Exercise sheet 3
1. Is the class of ordinals in bijection with the class of cardinals? Prove or
confute it.
2. Compute the cardinal numbers
1
2
0 , and ω · ℵω.
3. Compute the ordinal numbers ωω3
2 and (ω · ω)ω
· ω1.
4. Using Cantor normal forms, decide whether the ordinals ωω1
· ω1 and ωω1
1
are equal.
5. Define the function f from ordinals to cardinals as follows:
• f(0) = 0;
• f(α + 1) = ℵf(α);
• if α is a limit ordinal, f(α) = supβ<α f(β).
Show that a cardinal κ is weakly inaccessible if and only if f(κ) = κ.
6. Recall the definition of transitive closure of a set X as:
• X0 = X;
• Xn+1 = Xn;
• TC(X) = n∈N Xn.
Show that this really is the transitive closure of X, i.e. if Y is a transitive
set such that X ⊆ Y then TC(X) ⊆ Y .
Conclude that X is transitive if and only if X = TC(X).
7. Assume that there is a huge cardinal. Can we conclude that Vopˇenka’s
principle is provable in ZFC set theory?
1