On 4 Apr., 16:01, William Hughes <wpihug...@gmail.com> wrote: > On Apr 2, 10:39 pm, WM <mueck...@rz.fh-augsburg.de> wrote: > > > On 1 Apr., 15:30, William Hughes <wpihug...@gmail.com> wrote: > <snip> > > > You can only have no lines left if you > > > remove an infinite thing. > > > Piffle. > > So what non infinite thing do you remove?
I remove all finite things, namely all finite lines that are subject to induction.
> Sure you can remove any one finite line,
and its predecessor.
> but that does not leave no lines.
In actual infinity it leaves no lines. You can apply Cantor's argument to digonalize every line at a finite position n. Why don't you object in this case, that that does not leave no lines, i.e., that the diagonalization does not apply to the complete list including all lines?
> If you remove "every finite line" > your are removing an infinite thing > "an infinite collection of finite things"
If an infinite collection of infinite things exists actually, i.e., IF it is not only simple nonsense, to talk about an actually infinite set of finite numbers, then I can remove this infinite thing because it consists of only all finite things for which induction is valid.
> is an infinite thing. "every finite line" > is "an infinite collection of finite > things".
This is, by God, a very special trick to be successful. Induction holds for all natural numbers, but if they form an actually infinite thing, then induction does not hold for all of them. Cantor's argument, however, holds for this infinite thing, because set theory is defined this way.
Zermelo made the first mistake, when taking Dedekind's potential infinity as actual infinity. Set theorists now make the second mistake, when claiming that Cantor's diagonalization does not apply to line after line but to all lines simultaneously. And you make the third mistake, when claiming that induction does not apply to the set | N generated by this erroneous actual infinity, but only to a set that in priciple is not existing in set theory, namely a potential non- exhaustable set. Remember: The inductive set in set theory is actual and finished. Why can't we use it completely?