6
$\begingroup$

Background

The beth function is defined recursively by: $\beth_0 = \aleph_0$, $\beth_{\alpha + 1} = 2^{\beth_\alpha}$, and $\beth_\lambda = \bigcup_{\alpha < \lambda} \beth_\alpha$. Since the beth function is strictly increasing and continuous, it is guaranteed to have arbitrarily large fixed points by the fixed-point theorem on normal functions.

The cofinality of an ordinal $\alpha$ is the smallest ordinal $\beta$ such that there are unbounded increasing functions $f : \beta \to \alpha$.

$\endgroup$

2 Answers 2

5
$\begingroup$

Yes. The definable class $C$ of fixed points $\beth_\kappa = \kappa$ is closed and unbounded and therefore contains ordinals of all possible cofinalities. Specifically, let $\kappa_\alpha$ denote the $\alpha$-th element of $C$. Note that $\kappa_\delta = \sup_{\alpha<\delta} \kappa_\alpha$ for every limit ordinal $\delta$. If $\lambda$ is a regular cardinal then $\kappa_\lambda$ has cofinality $\lambda$.

$\endgroup$
6
$\begingroup$

Yes, the κth fixed point will have the same cofinality as κ, since the κ many earlier fixed points are cofinal in κ.

More generally, for any function f from the ordinals to the ordinals, it is not difficult to see that the collection of ordinals α for which f " α subset α, that is, the closure points of f, will form a closed unbounded class C. And for any limit ordinal β, the β-th element of C has the same cofinality as β, since it is the limit of the β many smaller elements of C.

In particular, if we consider the beth function, where α maps to Bethα, then β is a closure point of this function if and only if Bethα < β for all α < β, and this implies Bethβ = β. So the club C in this case consists of the Beth fixed points.

$\endgroup$
2
  • $\begingroup$ I think this is our record for closest back-to-back answers. :) $\endgroup$ Mar 6, 2010 at 18:08
  • $\begingroup$ I think you're right! $\endgroup$ Mar 6, 2010 at 18:19

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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