Virgil
Posts:
4,483
Registered:
1/6/11
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Re: Matheology � 162
Posted:
Nov 28, 2012 2:55 PM
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In article
<6f838fa9-0ae5-4949-8cb9-ea4b1ca21d74@bq2g2000vbb.googlegroups.com>,
WM <mueckenh@rz.fh-augsburg.de> wrote:
> On 28 Nov., 16:13, William Hughes <wpihug...@gmail.com> wrote:
> > On Nov 28, 10:59 am, WM <mueck...@rz.fh-augsburg.de> wrote:
> >
> > > On 28 Nov., 13:48, William Hughes <wpihug...@gmail.com> wrote:
> >
> > > > On Nov 28, 2:43 am, WM <mueck...@rz.fh-augsburg.de> wrote:
> >
> > > > > Induction proves also: Every set of natural numbers is finite.
> > > > > Why do you overlook this simple proof?
> >
> > > > No, what induction proves is that every set of natural numbers
> > > > with a largest number is finite.
> >
> > > And induction proves that every set of natural numbers has a largest
> > > number. For every finite n also n + 1 is finite
> >
> > Look! Over There! A Pink Elephant!
> >
> > >and the set containing
> > > both, n and n + 1 ist finite too.
> >
> > There is of course no such thing as
> > "the set containing both n and n+1".
>
> Here it is: {n, n+1}
But here too it is: {n, n+1, n+2}
And here too it is: {n, n+1, n+2, n+3}
And yet again it is {n, n+1, n+2, ...}
>
> > You need to prove that all such sets
> > have a largest number, not just the sets
> > you can get by starting with {} and
> > adding one element at a time.
>
> Induction does not say more than: 1 is natural number, and if n is a
> natural number, then n+1 is a natural number.
> If {1, 2, 3, ..., n} is a set of natural numbers, then
> {1, 2, 3, ..., n, n+1} is a set of natural numbers.
In every standard set theory, any union of sets all of whose members
have a given property is necessarily a set all of whose members have
that property.
Thus there exists in every standard set theory a set which is the union
of all sets whose members are natural numbers. Thus there is no natural
number which is NOT a member of such a set.
Those who reject that principle are rejecting set theory as a whole.
>
> Have you meanwhile convinced yourself that analysis is capable
> expanding infinity by a series of powers of 10? This is tantamount to
> a decimal representation.
Once one has an "infinity" as a limit of reals in in analysis, one
cannot "expand" it by powers of 10, or in any other way.
--
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