Problem 2
What is the minimal element of the set of all subsets of a given set \(X\), partially ordered by set inclusion. What is the maximal element?
Problem 4
By the greatest lower bound of two elements \(a\) and \(b\) of a partially ordered
set \(M\), we mean an element \(c \in M\) such that \(c \leqslant \mathrm{a}, c
\leqslant b\) and there is no element \(d \in \mathrm{M}\) such that \(c
Problem 7
Construct well-ordered sets with ordinals $$ \omega+n, \quad \omega+\omega, \omega+\omega+n, \quad \omega+\omega+\omega $$ Show that the sets are all countable.
Problem 8
Construct well-ordered sets with ordinals $$ \omega \cdot n, \quad \omega^{2}, \quad \omega^{2} \cdot n, \quad \omega^{3}, \ldots $$ Show that the sets are all countable.
Problem 10
Prove that the set \(W(\alpha)\) of all ordinals less than a given ordinal \(\alpha\) is well-ordered.
Problem 11
Prove that any nonempty set of ordinals is well-ordered,
Problem 12
Prove that the set \(\mathrm{M}\) of all ordinals corresponding to a countable set is itself uncountable.
