Constructions by transfinite recursion

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Problem 10 from Kirillov A.A., Gvishiani A.D. "Теоремы и задачи функционального анализа" [KiGv88]More problems in Section I.1 "Relations. The Axiom of Choice and Zorn's Lemma"

Two well-ordered countable sets are said to be equivalent if there is a monotonic one-to-one correspondence between them. Let $\mathcal{M}$ be the corresponding set of equivalence classes. Define a partial order relation on $\mathcal{M}$ by setting $\mu \geq \nu$ if the classes $\mu$ and $\nu$ have representatives $M$ and $N$ such that $M$ is equivalent to an initial segment of $N$ (i.e., a set of the form $N(n_0) = \left\{ n \in N : n \leq n_0\right\}$ ).

Prove that:

(a)

$\mathcal{M}$ contains a smallest element $\mu_0$ ;

(b)

any two elements of $\mathcal{M}$ are comparable;

(c)

$\mathcal{M}$ is well ordered;

(d)

$\mathcal{M}$ is uncountable;

(e)

any uncountable set contains a subset of the same cardinality as $\mathcal{M}$ .

Problem I.5.61 from Lavrov I.A., Maksimova L.L. "Задачи по теории множеств, математической логике и теории алгоритмов" [LaMa02]More problems in Chapter I.5 "Ordinal Numbers"

Prove that for any ordinal numbers $\alpha$ and $\beta$ , $\alpha^{\beta }$ exists and is unique.

Problem I.5.63 from Lavrov I.A., Maksimova L.L. "Задачи по теории множеств, математической логике и теории алгоритмов" [LaMa02]More problems in Chapter I.5 "Ordinal Numbers"

(a)(b)(c)

Prove that:

(a)

if $\alpha < \beta$ and $\gamma > 1$ , then $\gamma^{\alpha } < \gamma^{\beta }$ ;

(b)

$\alpha^{\beta +\gamma } = \alpha^{\beta } \cdot \alpha^{\gamma }$ ;

(c)

$(\alpha^{\beta })^{\gamma } = \alpha^{\beta \cdot \gamma }$ .

Problem I.5.65 from Lavrov I.A., Maksimova L.L. "Задачи по теории множеств, математической логике и теории алгоритмов" [LaMa02]More problems in Chapter I.5 "Ordinal Numbers"

A set $S$ is called transitive if the relation

X \leq Y \Leftrightarrow (X = Y \text{ or } X \in Y)

well-orders $S$ , $\emptyset \in S$ , and $X \in Y$ , $Y \in S$ imply $X \in S$ . Prove that for any ordinal number $\alpha \geq 0$ there exists a unique transitive set ordered by type $\alpha$ .

References

[KiGv88] Kirillov A.A., Gvishiani A.D. Теоремы и задачи функционального анализа. 2-е изд., Наука, 1988.

[LaMa02] Lavrov I.A., Maksimova L.L. Задачи по теории множеств, математической логике и теории алгоритмов. 5-е изд., Физматлит, 2002.