MTHE6003B
Version 1
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Submission: Once accessed, the submission point for this paper will remain
available for the duration set out in the timings section. It is recommended
that you submit your exam script as soon as completed.
All answers must be submitted as a single scanned pdf document.

You may use any result proved from the lecture notes, unless you are speci cally
asked to prove it. In all questions, marks are given for the quality of the
explanations of the answers, rather than the answers themselves. Work in ZF
unless otherwise speci ed.

Problem 1.

(i) Give an example, not appearing in the lecture notes, of a partial order
which is not linear.
(ii) Give an example of a linear order $(L, \leq)$ for which there are infinitely many order-isomorphisms $f:(L, \leq) \rightarrow(L, \leq)$.
iii) Is there any ordinal $\alpha>1$ which is dense in itself?
iv) Can an ordinal be order-isomorphic to a non-well-founded linear order?

Proof .

Problem 2.

In this question $\cdot$ and $+$ denote ordinal addition and multiplication.
(i) Let us say that an ordinal $\alpha$ is even if there is an ordinal $\beta$ such that
$\alpha=2 \cdot \beta$. Also, let us say that an ordinal $\alpha$ is odd if there is an ordinal
$\beta$ such that $\alpha=(2 \cdot \beta)+1$. Prove that an ordinal is even if and only
if it is not odd.
(ii) Is it true that $\alpha<2 \cdot \alpha$ for every nonzero ordinal $\alpha$ ? (iii) Find an example of an ordinal $\alpha$ such that $2 \cdot \alpha<\alpha \cdot 2$. (iv) Prove that $\alpha<\alpha+\beta$ holds for all ordinals $\alpha, \beta$ such that $\beta>0$.

Proof .

Problem 3.

Let $T=\left\langle\omega_{2}\right.$ be the set of all sequences $s: n \longrightarrow{0,1}$, for $n \in \omega$
and let $\Phi: T \longrightarrow \omega$ be a bijection. Given any function $b: \omega \longrightarrow 2$, let $X_{b}={\Phi(b \mid n) \mid n<\omega} \subseteq T$. Let $\mathcal{A}=\left{X_{b} \mid b: \omega \longrightarrow 2\right}$
(i) Prove that $\mathcal{A}$ is not a mad family of subsets of $\omega$.
(ii) (Working in ZFC ) Prove that there is a mad family $\mathcal{B} \subseteq \mathcal{P}(\omega)$ with $|\mathcal{B}|=2^{\aleph_{0}}$
iii) Prove that if $\mathcal{B} \subseteq \mathcal{P}(\omega)$ is a mad family, then there is some $X \subseteq \omega$
such that

• $Y \backslash X$ is infinite for every $Y \in \mathcal{B}$ and
• there is a mad family $\mathcal{B}^{\prime} \subseteq \mathcal{P}(\omega)$ such that $X \in \mathcal{B}^{\prime}$.

Problem 4.

(i) Prove that the following statements are equivalent.
(a) AC
(b) $V=\bigcup_{\kappa \in \text { Card }} H(\kappa)$
(ii) (Working in ZFC) Prove that there is a family of functions $\mathcal{F} \subseteq \omega_{\omega}$ satisfying both of $(\mathrm{a})$ and $(\mathrm{b})$ below.
(a) For any two distinct $f, g \in \mathcal{F}$ there are only finitely many $n \in \omega$
such that $f(n)=g(n)$.
(b) Given any function $f: \omega \rightarrow \omega$ there is some $g \in \mathcal{F}$ such that $g(n)=f(n)$ for infinitely many $n \in \omega$.
iii) Prove, in $\mathrm{ZF}$, that if a set $X$ has the property that $a \cap \mathrm{Ord} \neq \emptyset$ for every $a \in X$, then $X$ has a choice function.