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It’s at least credible that there is a standard or intended model for
Peano Arithmetic.  For instance, in Godel Escher Bach, Douglas
Hofstadter studies a sentence G with the standard meaning:
"this sentence has no formal proof in PA".

His argument goes along the lines that not(G) could be
added as an axion to PA, and if PA is consistent, then
PA + not(G) is also consistent.

But by not(G), G has a proof [in Godel numbers] but some of
these are are Godel-numbered by unusual whole numbers,
or "supernatural numbers".  This is an instance of
omega-inconsistency, as I understand it.

This difficult subject is discussed in Chapter XIV of
Hofstadter’ GEB book, and the chapter is entitled
“On Formally Undecidable Propositions of TNT and
   Related Systems ^(1) ".

Hofstadter writes that since G’s interpretation is true,
the interpretation of not(G) is false.  So that way
the standard way of extending PA is to admit G as a
new axiom.

—-

As for ZFC, or sub-systems of ZFC such as ZF or Z or ZC,
I have yet to see a convincing argument for a standard
model.

Cf.  the thread "Learning Logic and Set Theory",
http://groups.google.com/group/sci.logic/browse_thread/thread/75b8788…

If there is no consensus regarding V=L or the existence of some of
the larger large cardinal axioms, this makes it doubtful to me
that that there is an intended or standard model for ZFC
(at the present time).

What it suggests to me is that any models will be quite dissimilar
depending on accept/reject  V=L, accept/reject measurable cardinals,
etc. [ and also accept/reject CH, GCH, ... ]

David Bernier

- — -

lwal…@lausd.net wrote:
> On Nov 4, 7:41 pm, "amatur…@googlemail.com"
> <amatur…@googlemail.com> wrote:
> > Hello,
> > Is there any natural number with infinite number of digits
> > 4356456…..?

> There isn’t such a number in standard math.

> > If so, then how could it be constructed starting from 0
> > and successor function S(): 0, S(0), S(S(0)), ..?

> You just answered your own question. It can’t,
> which is why standard mathematics, such as
> Peano’s axioms, will reject it as a number.

> There can be nonstandard theories in which such
> numbers exist. Some of these theories, such as
> p-adics and hyperreals, have already been
> mentioned by others in this thread.

> Finally, some so-called "cranks," most notably
> Archimedes Plutonium, have attempted to invent
> new numbers with infinitely many digits. He
> calls his new numbers AP-adics. I’m still
> working on a theory to make AP-adics into a
> rigorous theory.

I should be back in doing the 2nd edition of that book — AP-adics
soon, perhaps in middle November.

You could have given a better answer to the person asking
whether a infinite integer can exist, by raising the question of
how solid of a reality is "finite integers and the Peano
axioms that justify them" Your answer to him would have
been better if you had said the finite integers and Peano
axioms would be like someone believing only as light
in physics as a wave. As if light and all of physics is wave
theory without any particle theory. Finite integers and Peano
axioms in mathematics would be similar to the progress of
physics by the 18th century.

No numbers in mathematics exists unless both infinite and finite
exist together. Calculus is nonexistent unless you provide for
infinity.

The Peano axioms are self contradictory and are as crude to
mathematics as what physics had in the time of Ben Franklin
when they thought that electricity was a form of fluid. Mathematicians
who believe in finite integers with their Peano axioms accompanyment
are like 21st century physicists who have never gone beyond
electricity as a fluid.

Recently I saw a TV show of the Mandelbrot Fractals on NOVA and I
realized for the first time that the whole enterprize of fractals is
based on
one concept– self similarity.

Well, mathematics has an older theory that can totally encompass all
of fractals–
Fourier Analysis, since sine and cosine are self similar, that Fourier
Analysis
swallows up all of Fractal theory.

And last night I was looking up that part of Fourier Analysis that
deals with the
surface of a sphere rather than the x to y axiis of 2nd dimensional
Fourier Analysis.
I had forgotten the name of that Fourier Analysis and it is called
Harmonic Analysis.

Anyway, to make a long story short, AP-adics are infinite-integers
that rest or lie
as the intrinsic points on a sphere or ellipsoid surface. The Reals
comprise points
in third dimension of the x,y,z axis and we can have a Fourier
Analysis of Reals
broken down into sine and cosine.

So if AP-adics are the native and natural points of the surface of the
sphere, I should
be able to show that Harmonic Analysis has a 2-dimensional analog that
is
equivalent to the Fourier Analysis of Reals.

So coming soon, is the 2nd edition of this book, which is the
historical equivalent of
what the Quantum Physics revolution was to physics by the 1900 to 1930
time period.
In other words, our present day mathematics is old musty crusty and
mostly fake
theory. Math never had a cleanup and a revolution. Comparing physics
to math by
2008, math is still stuck in the 18th century by comparison to
physics. The stuff that
math preaches today, would be like learning physics where they are
teaching that
electricity is a fluid. Modern day math is at least 2 centuries behind
the times.

Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

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Check out this useful website for Math, you can download e-worksheets
or can virtually practice Math online.  Visit www.mathebook.net

You will find Addition / subtraction / division / even odd numbers,
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Regards,

Tom

On 20 Okt., 20:54, Virgil <Vir…@gmale.com> wrote:

- — -

> In article
> <dd949f50-3dce-4980-a476-902418be8…@l64g2000hse.googlegroups.com>,

>  WM <mueck…@rz.fh-augsburg.de> wrote:
> > On 19 Okt., 21:18, Virgil <Vir…@gmale.com> wrote:

> > > > > Except that none of them are paths in the trees of interest, namely an
> > > > > infinite complete binary tree.

> > > > They continue with 0s.

> > > Which separates them forever from  the path 0.111… .

> > If all paths with a finite number of 1′s taken together contain all
> > 1′s of the path 0.111…,  why then does the following bijection
> > between the finite path and the initial segments of 0.111… fail?

> > 0.1, 0.11, 0.111, 0.1111, 0.11111, …,    ?
> >   |        |        |            |             |                |
> > 0.1, 0.11, 0.111, 0.1111, 0.11111, …, 0.111…

> WM is careful to overlook each of the uncountably many actual
> bijections, including, for example:

> 0.1,        0.11,     0.111     0.1111,     0.11111,     …,
>  |           |         |         |           |          
> 0.111…,   0.1,      0.11,     0.111,      0.1111,      ….

> Note well, WM,  that it is only for finite sets that existence of an
> injection which is not a surjection implies non-existence of a bijection.

The question is not the bijectability. That is even valid for sets
like N and Q. The problem is that my bijection shows the lack of
evidence for the existence of 0.111… as different from the union of
all finite paths.

Regards, WM

In article <645988d5-2374-4f06-8b3e-80f5868e6…@k16g2000hsf.googlegroups.com> WM <mueck…@rz.fh-augsburg.de> writes:
 > On 17 Okt., 15:07, "Dik T. Winter" <Dik.Win…@cwi.nl> wrote:
…
 > > By this definition the new sqeuence has been contructed if the
 > > antidiagonals have been contructed and if the rule to put them in
 > > a list has been defined.
 > >
 > > And because the antidiagonals have been constructed if the old sequence
 > > has been contructed and the replacement rule has been defined, the new
 > > sequence has been constructed if the old sequence has been constructed,
 > > the replacement has been defined en the rule to put the antidiagonals
 > > in a list has been defined.
 >
 > There is no rule to put them in a list.

Oh.  I thought some time earlier you gave a method to create the new list…

 > > So by this definition, as soon as you have the first list, the
 > > replacement rule and the rule to put antidiagonals in the list, you
 > > have defined *all* lists.  But now I wonder where the time element
 > > comes in play…
 >
 > Time does never end, therefore there is no "all lists".

Eh?  If every step takes no time?

 > However, I only wanted to show that "uncountability" does not imply
 > "many".

You have a pretty personal meaning attached to the word "uncountable".
—
dik t. winter, cwi, kruislaan 413, 1098 sj  amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn  amsterdam, nederland; http://www.cwi.nl/~dik/

In article <cc03c251-21d3-46b0-abbd-629cdef11…@p59g2000hsd.googlegroups.com> WM <mueck…@rz.fh-augsburg.de> writes:
 > On 17 Okt., 15:28, "Dik T. Winter" <Dik.Win…@cwi.nl> wrote:
…
 > >  > Constructed means constructed. There is no further definition
 > >  > possible, at most another definition.
 > >
 > > The first is a circular definition and so defines nothing.  Pray given
 > > another definition.
 >
 > Every definition is circular, strictly speaking.  But every man should
 > know what a construction is, from bridges, houses, etc.

I have some idea.  But if I have a list and a rule how to construct an
anti-diagonal I have *not* yet constructed the anti-diagonal.  The same
with a house.  If I have the bricks and a blueprint for a house I have
*not* yet contructed the house.

 > > I look at the common definition of *constructible* with respect to
 > > numbers and I only see that it is *not* related to what you mean.  So
 > > you must have something else in mind.
 >
 > Constructible means it is possible to construct. From 100 letters you
 > can construct a sentence. Bjut not every sentence constructible from
 > 100 letters has already been constructed – and most will never nbe
 > constructed – and all together will never be constructed
 > simultaneously.

What is the poit?

 > >  >                         Consider for instance all constructed bridges.
 > >  > They can be sequenced.
 > >
 > > Yes, so what?  diagonals and anti-diagonals are not bridges.
 > >
 > Really not?

And all contructed anti-diagonals with in a list can be put in a list.

 > >  > Then you should recognize that your sentence covers the real numbers.
 > >  > There is nothing what distinguishes their case from the constructed
 > >  > anti-diagonals.
 > >
 > > The distinction is that one set is countable and the other set is not
 > > countable.
 >
 > Sorry, a mistake. Both sets are uncountable.

Only with a pretty strange definition of "uncountable".
—
dik t. winter, cwi, kruislaan 413, 1098 sj  amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn  amsterdam, nederland; http://www.cwi.nl/~dik/

In article <d680a426-acc7-4ee0-b1b0-6b5e513c3…@s1g2000prg.googlegroups.com> WM <mueck…@rz.fh-augsburg.de> writes:
…
 > One side of the medal. Here is the other side:
 > Every node in the infinite path Q is in a finite path u together with
 > all its preceding nodes.

Actually: every node is with its preceding nodes in infinitely many paths u.

 > ==> There is no node in the infinite path Q that is not in a finite
 > path u togther with all its preceding nodes.

Actually: there is no node in the infinite path Q that is not in infinitely
many finite paths together with all its preceding nodes.

 > ==> All nodes are in a finite path u of U (together will all preceding
 > nodes).

Non sequitur.
—
dik t. winter, cwi, kruislaan 413, 1098 sj  amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn  amsterdam, nederland; http://www.cwi.nl/~dik/