Page 32 - Advanced Linear Algebra

16 Advanced Linear Algebra

We have already observed that (( ~L o . It can be shown that L is the smallest
infinite cardinal, that is,
0 L ¬ is a natural number
It can also be shown that the set s of real numbers is in one-to-one
correspondence with the power set Fo²³ of the natural numbers. Therefore,

s
(( ~ L
The set of all points on the real line is sometimes called the continuum and so
is sometimes called the power of the continuum
L and denoted by .
Theorem 0.14 shows that cardinal addition and multiplication have a kind of
“absorption” quality, which makes it hard to produce larger cardinals from
smaller ones. The next theorem demonstrates this more dramatically.

Theorem 0.16
1 Addition applied a countable number of times or multiplication applied a
)
, does not yield anything
finite number of times to the cardinal number L
L
more than . Specifically, for any nonzero  o , we have
and

Lh L ~ L L ~ L

2 Addition and multiplication applied a countable number of times to the
)
cardinal number L does not yield more than L . Specifically, we have
L
Lh L ~ and ² ³ L ~ L

L

Using this theorem, we can establish other relationships, such as
L
L  ²L ³ L  ² ³ L ~ L

¨
which, by the Schroder–Bernstein theorem, implies that
²L ³ L ~ L

We mention that the problem of evaluating in general is a very difficult one
and would take us far beyond the scope of this book.
We will have use for the following reasonable-sounding result, whose proof is
omitted.
2
Theorem 0.17 Let ¸( “  2¹ be a collection of sets, indexed by the set ,

with ((2~ . If ((  ( for all  2 , then
e (  e
2
Let us conclude by describing the cardinality of some famous sets.