Cardinality of real #s

May 2015
509
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Arlington, VA
Is 2 the minimum base for "the smallest possible infinite cardinality"? Why not less?
 
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topsquark

Math Team
May 2013
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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

Sep 2016
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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
May 2013
2,443
1,012
The Astral plane
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
 
May 2015
509
32
Arlington, VA
I think I understand why a power set has a base of 2 -- combinatorics?
 
Oct 2009
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You can take any base: $n^{\aleph_0} = 2^{\aleph_0}$ for $n>1$.
 
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May 2015
509
32
Arlington, VA
Just a fancy: is

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

akin to logarithms?
 
May 2015
509
32
Arlington, VA
What is the largest set or number which can divide?