Re: Relative Cardinality

In article <1121765409.315003.311630@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx wrote:

> Randy Poe wrote:
> > mueckenh@xxxxxxxxxxxxxxxxx wrote:
> > > Proginoskes wrote:
> > > > (You've mixed up quantifiers here. Pi can be defined by a sequence of
> > > > rational numbers a_n, but the condition on this sequence is that for
> > > > every positive epsilon, THERE EXISTS AN N such that |a_n - pi| <
> > > > epsilon when n >= N. The value of N changes when epsilon changes; you
> > > > do not need to have |a_n - pi| < epsilon for all n and for all epsilon.
> > >
> > > You misunderstood me. Perhaps my fault. It must be possible to satisfy
> > > this inequality for arbitrarily small epsilon.
> >
> > Yes.
> >
> > > But already for the
> > > comparatively large eps = 1/10^10^100 this inequality cannot be
> > > satisfied by any a_n.
> >
> > There are infinitely many rational numbers which are closer
> > than 1/10^10^100 to pi. For instance, all those which are
> > identical to pi in the first 10^10^100 digits but differ
> > by 2 in the 1+10^10^100-th digit, satisfy |a-pi| < eps
> > for your choice of eps.
> >
> In order to use the digits of pi, you must first know that pi does
> exist. But that can only be proved by this formalism.

Pi can be, and has been, defined by many different convergent infinite
sequences/series and infinite products, any one of which "proves" that
the number so defined does exist.

pi = 4/1 - 4/3 + 4/5 - 4/7 + ..., for example.
.