The power of a cardinal number to another cardinal number

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In summary, the conversation discusses the possibility of finding a formula to determine the power of a cardinal number to another cardinal number. It is mentioned that there are some partial answers, including the Hausdorff formula and a formula under the assumption of AC and GCH. However, GCH is not a widely accepted axiom. The conversation also recommends two books for studying the basics of cardinal numbers.
  • #1
yaa09d
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Hey there!


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


Thank you!
 
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  • #2
In general: certainly not. In fact, we can't even find a good formula for [tex]2^{\aleph_0}[/tex]!

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

[tex]\aleph_{\beta+1}{\aleph_\alpha}=\aleph_{\beta}^{\aleph_\alpha}\aleph_{\beta+1}[/tex].

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:

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

However, GCH is not a generally accept axiom under mathematicians. In fact, most mathematicians think GCH should not be accepted...
 
  • #3
Thank you for the detailed reply.

I thought if we accept CH, then [tex] 2^{\aleph_0}= \aleph_1 [/tex]

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.
 
  • #4
Yes, of course, under CH it is true that [tex]2^{\aleph_0}=\aleph_1[/tex]. But in general (thus without CH) there is no formula for [tex]2^{\aleph_0}[/tex].

And even if we accepted CH as true, then there would still be no formula for [tex]2^{\aleph_1}[/tex]... 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...
 
  • #5
Thank you very much.
 

1. What is the definition of the power of a cardinal number to another cardinal number?

The power of a cardinal number to another cardinal number, denoted as ab, is the number of ways to map each element of a set with b elements to each element of another set with a elements. In simpler terms, it represents the number of possible functions from one set to another.

2. How is the power of a cardinal number to another cardinal number calculated?

The power of a cardinal number to another cardinal number, ab, is calculated by multiplying the first number (a) by itself b times. For example, 23 is equal to 2 x 2 x 2, which is 8.

3. What is the significance of the power of a cardinal number to another cardinal number in mathematics?

The power of a cardinal number to another cardinal number is important in set theory and combinatorics. It allows us to quantify the number of possible combinations or arrangements of elements in a set, and is used in various mathematical proofs and calculations.

4. How does the power of a cardinal number to another cardinal number relate to exponentiation?

The power of a cardinal number to another cardinal number is essentially the same as exponentiation. In fact, the notation ab is often used interchangeably with ab in mathematics. The main difference is that the power of a cardinal number can also be applied to infinite sets, while exponentiation is limited to finite numbers.

5. Can the power of a cardinal number to another cardinal number ever be larger than the cardinality of the set of real numbers?

Yes, the power of a cardinal number to another cardinal number can be larger than the cardinality of the set of real numbers, which is known as the continuum hypothesis. This hypothesis states that there is no set with a cardinality between that of the natural numbers (denoted as ℵ0) and the cardinality of the real numbers (denoted as 20). However, this hypothesis has not been proven and is still a topic of debate among mathematicians.

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