Date: Mar 22, 2013 5:49 PM
Subject: Re: Matheology § 224
On 22 Mrz., 22:37, William Hughes <wpihug...@gmail.com> wrote:
> On Mar 22, 10:24 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> > On 22 Mrz., 21:54, William Hughes <wpihug...@gmail.com> wrote:
> > > On Mar 22, 7:24 pm, WM <mueck...@rz.fh-augsburg.de> wrote:> On 22 Mrz., 16:24, William Hughes <wpihug...@gmail.com> wrote:
> > > <snip>
> > > > > Infinite sets are different from finite sets
> > > > > but they do not contain anything
> > > > > "beyond any finite set".
> > > > Of course.
> > > We have now established that there
> > > are sets that do not contain anything
> > > "beyond any finite set" but
> > > are different from finite sets.
> > That is potential infinity.
> Nope, that is actual infinity. (In potential
> infinity you cannot have a set that is different from
> finite sets because all sets are finite)-
All sets are finite, but not fixed. There is no upper threshold,
contrary to every finite set.
Again, compare the decimal fraction of pi. It is not finite. And a
potentially infinite set will never reach farther than a rational
approximation. Only an actually infinite set would do.
Compare the Binary Tree. Try to find the difference between the tree
that contains only all finite paths and the tree that in addition
contains all actually infinite paths. Only the latter could be
uncountable. But it is impossible to distinguish it from the former.
In MathStackExchange, there were some experts who proposed to
establish the difference by the pot. tree with a number of levels <
omega and the act. tree with number of levels =< omega. That, however,
would require to determine the level at omega, so it is nonsense. But
the experts had enough support by other experts so that this question
has been deleted very soon. Fools stay together.