|
|
Re: The cardinality of the collection of all partitions of the
natural numbers equals that of the real numbers.
Posted:
Mar 17, 2011 5:17 PM
|
|
On Mar 17, 10:16 am, "Dave L. Renfro" <renfr...@cmich.edu> wrote:
> Kerry Soileau wrote:
> > Let P be the collection of all partitions of the natural numbers.
> > A partition of a set X is a pairwise disjoint collection of
> > sets whose union is X. I have a proof that the cardinality of P
> > is the cardinality of the real numbers. Is this well-known?
> > If so, would someone provide a reference?
>
> This seems to me to be around the average difficulty level of
> a problem in an undergraduate set theory or real analysis text,
> so it's not something anyone would bother citing a reference
> to if they wanted to use the result somewhere (and off-hand,
> I don't know of a specific reference, and all my books are at
> home and I'm elsewhere).
Exercise 6 on p. 76 of W. Sierpinski, _Cardinal and Ordinal Numbers_,
Second Edition Revised, Warszawa, 1965: "Prove that if X is the set of
all natural numbers, then the set Z of all partitions of the set N is
effectively of the power of the continuum." Here "effectively of the
power of the continuum" means that the equivalence is established by
an effectively definable bijection. The author provides a detailed
solution on the same page.
> To show that there are at least continuum many possible partitions
> of N, let E be the set of positive even integers. Then, for each
> subset B of E, the set {B, N - B} is a partition of N. Moreover,
> if B, B' are distinct subsets of E, then {B, N - B} is not equal
> to {B', N - B'}. Thus, there are at least continuum many partitions,
> since there are continuum many subsets of E.
Alternatively, you can associate with each real number a partition of
the set Q of rational numbers into two classes (Dedekind cuts), and
then use a bijection between Q and N.
|
|
