Re: abundance of irrationals!)


Virgil wrote:
>

> Consider the case where f(m,n) means m < n.
>
> ' A m in N: E n in N: f(m,n)' means that there is always a natural,
n,
> larger that any given natural, m.
>
> That this is true is seen but choosing n so that n = m+1.
>
> 'E n in N: A m in N: f(m,n)' means that there is a natural ,m, which
is
> larger than any natural, n.
>
> That this is false is seen by considering m = n.

The same proof holds for the cardinality of N: There is always a set of
n natural numbers which has at least one more element than a given set
of m elements. But that does not prove that there is a set which has
more elements than any set.

You have given a good argument that aleph_0 is less than an
imagination.

As I said: IFF N is infinite, then my list contains numbers with
infinitely many digits.

Regards, WM

.

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