Homework 7 (419)

Homework 7 (419):

1: Let X be a set of infinite cardinality κ, and call a set Y ⊂ X “small” if

there is a decomposition of X into subsets of cardinality κ each of which

intersects Y in at most one point. Then X is the union of two of its “small”

subsets.

2: Show that: If X is of cardinality κ ≥ ℵ0, then the following sets are of cardinality κ:

a) set of finite sequences of elements of X,

b) set of those functions that map a finite subset of X into X.

3: Show that: An infinite cardinal is regular if and only if κ is not the sum of fewer than

κ cardinals each of which is less than κ

4: Show that: A successor cardinal is regular.

5: Which are the smallest three singular (i.e., not regular) infinite cardinals?

6: Show that: If α is the cofinality of an ordered set, then α is a regular cardinal.

7: Show that: (Each part is one point)

(a) Every element of an N-set is an N-set.

(b) If x is an N-set, then y = x ∪ {x} is an N-set, and if z is an N-set

containing x, then y ⊂ z.

(c) If x is an N-set, y ∈ x, then y is an initial segment of x.

(d) If x is an N-set and Y ⊂ x is one of its initial segments, then Y is an

N-set, and either Y = x or Y ∈ x.

(e) If x, y are N-sets, then x = y or x ∈ y or y ∈ x.

(f) For N-sets x, y define x < y if x ∈ y. Then this is irreflexive, transitive

and trichotomous. Furthermore, if B is a nonempty set of N-sets, then

there is a smallest element of B with respect to < (“well order”).

(g) If x, y are different N-sets, then they are not similar.

(h) Every well-ordered set is similar to a unique N-set.

8: Show that: There is no infinite decreasing sequence of ordinals.

9: Show that: Arbitrary infinite sequence of ordinals includes an infinite nondecreasing

subsequence.

10: Show that: Addition among ordinals is monotonic in both arguments, and strictly

monotonic in the second argument. The same is true of multiplication

provided the first factor is nonzero.