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 7 Set theory - all with Video Answers

Educators

Chapter Questions

Problem 1

The notions of natural number, membership, function, and so on involved in this exercise are intuitive (not those of the universe $\mathcal{U}$ ). We will use them to construct a universe that satisfies some of the axioms of ZF.
Let $W$ denote the set of finite subsets of $\mathbb{N}$.
(a) Let $\phi$ be a bijection from $\mathbb{N}$ onto $W$ and let $\varepsilon_{\phi}$ be the binary relation on $\mathbb{N}$ defined as follows:
for all integers $x$ and $y, \quad x \varepsilon_{\phi} y$ if and only if $x \in \phi(y)$.
Show that the universe $\mathcal{M}_{\phi}=\left\langle\mathbb{N}, \varepsilon_{\phi}\right\rangle$ satisfies all the axioms of $Z F$ except the axiom of infinity. Show that if, for all $x, y \in \mathbb{N}, x \in \phi(y)$ implies $x<y$, then $\mathcal{M}_{\phi}$ also satisfies the axiom of foundation.
(b) Show that the mapping $\zeta$ whose value for $A \in W$ is $\sum_{a \in A} 2^{a}$ [with the convention that $\zeta(\emptyset)=0]$ is a bijection from $W$ onto $\mathbb{N}$. Let $\theta$ be the inverse mapping. Show that $\mathcal{M}_{\theta}$ is a model of $\mathrm{ZF}^{-}$and of AF.
(c) Find a bijection $\phi$ from $\mathbb{N}$ onto $W$ such that $\mathcal{M}_{\phi}$ does not satisfy AF.

Angelo Rendina

Numerade Educator

05:12

Problem 2

Show that the class $O n^{\prime}$ defined in $U$ by the formula
$$
\forall y((y \subseteq x \wedge \neg y=x \wedge y \text { is transitive }) \Rightarrow y \in x)
$$
is the class of ordinals.

Ibrahima Barry

Numerade Educator

02:27

Problem 3

Let $x$ be a set and $\Gamma(x)$ be the class of ordinals that are subpotent to $x$.
Show that $\Gamma(x)$ is an ordinal, that it is the least ordinal not subpotent to $x$, and that it is a cardinal. We call this Hartog's cardinality of $x$.
Characterize $\Gamma(x)$ assuming that $U$ satisfies the axiom of choice.

Angelo Rendina

Numerade Educator

00:59

Problem 4

This exercise is devoted to some statements that are equivalent to the axiom of choice.

A choice function on a set $a$ is a mapping $\phi$ from the set of non-empty subsets of $a$ into $a$ such that, for every non-empty subset $x \subseteq a, \phi(x) \in x$.

Show that $A C$ is equivalent (in the theory $Z F$ ) to each of the following statements:
(a) For every set $a$, there exists at least one choice function on $a$.
(b) If $x$ and $y$ are sets and if $g$ is a surjective mapping from $x$ onto $y$, then there exists a mapping $h$ from $y$ into $x$ such that $g \circ h$ is the identity mapping from $y$ into itself. (c) For every set $a$ whose elements are non-empty and pairwise disjoint, there exists a set $b$ whose intersection with each of the elements of $a$ is a singleton.
(d) For any sets $a$ and $b$, either $a$ is subpotent to $b$ or $b$ is subpotent to $a$.
[Exercise 3 may be used in proving the equivalence between (d) and $\mathrm{AC}$ ]. Property (d) is known as the trichotomy of cardinals because it can also be expressed in the following way: given two cardinal classes $\lambda$ and $\mu$, one and only one of the three possibilities holds:
$$
\lambda=\mu \quad \text { or } \quad \lambda<\mu \quad \text { or } \quad \mu<\lambda .
$$
More simply, trichotomy is satisfied if and only if the ordering of the cardinal classes is a total ordering.

Srilakshmi E K

Numerade Educator

01:29

Problem 5

Show that in the theory ZF $+\mathrm{AF}$, the axiom of choice is equivalent to each of the following three statements:
(a) If the set $x$ has a well-ordering, then $\wp(x)$ has a well-ordering.
(b) For every ordinal $\alpha, \wp$ ( $\alpha$ ) has a well-ordering.
(c) Every totally ordered set has a well-ordering.
6. Without using the axiom of choice, show that for every non-empty set $a$, the following properties are equivalent:
(I) $a$ includes a denumerable subset.
(2) $a$ includes a denumerable subset $b$ such that $a$ and $a-b$ are equipotent.
(3) For every denumerable set $b, a$ and $a \cup b$ are equipotent.
(4) For every finite set $x, a$ and $a \cup x$ are equipotent.
(5) For every finite subset $x$ of $a, a$ and $a-x$ are equipotent.

Mohamed Mohamed

Numerade Educator

02:20

Problem 6

There exists a non-zero integer $n$ such that, for every subset $x$ of $a$ that is subpotent to $n, a$ and $a-x$ are equipotent.

James Chok

Numerade Educator

04:57

Problem 6

Without using the axiom of choice, show that for every non-empty set $a$, the following properties are equivalent:
(1) $a$ includes a denumerable subset.
(2) $a$ includes a denumerable subset $b$ such that $a$ and $a-b$ are equipotent.
(3) For every denumerable set $b, a$ and $a \cup b$ are equipotent.
(4) For every finite set $x, a$ and $a \cup x$ are equipotent.
(5) For every finite subset $x$ of $a, a$ and $a-x$ are equipotent.
(6) There exists a non-zero integer $n$ such that, for every subset $x$ of $a$ that is subpotent to $n, a$ and $a-x$ are equipotent.
(7) There exists a non-zero integer $n$ such that, for every set $x$ of cardinality $n$, $a$ and $a \cup x$ are equipotent.
(8) for every $t, a$ and $a \cup\{t\}$ are equipotent.
(9) There exists an element $t \in a$ such that $a$ and $a-\{t\}$ are equipotent.
(10) There exists a subset of $a$ that is non-empty, different from $a$, and equipotent to $a$.
(11) There exists a subset $b \subseteq a$ that is non-empty, different from $a$, and such that $a$ is subpotent to $b$.

Angelo Rendina

Numerade Educator

01:17

Problem 7

There exists a non-zero integer $n$ such that, for every set $x$ of cardinality $n$, $a$ and $a \cup x$ are equipotent.

Nick Johnson

Numerade Educator

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Problem 7

Determine the cardinality of each of the following sets:
$$
\begin{aligned}
&x_{1}=\left\{f \in \mathbb{N}^{\mathbb{N}}:(\forall n \in \mathbb{N})(\forall p \in \mathbb{N})(n<p \Rightarrow f(n)<f(p))\right\} ; \\
&x_{2}=\left\{f \in \mathbb{N}^{\mathbb{N}}:(\exists p \in \mathbb{N})(\forall n \in \mathbb{N})(f(n) \leq p)\right\} ;
\end{aligned}
$$ $$
\begin{aligned}
&x_{3}=\left\{f \in \mathbb{Q}^{\mathbb{N}}:(\forall n \in \mathbb{N})(\forall p \in \mathbb{N})(n<p \Rightarrow f(n)<f(p))\right\} \\
&x_{4}=\left\{f \in \mathbb{Q}^{\mathbb{N}}:(\exists p \in \mathbb{Q})(\forall n \in \mathbb{N})(f(n) \leq p)\right\} ; \\
&x_{5}=x_{3} \cap x_{4} ; \\
&x_{6}=\left\{f \in \mathbb{Q}^{\mathbb{N}}:(\exists n \in \mathbb{N})(\forall p \in \mathbb{N})(n \leq p \Rightarrow f(n)=f(p))\right\} ; \\
&x_{7}=\left\{f \in \mathbb{R}^{\mathbb{N}}:(\forall r \in \mathbb{R})(\exists n \in \mathbb{N})(f(n) \geq r)\right\} .
\end{aligned}
$$

Donna Densmore

Numerade Educator

03:49

Problem 8

for every $t, a$ and $a \cup\{t\}$ are equipotent.

Ramesh Singh

Numerade Educator

03:20

Problem 8

Determine the cardinality of each of the following sets:
$$
\begin{aligned}
&E_{0}=\text { the set } \mathbb{Q}^{\mathbb{N}} \text { of sequences of rational numbers; } \\
&E_{1}=\text { the set } \mathbb{R}^{\mathbb{N}} \text { of sequences of real numbers; } \\
&E_{2}=\text { the set of sequences of rationals that converge to } 0 ; \\
&E_{3}=\text { the set of convergent sequences of rationals; } \\
&E_{4}=\text { the set of bounded sequences of rationals; } \\
&E_{5}=\text { the set of unbounded sequences of rationals; } \\
&E_{6}=\text { the set } \mathbb{R}^{\mathbb{Q}} \text { of mappings from } \mathbb{Q} \text { into } \mathbb{R} ; \\
&E_{7}=\text { the set of continuous mappings from } \mathbb{R} \text { into } \mathbb{R} ; \\
&E_{8}=\text { the set of open intervals of } \mathbb{R} ; \\
&E_{9}=\text { the set of open subsets of } \mathbb{R} \text { (with the usual topology). }
\end{aligned}
$$

Madi Sousa

Numerade Educator

02:03

Problem 9

There exists an element $t \in a$ such that $a$ and $a-\{t\}$ are equipotent.

Nick Johnson

Numerade Educator

04:02

Problem 9

Determine the cardinality of each of the following sets:
$$
\begin{aligned}
&a_{1}=\left\{f \in \omega^{\omega}:(\forall n \in \omega)(\forall p \in \omega)(f(n) \leq p)\right\} \\
&a_{2}=\left\{f \in \omega^{\omega}:(\forall n \in \omega)(\exists p \in \omega)(f(n) \leq p)\right\} \\
&a_{3}=\left\{f \in \omega^{\omega}:(\exists n \in \omega)(\forall p \in \omega)(f(n) \leq p)\right\} \\
&a_{4}=\left\{f \in \omega^{\omega}:(\exists n \in \omega)(\exists p \in \omega)(f(n) \leq p)\right\} \\
&a_{5}=\left\{f \in \omega^{\omega}:(\exists p \in \omega)(\forall n \in \omega)(f(n) \leq p)\right\} \\
&a_{6}=\left\{f \in \omega^{\omega}:(\forall p \in \omega)(\exists n \in \omega)(f(n) \leq p)\right\} \\
&b_{1}=\left\{f \in \omega^{\omega}:(\forall n \in \omega)(\forall p \in \omega)(f(n) \geq p)\right\} \\
&b_{2}=\left\{f \in \omega^{\omega}:(\forall n \in \omega)(\exists p \in \omega)(f(n) \geq p)\right\} \\
&b_{3}=\left\{f \in \omega^{\omega}:(\exists n \in \omega)(\forall p \in \omega)(f(n) \geq p)\right\} \\
&b_{4}=\left\{f \in \omega^{\omega}:(\exists n \in \omega)(\exists p \in \omega)(f(n) \geq p)\right\} \\
&b_{5}=\left\{f \in \omega^{\omega}:(\exists p \in \omega)(\forall n \in \omega)(f(n) \geq p)\right\} \\
&b_{6}=\left\{f \in \omega^{\omega}:(\forall p \in \omega)(\exists n \in \omega)(f(n) \geq p)\right\}
\end{aligned}
$$

Mengchun Cai

Numerade Educator

00:59

Problem 10

Assume that the universe satisfies the axiom of choice. Let $a$ and $b$ be two infinite sets whose cardinalities are $\lambda$ and $\mu$, respectively. Assume $\lambda>\mu$. Let $g$ be an injective mapping from $b$ into $a$. The reader should determine the cardinalities Assume that the universe satisfies the axiom of choice. Let $a$ and $b$ be two infinite sets whose cardinalities are $\lambda$ and $\mu$, respectively. Assume $\lambda>\mu$. Let $g$ be an injective mapping from $b$ into $a$. The reader should determine the cardinalities of each of the following sets:
$$
\begin{aligned}
&y_{1}=\left\{f \in b^{a}: \operatorname{card}(\vec{f}(a))=1\right\} \\
&y_{2}=\left\{f \in b^{a}:(\mathrm{V} x \in \wp(a))(\mathrm{card}(\bar{f}(x)) \leq 1)\right\} \\
&y_{3}=\left\{f \in b^{a}: \operatorname{card}\left(\bar{f}^{-1}(b)\right)=\lambda\right\} \\
&y_{4}=\left\{f \in b^{a}: \operatorname{card}(\bar{f}(a))=2\right\} \\
&y_{5}=a-\bar{g}(b) ; \\
&y_{6}=\left\{f \in b^{a}:(\forall y \in b)(f(g(y))=y)\right\} \\
&y_{7}=\left\{f \in b^{a}: \operatorname{card}(\bar{f}(a))=\mu\right\}
\end{aligned}
$$
[Recall that if $f \in b^{a}$, then $\bar{f}$ and $\bar{f}^{-1}$ respectively denote the direct image mapping induced by $f$ from $\wp(a)$ into $\rho(b)$ and the inverse image mapping from $\wp(b)$ into $\wp(a)$.]

Srilakshmi E K

Numerade Educator

10:45

Problem 10

There exists a subset of $a$ that is non-empty, different from $a$, and equipotent to $a$.

Paul A.

California State Polytechnic University, Pomona

01:43

Problem 11

Assume that the universe satisfies the axiom of choice. Lct $a$ be an infinite set and let $\lambda$ be its cardinality. Set
$$
\wp^{*}(a)=\{x \in \wp(a): \operatorname{card}(x)=\operatorname{card}(a-x)\}
$$
(a) Show that for every integer $n$, if $n \neq 0$, we can find sets $a_{1}, a_{2}, \ldots, a_{n}$, each of cardinality $\lambda$, that constitute a partition of $a$ (i.c. these sets are pairwise disjoint and $\left.\bigcup_{1 \leq i \leq n} a_{i}=a\right)$.
(b) Determine the cardinality of each of the elements of $g^{*}(a)$.
(c) Use the result from (a) for $n=3$ to determine the cardinality of $\wp^{*}(a)$.
(d) Show that, for every set $a_{1} \in \rho^{*}(a)$, there exists a bijection $f$ from $a$ onto $a$ such that, for every $x \in a, f(x)=x$ if and only if $x \in a_{1}$.
(e) Determine the cardinality of the set of bijections from $a$ onto $a$.
(f) Let $b$ be an element of $\wp^{*}(a)$. Determine the cardinality of the set of bijections from $a$ onto $a$ whose restriction to $b$ is the identity on $b$.
(g) What is the cardinality of the set of injections from $a$ into $\rho(a) ?$

Mohamed Mohamed

Numerade Educator

14:56

Problem 12

Assume the universe satisfies the axiom of choice. We are given an infinite cardinal $\lambda$, an ordinal $\alpha$, and a family of sets $\left(X_{\beta}\right)_{b \in \alpha}$ indexed by $\alpha$ that satisfies
$$
\text { for every } \beta \in \alpha, \quad \operatorname{card}\left(X_{\beta}\right)<\lambda,
$$
and
for every $\beta \in \alpha$ and $\gamma \in \alpha$, if $\beta<\gamma, X_{\beta} \subseteq X_{\gamma} .$
Show that card $\left(\bigcup_{\beta \in \alpha} X_{\beta}\right) \leq \lambda .$

Chris Trentman

Numerade Educator

06:04

Problem 13

Assume that the universe satisfies the axiom of choice. Show that for every family $\left(\lambda_{\alpha}\right)_{\alpha \in \kappa}$ of non-zero cardinals indexed by an infinite cardinal $x$, we have
$$
\sum_{\alpha \in K} \lambda_{\alpha}=\sup \left(\kappa, \sup _{\alpha \in K}\left(\lambda_{\alpha}\right)\right)
$$

Mengchun Cai

Numerade Educator

06:53

Problem 14

Suppose that the universe satisfies the axiom of choice.
Let $\mu$ be an infinite cardinal. By induction on the integers, we define a sequence of cardinal $\left(\lambda_{n}\right)_{n \in \omega}$ by setting
- $\lambda_{0}=\mu$;
- for every $n \in \omega, \lambda_{n+1}=2^{\lambda_{n}} .$
Set $\lambda=\sum_{n \in \omega} \lambda_{n} .$
(a) Show that $\lambda^{\mu}=\mu^{2}=\lambda^{\lambda}=2^{\lambda}$.
(b) Show that, for every cardinal $\gamma$,
$$
\begin{aligned}
&\text { if } \aleph_{0} \leq \gamma \leq \lambda, \quad \text { then } \lambda^{\gamma_{0}}=\lambda^{\gamma}=\lambda^{\lambda} \\
&\text { if } \gamma \geq \lambda, \quad \text { then } \lambda^{\gamma}=2^{\gamma}
\end{aligned}
$$
(c) Show that there exist cardinals $\alpha, \beta, \gamma$, and $\delta$ such that
$$
\alpha<\beta, \quad \gamma<\delta, \quad \text { and } \quad \alpha^{\gamma}=\beta^{\delta} .
$$

Mohamed Mohamed

Numerade Educator

05:28

Problem 15

Let $\alpha$ and $\beta$ be two ordinals. By definition, $\beta$ is colinal with $\alpha$ if and only if there exists a strictly increasing mapping $f$ from $\beta$ into $\alpha$ whose image does not have a strict upper bound. More precisely, this means that
- for all ordinals $\gamma$ and $\delta$ belonging to $\beta$, if $\gamma<\delta$, then $f(\gamma)<f(\delta)$, and - for every ordinal $\xi \in \alpha$, there exists $\gamma \in \beta$ such that $f(\gamma) \geq \xi$.
Remark This must not be confused with the notion of cofinal in : a subset $Y$ of an ordered set $\langle X, \leq\rangle$ is cofinal in $X$ if, for every $x \in X$, there exists $y \in Y$ such that $x \leq y$. Thus, for example, while $\omega$ is clearly not cofinal in $s_{\omega}$, the mapping $n \mapsto \aleph_{n}$ witnesses that $\omega$ is cofinal with $\aleph_{\omega}$.
(a) Show that the (meta-)relation 'is cofinal with' defined on the class $O n$ is reflexive, transitive, and is not symmetric. With which ordinals is the ordinal 1 cofinal?
(b) Show that for every ordinal $\alpha$, the class of ordinals $\beta$ such that $\beta$ is cofinal with $\alpha$ is a non-empty sct. The least ordinal belonging to this set is called the cofinality of $\alpha$ and is denoted by cof $\alpha$. An ordinal that satisfies $\operatorname{cof}(\alpha)=\alpha$ is called a regular ordinal.

Show that for every ordinal $\alpha, \operatorname{cof}(\alpha) \leq \alpha$ and that $\operatorname{cof}(\alpha)$ is a regular ordinal.
(c) Show that for all ordinals $\alpha$ and $\beta, \beta<c o f(\alpha)$ if and only if every mapping from $\beta$ into $\alpha$ is strictly bounded in $\alpha$. (d) Show that every regular ordinal is a cardinal. Show that for every cardinal $\lambda_{1}$ $\lambda$ is a regular ordinal if and only if it is a regular cardinal in the sense of Definition $7.87 .$
(e) Assume that the universe satisfies the axiom of choice. Show that for every ordinal $\alpha_{1} \kappa_{\alpha+1}$ is regular. Show that if $\alpha$ is a limit ordinal, then $\operatorname{cof}\left(\kappa_{\alpha}\right)=$ $\operatorname{cof}(\alpha) .$
(f) Determine the first ordinal (respectively, the first cardinal) strictly greater than $\omega$ with which $\omega$ is cofinal.

Carolyn Behr-Jerome

Numerade Educator

14:56

Problem 16

Assume that the universe satisfies the axiom of choice. This exercise presupposes the concepts and results from the previous exercise.
(a) Show that for every cardinal $\lambda, \lambda^{c o f(\lambda)}>\lambda$ (use König's theorem).
(b) Show that $\omega$ is not cofinal with card $\left(2^{\omega}\right)$.
(c) Suppose that the universe satisfies the generalized continuum hypothesis $(\mathrm{GCH})$, i.e. that for every ordinal $\alpha, 2^{\mathrm{K}_{\alpha}}=\mathbb{N}_{\alpha+1}$.

Let $\lambda$ be an infinite cardinal. Show that for every cardinal $\mu$ other than 0 , we have
$$
\lambda^{\mu}= \begin{cases}\lambda & \text { if } \mu<\operatorname{cof}(\lambda) \\ 2^{\lambda} & \text { if } \operatorname{cof}(\lambda) \leq \mu \leq \lambda \\ 2^{\mu} & \text { if } \lambda<\mu\end{cases}
$$

Chris Trentman

Numerade Educator

01:06

Problem 17

Let $\Phi$ be a definable strictly increasing function from the class $O n$ of ordinals into itself. We say that $\Phi$ is continuous at a limit ordinal $\alpha$ if $\Phi(\alpha)=$ $\sup _{\beta \in \alpha} \Phi(\beta) .$ Such a function $\Phi$ is called continuous if it is continuous at all limit ordinals.
An ordinal $\alpha$ such that $\Phi(\alpha)=\alpha$ is called a fixed point of $\Phi$.
(a) Show that every strictly increasing function $\Phi$ from $O n$ into $O n$ has the following property:
for every ordinal $\alpha, \quad \Phi(\alpha) \geq \alpha$.
(b) Show that if $\Phi$ is a strictly increasing function from $O n$ into $O n$ that is continuous at every limit ordinal whose cofinality is $\omega$, then for every ordinal $\alpha, \Phi$ has a fixed point that is greater than $\alpha$.
(c) Show that if $\Phi$ and $\Psi$ are two strictly increasing functions from $O n$ into $O n$ that are continuous at every limit ordinal whose cofinality is $\omega$, then for every ordinal $\alpha, \Phi$ and $\Psi$ have a common fixed point greater than $\alpha$.
(d) Suppose the universe satisfies the axiom of choice. Show that for every ordinal $\alpha$, there exists an ordinal $\beta>\alpha$ such that card $\left(V_{\beta}\right)=\aleph_{\beta}=\beta$.

Carson Merrill

Numerade Educator

01:03

Problem 18

Assume the universe satisfies $\mathrm{AC}+\mathrm{GCH}$. (In fact, it can be proved that $\mathrm{AC}$ is true in every model of $\mathrm{ZF}+\mathrm{GCH}$.) (a) Consider the function from $O n$ into $O n$ whose value at an ordinal $\alpha$ is $\aleph_{0}^{K_{\alpha}} .$ Is this function continuous at every limit ordinal? (For the definition of continuity, see Exercise 17.) Answer the same question for the function whose value at an ordinal $\alpha$ is $\kappa_{\alpha}^{\mathrm{K}_{0}}$.
(b) Let $\delta$ be an ordinal. Is the function whose value at an ordinal $\alpha$ is $\delta+\alpha$ (ordinal sum) continuous at every limit ordinal? Answer the same question for the functions whose value at an ordinal $\alpha$ are respectively $\alpha+\delta, \alpha \cdot \delta$, and $\delta \cdot \alpha$ (these ordinal operations are explained in Definitions $7.31$ and $7.32$ ).

Carson Merrill

Numerade Educator

01:55

Problem 19

Show that the axiom of foundation is equivalent (in the presence of the axioms of ZF) to the following axiom scheme;
for every formula $F$ with one free variable in the language $\{\in, \simeq\}$,
$$
\exists v_{0} F\left[v_{0}\right] \Rightarrow \exists v_{0}\left(F\left[v_{0}\right] \wedge \forall v_{1}\left(v_{1} \in v_{0} \Rightarrow \neg F\left[v_{1}\right]\right)\right)
$$

Linh Vu

Numerade Educator

05:28

Problem 20

In this exercise, we assume the axiom of choice. Let $\lambda$ be an uncountable regular cardinal (see Definition 7.87). A subset $X$ of $\lambda$ is closed cofinal if
(1) it is closed: this means that for every subset $X_{0}$ of $X$ that satisfies card $\left(X_{0}\right)<\lambda_{1} \sup \left(X_{0}\right) \in X$ [note that because $\lambda$ is regular, $\sup \left(X_{0}\right)$ is an ordinal that is strictly less than $\lambda$ ].
(2) it is colinal in $\lambda_{i}$ this means that for every $\alpha \in \lambda$, there exists $\beta \in X$ such that $\alpha \leq \beta$.
(a) Show that the collection of closed cofinal subsets of $\lambda$ forms a filterbase on $\lambda$ (see Chapter 2).
(b) Show that if $I$ is a non-empty set whose cardinality is strictly less than $\lambda$ and if $\left(X_{i}: i \in I\right)$ is a family of closed cofinal subsets of $\lambda$, then $\bigcap_{i \in I} X_{i}$ is a closed cofinal subset of $\lambda$.
(c) A subset $Y$ of $\lambda$ is called stationary if it intersects every closed cofinal subset. Show that the following three properties are equivalent:
(1) there exists a pair of disjoint stationary sets;
(2) there exists at least one stationary set that does not include a closed cofinal set;
(3) the filter $\mathcal{F}$ generated by the collection of closed cofinal subsets is not an ultrafilter.
(d) Let $X=\left(X_{a}: \alpha \in \lambda\right)$ be a sequence of subsets of $\lambda$. The set
$$
\left\{\alpha \in \lambda: \alpha \in X_{\alpha}\right\}
$$
is called the diagonal intersection of $X$ and is denoted by $\Delta(X)$. Show that if $X$ satisfies the following three conditions,
(1) for every $\alpha \in \lambda, X_{\alpha}$ is closed cofinal;
(2) for every $\alpha \in \lambda$ and $\beta \in \lambda$, if $\alpha \in \beta$, then $X_{\beta} \subseteq X_{\alpha}$;
(3) for every $\alpha \in \lambda$, if $\alpha$ is a limit ordinal, then $X_{\alpha \alpha}=\bigcap_{\beta \in \alpha} X_{\beta}$; then $\Delta(X)$ is closed cofinal. (e) Prove the following theorem (known as Fodor's theorem).
Theorem 7.93 Let $f$ be a mapping from $\lambda$ into $\lambda$ such that $\{\alpha \in \lambda:$ $f(\alpha)<\alpha\}$ is stationary. Then there exists $\gamma \in \lambda$ such that $\bar{f}^{-1}(\gamma)$, the inverse image of $\gamma$ under $\bar{f}_{1}$ is stationary.
(f) Assume that $\lambda \geq N_{2}$. Show that the set of ordinals in $\lambda$ that have cofinality $\aleph_{0}$ (see Exercise 15 ) is stationary. Show that the set of ordinals in $\lambda$ that have cofinality $\mathbb{N}_{1}$ is also stationary and is disjoint from the preceding set.
(g) Part (f) shows that for every regular cardinal $\lambda$ strictly greater than $\aleph{1}_{1}$, the conditions from part (c) are satisfied. The argument that we will offer in the next paragraph shows that these conditions are satisfied for $\aleph_{1}$ (indeed, the argument works for any successor cardinal).
For every denumerable ordinal $\alpha$, let $f_{\alpha}$ be a surjective mapping from $\omega$ onto $\alpha$ and, for every $n \in \omega_{1}$ let $h_{n}$ be the mapping from $\aleph_{1}$ into $N_{1}$ defined by $h_{n}(0)=0$ and $h_{n}(\alpha)=f_{\alpha}(n)$ if $\alpha \neq 0$. Show that, for every $n \in \omega$, there exists $\beta_{n} \in \aleph_{1}$ and a stationary subset $Y_{n}$ of $\aleph_{1}$ such that, for every $\gamma \in Y_{n}, f_{\gamma}(n)=\beta_{n} .$ Show that there exists an integer $n$ such that $Y_{n}$ does not include a closed cofinal set.

Carolyn Behr-Jerome

Numerade Educator

01:10

Problem 21

. (a) Show that if $\alpha$ is a limit ordinal strictly greater than $\omega$, then $\left\langle V_{\alpha}, \in\right\rangle$ is a model of the theory $Z$.
(b) Conclude from (a) that if $Z$ is a consistent theory, then the axioms of ZF cannot all be derivable from those of $Z$

Adriano Chikande

Numerade Educator

01:49

Problem 22

Suppose the universe satisfies the axiom of choice.
Consider the class $W$ of sets $x$ such that $\mathrm{Cl}(x)$ (the transitive closure of $x$ ) is denumerable. Show that $\langle W, \in\rangle$ is a model of all the axioms of $Z F$ except the power set axiom.

Diwakar Mandilwar

Numerade Educator

01:00

Problem 23

Show directly, using a diagonal argument, that the half-open interval of reals, $\{x \in \mathbb{R}: 0<x \leq 1\}$, is not denumerable.

Ankit Singh

Numerade Educator