# Cardinal numbers

• Jul 14th 2010, 10:25 PM
Cardinal numbers
This is the question: Define Aleph0, C and F (this three are cardinal numbers) and also i want to know what is the relationship between them.

Explanation --> Aleph0 = Aleph-null = Aleph-zero = Aleph-naught
Explanation --> C = cardinality of the continuum = power of the continuum

There is a short story from Jorge Luis Borges called "El Aleph" and i would like to know if this story has some connection with the number Aleph0. Thanks for your help.

• Jul 15th 2010, 02:59 AM
Ackbeet
Correct me if I'm wrong, but $\aleph_{0}$ is regarded as the cardinality of the integers. It's the "first rung" of the infinities. The continuum hypothesis would claim that $2^{\aleph_{0}}=\aleph_{1}=\mathfrak{c}.$
That is, taking the power set of an infinite set means you exponentiate the cardinality of the original set in order to get the cardinality of the power set. (This always works for finite sets - it's only the infinite set we're talking about here.)

I'm not exactly sure what your F is, but if it corresponds to the cardinality of the power set of the reals, then the continuum hypothesis (which has not been proven) would say that $2^{\aleph_{1}}=|P(\mathbb{R})|.$

The wiki on El Aleph says that the aleph numbers and infinite sets "are related" to El Aleph. It seems to me, though, that the author was more interested in Jewish themes, and incorporated those more than the mathematical side of things, especially since, apparently, he wrote more than one story connected with Judaism.
• Jul 16th 2010, 01:25 PM
Matt Westwood
Quote:

Originally Posted by Ackbeet
Correct me if I'm wrong, but $\aleph_{0}$ is regarded as the cardinality of the integers. It's the "first rung" of the infinities. The continuum hypothesis would claim that $2^{\aleph_{0}}=\aleph_{1}=\mathfrak{c}.$
That is, taking the power set of an infinite set means you exponentiate the cardinality of the original set in order to get the cardinality of the power set. (This always works for finite sets - it's only the infinite set we're talking about here.)

I'm not exactly sure what your F is, but if it corresponds to the cardinality of the power set of the reals, then the continuum hypothesis (which has not been proven) would say that $2^{\aleph_{1}}=|P(\mathbb{R})|.$

The wiki on El Aleph says that the aleph numbers and infinite sets "are related" to El Aleph. It seems to me, though, that the author was more interested in Jewish themes, and incorporated those more than the mathematical side of things, especially since, apparently, he wrote more than one story connected with Judaism.

My quick two-cents' worth:

F, if I'm not mistaken, is the cardinal number which denotes the number of real functions there are, and is indeed the powerset of the reals, as Ackbeet's post suggests.