\left[v_{0}\right]$ and $\neg \exists v_{0} F\left[v_{0}\right]$ is not derivable in $\mathcal{P}$. Conclude from this that, for every formula $G\left[v_{0}, v_{1}, \ldots, v_{n}\right]$, there exists a formula $H\left[v_{0}, v_{1}, \ldots, v_{n}\right]$ such that
$$
\forall v_{0} \forall v_{1} \ldots \forall v_{n}\left(G\left[v_{0}, v_{1}, \ldots, v_{n}\right] \Leftrightarrow H\left[v_{0}, v_{1}, \ldots, v_{n}\right]\right)
$$
is true in $\mathbb{N}$ but is not derivable in $\mathcal{P}$.

Nick Johnson
Nick Johnson
Numerade Educator
04:43

Problem 9

Show that if $F$ is a closed formula and if
$$
\mathcal{P} \vdash \exists v_{0} \operatorname{Drv}\left[\# F, v_{0}\right] \Rightarrow F,
$$
then
$$
\mathcal{P} \vdash F
$$
(See Exercise 7; we could apply the second incompleteness theorem to the theory $\mathcal{P} \cup\{\neg F\}$.)

Stanley Enemuo
Stanley Enemuo
Numerade Educator
01:35

Problem 10

This exercise uses the notion of elementary extension, which will be introduced in Chapter 8 on model theory.

Let $\mathcal{M}$ be a non-standard model of $\mathcal{P}$, and let $A$ be a subset of the underlying set, $M$, of $\mathcal{M}$. We say that a function $f$ from $M^{p}$ into $M$ is definable with parameters from $A$ if there exists a formula $F\left[v_{0}, v_{1}, \ldots, v_{p}\right]$ of $\mathcal{L}_{0}$ with parameters in $A$ such that, for all $a_{1}, a_{2}, \ldots, a_{p}$ belonging to $M$, we have
$$
\mathcal{M} \vDash \forall v_{0}\left(F\left[a_{0}, a_{1}, \ldots, a_{p}\right] \Leftrightarrow v_{0} \simeq f\left(a_{0}, a_{1}, \ldots, a_{p}\right)\right)
$$
(a) Let $\mathcal{N}$ be a substructure of $\mathcal{M}$ whose underlying set, $N$, is closed under functions that are definable with parameters from $N$; in other words, it is such that for all $p \in \mathbb{N}$, for every function $f$ from $M^{p}$ into $M$ that is definable with parameters from $N$, and for all $a_{1}, a_{2}, \ldots, a_{p}$ belonging to $N$,
$$
f\left(a_{1}, \ldots, a_{p}\right) \in N
$$
Show that $\mathcal{N}$ is an elementary substructure of $\mathcal{M}$ (and is therefore a model of $\mathcal{P}$ ).
(b) Next, we say that a subset $X$ of $M^{p}$ is definable with parameters from $A$ if there exists a formula $G\left[v_{1}, \ldots, v_{p}\right]$ of $\mathcal{L}_{0}$ with parameters in $A$ such that, for all $a_{1}, a_{2}, \ldots, a_{p}$ belonging to $M$,
$$
\left(a_{0}, a_{1}, \ldots, a_{p}\right) \in X \quad \text { if and only if } \quad \mathcal{M} \models G\left[a_{1}, \ldots, a_{p}\right]
$$
Show that the collection of subsets of $M$ that are definable with parameters from $A$ forms a Boolean subalgebra of the algebra of all subsets of $M$.

Show that if $f$ and $g$ are functions from $M$ into $M$ that are definable with parameters from $A$, then the set $\{a \in M: f(a)=g(a)\}$ is definable with parameters from $A$.
(c) Let $F$ and $G$ be two maps from $M$ into $M$. Define the maps $S f, f+g$, and $f \times g$ from $M$ into $M$ by
$$
\begin{aligned}
S f(x) &=f(x)+1 \\
(f+g)(x) &=f(x)+g(x) \\
(f \times g)(x) &=f(x) \times g(x)
\end{aligned}
$$
Show that the set of functions definable with parameters from $A$ is closed under these operations.
(d) Let $\mathcal{B}$ be the Boolean algebra of subsets of $M$ that are definable with parameters from $A$, let $\mathcal{U}$ be an ultrafilter on this algebra, and let $\mathcal{F}$ be the set of functions from $M$ into $M$ that are definable with parameters from $M$.
Show that the relation $\approx$ on $\mathcal{F}$ defined by
$f \approx g \quad$ if and only if $\quad\{a \in M: f(a)=g(a)\} \in \mathcal{U}$
is an equivalence relation and that if $f \approx f^{\prime}$ and $g \approx g^{\prime}$, then
$$
S f \approx S f^{\prime}, \quad f+g \approx f^{\prime}+g^{\prime}, \quad f \times g \approx f^{\prime} \times g^{\prime} .
$$
If $f \in \mathcal{F}$, we let $f / \mathcal{U}$ denote the equivalence class of $f$ relative to $\approx$ and we let $\mathcal{F} / \mathcal{U}$ denote the set of equivalence classes relative to $\approx$.We then observe that it is possible to define the operations $S,+$, and $\times$ on $\mathcal{F} / \mathcal{U}$. The zero element, 0 , of $\mathcal{F} / \mathcal{U}$ will be, by definition, the equivalence class of the constant function equal to 0 . This allows us to treat $\mathcal{F} / \mathcal{U}$ as an $\mathcal{L}_{0}$-structure.
(e) For every element $a \in M$, let $\bar{a}$ denote the element of $\mathcal{F} / \mathcal{U}$ that is the equivalence class relative to $\approx$ of the constant function equal to $a$.

Show that the map from $\mathcal{M}$ into $\mathcal{F} / \mathcal{U}$ that sends $a \in M$ into $\bar{a} \in \mathcal{F} / \mathcal{U}$ is a homomorphism of $\mathcal{L}_{0}$-structures.
(f) Show that for every $p \in \mathbb{N}$, for every formula $F\left[v_{1}, \ldots, v_{p}\right]$ of $\mathcal{L}_{0}$, and for all $f_{1}, f_{2}, \ldots, f_{p}$ in $\mathcal{F}$, we have
$$
\mathcal{F} / \mathcal{U} \vDash F\left[f_{1} / \mathcal{U}, f_{2} / \mathcal{U}, \ldots, f_{p} / \mathcal{U}\right]
$$
if and only if
$$
\left\{a \in M: \mathcal{M} \vDash F\left[f_{1}(a), f_{2}(a), \ldots, f_{p}(a)\right]\right\} \in \mathcal{U}
$$
Conclude that the map from $\mathcal{M}$ into $\mathcal{F} / \mathcal{U}$ that sends $a \in M$ into $\bar{a} \in \mathcal{F} / \mathcal{U}$ is elementary (see Chapter 8).
(g) Suppose that $\mathbb{N}$ is an elementary substructure of $\mathcal{M}$. Show that if $f$ is a function from $M$ into $M$ that is definable with parameters from $\mathcal{M}$ and if $a \in M$, then there exists $b \in M$ such that
$$
\mathcal{M} \vDash \forall v_{0}\left(v_{0}<a \Rightarrow f\left(v_{0}\right)<b\right)
$$
(h) Let $\mathcal{M}$ be a proper elementary extension of $\mathbb{N}$. Show that there exists a proper elementary extension $\mathcal{N}$ of $\mathcal{M}$, with base set $N$, which satisfies
for all $a \in N$, there exists $b \in M$ such that $\mathcal{N} \vDash a<b$.

Manik Pulyani
Manik Pulyani
Numerade Educator
23:32

Problem 11

Let $\mathcal{L}$ be a finite language and let $\mathcal{M}$ be an $\mathcal{L}$-structure whose underlying set is $M$. We say that $\mathcal{M}$ is strongly undecidable if every theory in the language $\mathcal{L}$ that has $\mathcal{M}$ as a model is undecidable.
(a) Show that $\mathbb{N}$ is strongly undecidable.
(b) Let $G_{0}\left[v_{0}\right], G_{1}\left[v_{0}\right], G_{2}\left[v_{0}, v_{1}\right], G_{3}\left[v_{0}, v_{1}, v_{2}\right], G_{4}\left[v_{0}, v_{1}, v_{2}\right]$ be five fixed formulas of $\mathcal{L}$ and consider the theory $T_{0}$ that has the following formulas as axioms:
(1) $\forall v_{0}\left(G_{1}\left[v_{0}\right] \Rightarrow G_{0}\left[v_{0}\right]\right)$;
(2) $\forall v_{0} \forall v_{1}\left(G_{2}\left[v_{0}, v_{1}\right] \Rightarrow\left(G_{0}\left[v_{0}\right] \wedge G_{0}\left[v_{1}\right]\right)\right)$;
(3) $\forall v_{0} \forall v_{1} \forall v_{2}\left(G_{3}\left[v_{0}, v_{1}, v_{2}\right] \Rightarrow\left(G_{0}\left[v_{0}\right] \wedge G_{0}\left[v_{1}\right] \wedge G_{0}\left[v_{2}\right]\right)\right) ;$
(4) $\forall v_{0} \forall v_{1} \forall v_{2}\left(G_{4}\left[v_{0}, v_{1}, v_{2}\right] \Rightarrow\left(G_{0}\left[v_{0}\right] \wedge G_{0}\left[v_{1}\right] \wedge G_{0}\left[v_{2}\right]\right)\right) ;$
(5) $\exists ! v_{0} G_{1}\left[v_{0}\right] ;$
(6) $\forall v_{1}\left(G_{0}\left[v_{1}\right] \Rightarrow \exists ! v_{0} G_{2}\left[v_{0}, v_{1}\right]\right)$;
(7) $\forall v_{1} \forall v_{2}\left(\left(G_{0}\left[v_{1}\right] \wedge G_{0}\left[v_{2}\right]\right) \Rightarrow \exists ! v_{0} G_{3}\left[v_{0}, v_{1}, v_{2}\right]\right)$;
(8) $\forall v_{1} \forall v_{2}\left(\left(G_{0}\left[v_{1}\right] \wedge G_{0}\left[v_{2}\right]\right) \Rightarrow \exists ! v_{0} G_{4}\left[v_{0}, v_{1}, v_{2}\right]\right)$.
If $\mathcal{M}$ is a model of $T_{0}$, we define the $\mathcal{L}_{0}$-structure $\mathcal{N}$ in the following way:
- the base set of $\mathcal{N}$ is the set $N=\left\{a \in M: \mathcal{M} \vDash G_{0}[a]\right\}$
- the constant symbol $\underline{0}$ is interpreted by the unique element $a$ of $\mathcal{M}$ that satisfies $G_{1}[a]$;
- the symbol $\underline{S}$ is interpreted by the function that associates with each $a \in N$ the unique element $b$ such that $\mathcal{M} \vDash G_{2}[b, a]$;
- the symbol $\pm$ is interpreted by the function that associates, with two elements $a$ and $b$ of $N$, the unique element $c$ such that $\mathcal{M} \vDash G_{3}[c, a, b]$;
- the symbol $\times$ is interpreted by the function that associates, with two elements $a$ and $b$ of $N$, the unique element $c$ such that $\mathcal{M} \vDash G_{4}[c, a, b]$.
We will say that $\mathcal{N}$ is definable in $\mathcal{M}$ (one must take care not to confuse this with the notion of a definable subset).

Show that, for every formula $F\left[v_{1}, v_{2}, \ldots, v_{p}\right]$ of $\mathcal{L}_{0}$, there exists a formula $F^{*}\left[v_{1}, v_{2}, \ldots, v_{p}\right]$ of $\mathcal{L}$ such that if $\mathcal{M}$ is a model of $T_{0}$ and $\mathcal{N}$ is the $\mathcal{L}_{0}$-structure defined in $\mathcal{M}$ and if $a_{1}, a_{2}, \ldots, a_{p}$ are elements of $N$, then
$\mathcal{N} \vDash F\left[a_{0}, a_{1}, \ldots, a_{p}\right] \quad$ if and only if $\quad \mathcal{M} \vDash F^{*}\left[a_{0}, a_{1}, \ldots, a_{p}\right]$
Show that $F^{*}$ can be determined from $F$ in an effective way [which means that there exists a primitive recursive function $\alpha$ such that if $n=\# F$, then $\left.\alpha(n)=\# F^{*}\right] .$
(c) Let $T$ be a theory of $\mathcal{L}$ that includes $T_{0}$. Set
$$
T^{-}=\left\{F: F \text { is a closed formula of } \mathcal{L}_{0} \text { and } T \vdash F^{*}\right\} .
$$
Show that if $G$ is a closed formula of $\mathcal{L}_{0}$, the following three conditions are equivalent:
(1) $G \in T^{-}$;
(2) $T^{-} \vdash G$;
(3) $T \vdash G^{*}$.
(d). Show that if $\mathbb{N}$ is definable in $\mathcal{M}$, then $\mathcal{M}$ is strongly undecidable.
(e) Show that the structure $\mathbb{Z}$ in the language $\mathcal{L}=\{\underline{0}, \pm, \underline{x}\}$ of ring theory is strongly undecidable. (Use Lagrange's theorem that every positive integer is the sum of four squares.) Show that the following theories are undecidable: the theory of rings, the theory of commutative rings, the theory of integral domains.
(f) Let $\mathcal{L}$ be the language that contains only the binary predicate symbol $R$. Consider the $\mathcal{L}$-structure $\mathcal{M}$ whose underlying set is $M=\mathbb{N} \cup(\mathbb{N} \times \mathbb{N})$ and in which $R^{\mathcal{M}}$ is equal to
$$
\begin{gathered}
\{(a,(a, b)): a \in \mathbb{N}, b \in \mathbb{N}\} \cup\{((a, b), b): a \in \mathbb{N}, b \in \mathbb{N}\} \\
\cup\{((a, b),(a+b, a \cdot b)): a \in \mathbb{N}, b \in \mathbb{N}\}
\end{gathered}
$$
Show that $\mathbb{N}$ is interpretable in $\mathcal{M}$. Show that the set of universally valid formulas of the language $\mathcal{L}$ is not recursive.
(g) This time, $\mathcal{L}$ is the language that contains a binary predicate symbol $D$ and a binary function symbol $\pm$. Let $\mathcal{M}$ be the $\mathcal{L}$-structure whose underlying set is $\mathbb{N}$ and in which $\pm$ is interpreted by addition and $D$ by the relation 'divides' (Dxy is true if and only if $x$ divides $y$ ).
Show that the element 1 and the relation $x=y \cdot(y+1)$ are definable in $\mathcal{M}$. Show that $\mathcal{M}$ is strongly undecidable.

Chris Trentman
Chris Trentman
Numerade Educator
18:05

Problem 12

Let $f$ be a total recursive function from $\mathbb{N}$ into $\mathbb{N}$ and let $F\left[v_{0}, v_{1}\right]$ be a $\Sigma$ formula that represents it and is such that
$$
\mathcal{P} \vdash \forall v_{1} \exists v_{0} F\left[v_{0}, v_{1}\right] .
$$
The purpose of this exercise is to show that there exist total recursive functions that are not provably total.
(a) Let $F\left[v_{0}, v_{1}, \ldots, v_{k}\right]$ be a $\Sigma$ formula. Show that the set
$$
\left\{\left(n_{0}, n_{1}, \ldots, n_{k}\right): \mathbb{N} \vDash F\left[n_{0}, n_{1}, \ldots, n_{k}\right]\right\}
$$
is recursively enumerable.
(b) Let $f$ be a total function from $\mathbb{N}$ into $\mathbb{N}$. Show that the following two conditions are equivalent:
(i) $f$ is recursive;
(ii) there exists a $\Sigma$ formula that represents $f$.
(c) Show that there exists a partial recursive function $h$ of two variables such that, for every integer $n$,
- if $a$ is the Gödel number of $\Sigma$ formula, say $F\left[v_{0}, v_{1}\right]$, and if there exists an integer $m$ such that $\mathcal{P} \vdash F[\underline{m}, \underline{n}]$, then
$$
\mathcal{P} \vdash F[\underline{h(a, n)}, \underline{n}]
$$
- if $a$ is the Gödel number of the formula $F\left[v_{0}, v_{1}\right]$ and if there does not exist an integer $m$ such that $\mathcal{P} \vdash F[\underline{m}, \underline{n}]$, then $h(a, n)$ is not defined;
- $h(a, n)=0$ otherwise.
(d) We now define a function $g$ from $\mathbb{N}^{3}$ into $\mathbb{N}$ in the following way: for every integer $n$,
- if $a$ is the Gödel number of a $\Sigma$ formula $F\left[v_{0}, v_{1}\right]$ and if $b$ is the Gödel number of a derivation in $\mathcal{P}$ of the formula
$\forall v_{1} \exists v_{0} F\left[v_{0}, v_{1}\right]$
then $g(a, b, n)=h(a, n)$;
- $g(a, b, n)=0$ otherwise.
Show that $g$ is a total recursive function.
(e) Show that there exist total recursive functions that are not provably total.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
02:26

Problem 13

This exercise should be worked after reading Chapter 7 on set theory. In particular, one needs to know what the cardinal number $2^{\kappa_{0}}$ is.
(a) Show that if $T$ is a consistent theory obtained by adjoining a finite number of formulas to $\mathcal{P}$, then $T$ is not complete.
(b) For every integer $n$ and every sequence
$$
s=(s(0), s(1), \ldots, s(n-1)) \in\{0,1\}^{n},
$$
construct a closed formula $F_{s}$ such that, for all $s$,
(i) $F_{(s(0), s(1), \ldots, s(n-1), 1)}=\neg F_{(s(0), s(1), \ldots, s(n-1), 0)}$;
(ii) $\mathcal{P} \cup\left\{F_{\emptyset}, F_{(s(0))}, F_{(s(0), s(1))}, \ldots, F_{(s(0), s(1), \ldots, s(n-1))}\right\}$ is a consistent theory.
(c) Show that there exist $2^{\kappa_{0}}$ theories that include $\mathcal{P}$ and that are pairwise inequivalent.

Adam Dehollander
Adam Dehollander
Numerade Educator
07:00

Problem 14

Some notions from set theory are required for this exercise as well. Knowledge of a certain amount of model theory is also required (elementary extensions and the method of diagrams).
Let $\mathcal{M}$ be an elementary extension of $\mathbb{N}$ and let $X$ be a subset of $\mathbb{N}$. Recall (Chapter 3, Definition $3.96$ that $X$ is definable in $\mathcal{M}$ if there exists a formula $F$ of $\mathcal{L}_{0}$ with one free variable and with parameters from $\mathcal{M}$ such that, for all $n \in \mathbb{N}$,
$n \in X \quad$ if and only if $\quad \mathcal{M} \vDash F[n] .$
(a) Show that if $\mathcal{M}$ is countable, then the set of subsets of $\mathbb{N}$ that are definable in $\mathcal{M}$ is countable.
(b) Show that for every subset $X$ of $\mathbb{N}$, there exists a countable elementary extension $\mathcal{M}$ of $\mathbb{N}$ in which $X$ is definable.
(c) Show that there exist $2^{\kappa_{0}}$ countable elementary extensions of $\mathbb{N}$ that are pairwise non-isomorphic.

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
01:00

Problem 15

(a) What is there that is paradoxical in the statement of Epimenides (see the introduction to this chapter).
(b) In a Carpathian village, there lives a barber who shaves all the men who do not shave themselves and only these. What can you say about this barber?

Nick Johnson
Nick Johnson
Numerade Educator