PlanetMath: cardinal exponentiation under GCH

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (200)
Orphanage
Unclass'd
Unproven (511)
Corrections (9)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

cardinal exponentiation under GCH

(Theorem)

Many results about cardinal exponentiation can neither be proved nor disproved in ZFC. If, however, we allow ourselves to use GCH in addition to ZFC, then we have the following theorem, which gives an essentially complete description of the way cardinal exponentiation involving infinite cardinals works.

Theorem Assume the Generalized Continuum Hypothesis holds. Let $\kappa$ and $\lambda$ be cardinals, at least one of which is infinite, and such that $\kappa>0$ and $\lambda>1$ . Then

$\begin{displaymath} % latex2html id marker 158 \lambda^\kappa= \begin{cases} \ka... ...ambda,&\text{if}\quad\kappa<{\mathrm{cf}}(\lambda). \end{cases}\end{displaymath}$

Here, $% latex2html id marker 160 $ {\mathrm{cf}}(\lambda)$$ is the cofinality of $\lambda$ , and $\lambda^+$ is the cardinal successor of $\lambda$ .

"cardinal exponentiation under GCH" is owned by yark.

(view preamble)

Keywords:

GCH, exponentiation

This object's parent.

Log in to rate this entry.
(view current ratings)

Cross-references: cardinal successor, cofinality, cardinals, infinite, GCH, ZFC, cardinal exponentiation
There is 1 reference to this entry.

This is version 6 of cardinal exponentiation under GCH, born on 2004-12-15, modified 2007-01-18.
Object id is 6584, canonical name is CardinalExponentiationUnderGCH.
Accessed 2007 times total.

Classification:

AMS MSC:

03E10 (Mathematical logic and foundations :: Set theory :: Ordinal and cardinal numbers)

Pending Errata and Addenda

None.[ View all 1 ]

Discussion

forum policy

No messages.

Interact