# Cardinality of real #s

#### Loren

Is 2 the minimum base for "the smallest possible infinite cardinality"? Why not less?

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#### Maschke

Is 2 the minimum base for "the smallest possible infinite cardinality"? Why not less?
What do you mean?

#### topsquark

Math Team
Is 2 the minimum base for "the smallest possible infinite cardinality"? Why not less?
Say we have a set G, with cardinality g. Then the cardinality of the power set of G is denoted as $$\displaystyle 2^{g}$$, which you can verify for finite cardinals. The cardinality of the reals is equal to the cardinality of the power set of the natural numbers, ie. $$\displaystyle 2^{ \aleph _0 }$$. There may be some higher meaning to this with infinite cardinals but if you like you can simply call the cardinality of the reals $$\displaystyle \aleph _1$$.

-Dan

#### SDK

but if you like you can simply call the cardinality of the reals $$\displaystyle \aleph _1$$.
This is true only if the continuum hypothesis is true. In fact, this statement is exactly the continuum hypothesis.

#### topsquark

Math Team
This is true only if the continuum hypothesis is true. In fact, this statement is exactly the continuum hypothesis.
Hmmmm.... I overstepped then. Sorry!

-Dan

#### Loren

I think I understand why a power set has a base of 2 -- combinatorics?

#### [email protected]

You can take any base: $n^{\aleph_0} = 2^{\aleph_0}$ for $n>1$.

Loren

#### Loren

Just a fancy: is

(1+1/(aleph-naught))^(aleph-naught)

akin to logarithms?

#### [email protected]

Division of cardinals is not defined. Something like $\frac{1}{\aleph_0}$ doesn't exist.

#### Loren

What is the largest set or number which can divide?