# Ordinal Problem.

• Nov 23rd 2010, 10:56 AM
skamoni
Ordinal Problem.
Hi, the question posed to me is as follows:

Prove or find a counterexample to:

1. $\alpha\cup\{\alpha \}$ is an ordinal $\Rightarrow \alpha$ is an ordinal
2. $\alpha\cup\bigcup{\alpha}$ is an ordinal $\Longleftrightarrow \alpha$ is an ordinal

I've found proofs to both of these questions but I'm not entirely sure that they are both correct. Can anyone see a way these implications do not hold?

Thank you.
• Nov 23rd 2010, 11:17 AM
MoeBlee
a u {a} is an ordinal <-> a is an ordinal.

a is an ordinal -> a u Ua is an ordinal.

But it it does not hold for all a that

a u Ua is an ordinal -> a is an ordinal.

You should find a simple example of an a such that a u Ua is an ordinal but a is not an ordinal.