Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I am curious whether $\omega_1^{CK} - \omega$ would result in a finite set or infinite set. Does anyone know what happens?

Edit: OK, let me add one more question: Suppose that we take $\omega \cdot \kappa$ where $\kappa$ is some ordinal. What would be $\kappa$ that is the least ordinal so that $\omega_1^{CK} < \omega \cdot \kappa$? (.. and this is already answered.)

share|improve this question

2 Answers 2

up vote 1 down vote accepted

$\omega_1^{\text{CK}}$ cannot be a successor ordinal, and therefore $\omega + n \in \omega_1^{\text{CK}} \setminus \omega$ for all $n \in \omega$.

share|improve this answer
    
Oh.. right. I forgot that it cannot be a successor ordinal... Haha. Thanks. –  Mark Zwazingker Mar 2 '13 at 4:29

It is not only it is not a successor, but also $\omega_1^{CK}\setminus\omega$ is isomorphic to $\omega_1^{CK}$.

It is not hard to show that $\omega_1^{CK}$ is closed under products, so $\kappa=\omega_1^{CK}+1$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.