Cofinality of the continuum

From: Daryl McCullough (stevendaryl3016_at_yahoo.com)
Date: 02/08/05 

Date: 8 Feb 2005 08:39:31 -0800

I tracked down the proof that the cofinality of the continuum cannot
be omega, and it seems simple enough to post.

First, some facts that I'll need, which I won't be proving. Let C be
the cardinality of the continuum, and let omega -> C be the set of
functions from omega into the ordinals less than C (or the set of
countably infinite lists of reals) Let |A| be the
cardinality of A. Let omega * A be the set of pairs <i,x> such that
i is an element of omega and x is an element of A.

1. |omega -> C| = C

2. for any set A, if |A| < C, then |omega * A| < C.

The next lemma I will actually prove:

3. If A_0, A_1, ... is an omega-sequence of sets such that |A_i| < C,
and A = the union of all the A_i, then |A| < |omega -> C|.

Proof: We show by diagonalization, that for any function
g from A into omega -> C, there is an element of omega -> C
that is not in the image of g.

Let g be a function from A into omega -> C. Then for each
a in A, g(a) is a function from omega -> C. For each j,
let x_j be some element of C such that for all a in A_j,
x_j is not in the image of g(a). (There has to be such
an x_j, because the set
{ x | exists a in A_j, exists i in omega, g(a)(i) = x }
has cardinality less than or equal to |omega * A_j|, which
is less than C).

Let l = [x_0, x_1, ...] be the element of omega -> C formed from
all the x_j. Then l is not in the image of g.

4. From 1 & 3, it follows that |A| < C. Therefore, C cannot equal A.
Therefore, C cannot equal the union of countably sets of size less than
C. Therefore, the cofinality of C cannot be omega. 

--
Daryl McCullough

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