Formula for exponentiation of cardinal numbers

[yaa09d]

Hey there!


Is there any formula to determine the power of a cardinal number to another cardinal number?


Thank you!

 

[micromass]

In general: certainly not. In fact, we can't even find a good formula for $$2^{\aleph_0}$$!

However, there are some partial answers. One of these is the so-called Hausdorff formula which states:

$$\aleph_{\beta+1}{\aleph_\alpha}=\aleph_{\beta}^{\aleph_\alpha}\aleph_{\beta+1}$$.

There are some other answers. I'll provide a reference for them: staff.science.uva.nl/~vervoort/AST/ast.ps on page 46 and page 50. (to read the file, you'll need to be able to read .ps file though).

The most satisfying answer to this question happens when you assume AC (axiom of choice) and GCH (generalized continuum hypothesis). In that case, there IS a nice formula for exponentiation:

$$\aleph_\alpha^{\aleph_\beta}=\left\{\begin{array}{ll}<br /> \aleph_\alpha & \text{if}~\aleph_\beta<cf(\aleph_\alpha)\\<br /> \aleph_{\alpha+1} & \text{if}~cf(\aleph_\alpha}\leq \aleph_\beta\leq \aleph_\alpha\\<br /> \aleph_{\beta+1} & \text{if}~\aleph_\alpha<\aleph_\beta<br /> \end{array}\right.$$

However, GCH is not a generally accept axiom under mathematicians. In fact, most mathematicians think GCH should not be accepted...

 

[yaa09d]

Thank you for the detailed reply.

I thought if we accept CH, then $$2^{\aleph_0}= \aleph_1$$

Can you recommend me a book to study the basics of cardinal numbers, please? I am a first year grad student. I am not familiar enough with cardinals.

 

[micromass]

Yes, of course, under CH it is true that $$2^{\aleph_0}=\aleph_1$$. But in general (thus without CH) there is no formula for $$2^{\aleph_0}$$.

And even if we accepted CH as true, then there would still be no formula for $$2^{\aleph_1}$$... The situation for exponentiation is really complex and it's still a very active field of study!

As for books: there are two books which I highly recommend:
- Introduction to set theory by Hrbaced and Jech: this is a great book which is made specially for the beginner. It contains everything of set theory that an average mathematician should know. The discussion of exponentiation is on page 164.
- Set theory by Jech: this is hands-down the best and most comprehensive book on set theory. Unfortunately it is not suited for somebody who is not yet acquainted with some logic and some set theory. So it could be quite tough...

 

[yaa09d]

Thank you very much.