Google Groups no longer supports new Usenet posts or subscriptions. Historical content remains viewable.
Dismiss
Matheology § 018
10 views
Skip to first unread message
WM
May 25, 2012, 8:00:24 AM5/25/12
to
Feferman and Levy showed that one cannot prove that there is any non-
denumerable set of real numbers which can be well ordered. Moreover,
they also showed that the statement that the set of all real numbers
is the union of a denumerable set of denumerable sets cannot be
refuted.
[Abraham A. Fraenkel, Yehoshua Bar-Hillel, Azriel Levy: "Foundations
of Set Theory", North Holland, Amsterdam (1973) p. 62]
http://www.amazon.de/gp/product/0720422701/ref=sib_rdr_dp
Regards, WM
Tonico
May 25, 2012, 8:32:27 AM5/25/12
to
PRECISELY and exactly Feferman-Levy's results imply and/or come from,
yet he insists in his own, personal cranky mantra.
Tonio
Ps..BTW, and for anyone interested, there's no use in trying to
convince WM with mathematics and reasons. He's utterly impermeable to
these both.
WM
May 25, 2012, 11:55:13 AM5/25/12
to
On 25 Mai, 14:32, Tonico <Tonic...@yahoo.com> wrote:
> WM was already explained what
> PRECISELY and exactly Feferman-Levy's results imply and/or come from,
> yet he insists in his own, personal cranky mantra.
>
Sorry, where can I read this mantra? I am not aware of having
> WM was already explained what
> PRECISELY and exactly Feferman-Levy's results imply and/or come from,
> yet he insists in his own, personal cranky mantra.
>
commented that post. Nevertheless, thank you for your, as usual, very
constructive contribution.
Regards, WM
Tonico
May 25, 2012, 1:54:36 PM5/25/12
to
Tonio
Uirgil
May 25, 2012, 4:08:37 PM5/25/12
to
In article
<76d52725-0101-4090...@q2g2000vbv.googlegroups.com>,
falsehoods that cannot be refuted.
<76d52725-0101-4090...@q2g2000vbv.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
> Feferman and Levy showed that one cannot prove that there is any non-
> denumerable set of real numbers which can be well ordered. Moreover,
> they also showed that the statement that the set of all real numbers
> is the union of a denumerable set of denumerable sets cannot be
> refuted.
And Godel showed that there must be truths that cannot be proved and
> Feferman and Levy showed that one cannot prove that there is any non-
> denumerable set of real numbers which can be well ordered. Moreover,
> they also showed that the statement that the set of all real numbers
> is the union of a denumerable set of denumerable sets cannot be
> refuted.
falsehoods that cannot be refuted.
WM
May 25, 2012, 4:13:15 PM5/25/12
to
On 25 Mai, 22:08, Uirgil <uir...@uirgil.ur> wrote:
> In article
> <76d52725-0101-4090-bd8c-db81043e1...@q2g2000vbv.googlegroups.com>,
exists. This premise is false as I have shown:
Every rational is the center of an interval: q_n is the center of
interval I_n. But there remain uncountably many irrationals (of the
remaining 8/9 or more of the unit interval) such that not two of them
are "connected", i.e. each pair is separated by at least one interval
I_n.
Regards, WM
> In article
> <76d52725-0101-4090-bd8c-db81043e1...@q2g2000vbv.googlegroups.com>,
>
> WM <mueck...@rz.fh-augsburg.de> wrote:
> > Feferman and Levy showed that one cannot prove that there is any non-
> > denumerable set of real numbers which can be well ordered. Moreover,
> > they also showed that the statement that the set of all real numbers
> > is the union of a denumerable set of denumerable sets cannot be
> > refuted.
>
> And Godel showed that there must be truths that cannot be proved and
> falsehoods that cannot be refuted.
He showed it under the premise that the hierarchy of infinities
> WM <mueck...@rz.fh-augsburg.de> wrote:
> > Feferman and Levy showed that one cannot prove that there is any non-
> > denumerable set of real numbers which can be well ordered. Moreover,
> > they also showed that the statement that the set of all real numbers
> > is the union of a denumerable set of denumerable sets cannot be
> > refuted.
>
> And Godel showed that there must be truths that cannot be proved and
> falsehoods that cannot be refuted.
exists. This premise is false as I have shown:
Every rational is the center of an interval: q_n is the center of
interval I_n. But there remain uncountably many irrationals (of the
remaining 8/9 or more of the unit interval) such that not two of them
are "connected", i.e. each pair is separated by at least one interval
I_n.
Regards, WM
Uirgil
May 25, 2012, 7:13:12 PM5/25/12
to
In article
<04c07ba4-2451-4e00...@n16g2000vbn.googlegroups.com>,
<04c07ba4-2451-4e00...@n16g2000vbn.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
> n 25 Mai, 22:08, Uirgil <uir...@uirgil.ur> wrote:
> > In article
> > <76d52725-0101-4090-bd8c-db81043e1...@q2g2000vbv.googlegroups.com>,
> >
> > WM <mueck...@rz.fh-augsburg.de> wrote:
> > > Feferman and Levy showed that one cannot prove that there is any non-
> > > denumerable set of real numbers which can be well ordered. Moreover,
> > > they also showed that the statement that the set of all real numbers
> > > is the union of a denumerable set of denumerable sets cannot be
> > > refuted.
> >
> > And Godel showed that there must be truths that cannot be proved and
> > falsehoods that cannot be refuted.
>
> He showed it under the premise that the hierarchy of infinities
> exists. This premise is false as I have shown
You will have to argue out that with Godel.
> n 25 Mai, 22:08, Uirgil <uir...@uirgil.ur> wrote:
> > In article
> > <76d52725-0101-4090-bd8c-db81043e1...@q2g2000vbv.googlegroups.com>,
> >
> > WM <mueck...@rz.fh-augsburg.de> wrote:
> > > Feferman and Levy showed that one cannot prove that there is any non-
> > > denumerable set of real numbers which can be well ordered. Moreover,
> > > they also showed that the statement that the set of all real numbers
> > > is the union of a denumerable set of denumerable sets cannot be
> > > refuted.
> >
> > And Godel showed that there must be truths that cannot be proved and
> > falsehoods that cannot be refuted.
>
> He showed it under the premise that the hierarchy of infinities
> exists. This premise is false as I have shown
WM
May 26, 2012, 8:56:04 AM5/26/12
to
On 26 Mai, 01:13, Uirgil <uir...@uirgil.ur> wrote:
> In article
> <04c07ba4-2451-4e00-93dd-eee0b5fda...@n16g2000vbn.googlegroups.com>,
> In article
Nothing to argue.
"Der wahre Grund für die Unvollständigkeit, welche allen formalen
Systemen der Mathematik anhaftet, liegt, wie im lI. Teil dieser
Abhandlung gezeigt werden wird, darin,
daß die Bildung immer höherer Typen sich ins Transfinite fortsetzen
läßt [...] während in jedem formalen System höchstens abzählbar viele
vorhanden sind. Man kann nämlich zeigen, daß die hier aufgestellten
unentscheidbaren Sätze durch Adjunktion passender höherer Typen (z.
B.
des Typus omega zum System P) immer entscheidbar werden. Analoges
gilt
auch für das Axiomensystem der Mengenlehre." [p. 191]
[Kurt Gödel: "Über formal unentscheidbare Sätze der Principia
Mathematica und verwandter Systeme I", Monatshefte für Mathematik und
Physik 38 (1931) S.173–198.]
Regards, WM
Uirgil
May 26, 2012, 4:05:39 PM5/26/12
to
In article
<c261f5c5-651a-493c...@x21g2000vbc.googlegroups.com>,
logic there is one above it, which is, in effect, your own merely
potential infiniteness.
<c261f5c5-651a-493c...@x21g2000vbc.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
> On 26 Mai, 01:13, Uirgil <uir...@uirgil.ur> wrote:
> > In article
> > <04c07ba4-2451-4e00-93dd-eee0b5fda...@n16g2000vbn.googlegroups.com>,
> >
> >
> >
> >
> >
> > WM <mueck...@rz.fh-augsburg.de> wrote:
> > > n 25 Mai, 22:08, Uirgil <uir...@uirgil.ur> wrote:
> > > > In article
> > > > <76d52725-0101-4090-bd8c-db81043e1...@q2g2000vbv.googlegroups.com>,
> >
> > > > WM <mueck...@rz.fh-augsburg.de> wrote:
> > > > > Feferman and Levy showed that one cannot prove that there is any non-
> > > > > denumerable set of real numbers which can be well ordered. Moreover,
> > > > > they also showed that the statement that the set of all real numbers
> > > > > is the union of a denumerable set of denumerable sets cannot be
> > > > > refuted.
> >
> > > > And Godel showed that there must be truths that cannot be proved and
> > > > falsehoods that cannot be refuted.
> >
> > > He showed it under the premise that the hierarchy of infinities
> > > exists. This premise is false as I have shown
> >
> > You will have to argue out that with Godel.-
>
> Nothing to argue.
Because you have no case! All Godel needs is that for each level of
> On 26 Mai, 01:13, Uirgil <uir...@uirgil.ur> wrote:
> > In article
> > <04c07ba4-2451-4e00-93dd-eee0b5fda...@n16g2000vbn.googlegroups.com>,
> >
> >
> >
> >
> >
> > WM <mueck...@rz.fh-augsburg.de> wrote:
> > > n 25 Mai, 22:08, Uirgil <uir...@uirgil.ur> wrote:
> > > > In article
> > > > <76d52725-0101-4090-bd8c-db81043e1...@q2g2000vbv.googlegroups.com>,
> >
> > > > WM <mueck...@rz.fh-augsburg.de> wrote:
> > > > > Feferman and Levy showed that one cannot prove that there is any non-
> > > > > denumerable set of real numbers which can be well ordered. Moreover,
> > > > > they also showed that the statement that the set of all real numbers
> > > > > is the union of a denumerable set of denumerable sets cannot be
> > > > > refuted.
> >
> > > > And Godel showed that there must be truths that cannot be proved and
> > > > falsehoods that cannot be refuted.
> >
> > > He showed it under the premise that the hierarchy of infinities
> > > exists. This premise is false as I have shown
> >
> > You will have to argue out that with Godel.-
>
> Nothing to argue.
logic there is one above it, which is, in effect, your own merely
potential infiniteness.
WM
May 26, 2012, 4:09:30 PM5/26/12
to
On 26 Mai, 22:05, Uirgil <uir...@uirgil.ur> wrote:
> In article
> <c261f5c5-651a-493c-823d-dbab32ad5...@x21g2000vbc.googlegroups.com>,
> In article
No, there is no infinity above the first one, because the first one
already is potential only.
But it is correct that, if the hierarchy would exist, it would be
potential.
Regards, WM
0 new messages