Review of Mueckenheims book.
Dik T. Winter
the basic part. The full review will be available at:
<http://www.cwi.nl/~dik/english/mathematics/mueck>
Book:
Die Mathematik des Unendlichen
Wolfgang Mückenheim
Shaker Verlag, Aachen, Germany, 2006, ISBN 3-8322-5587-7.
The book contains two parts.
The first part are the initial eight chapters, the second part
contains the last two chapters. They are quite different in content, so
I do split the review in two parts (note that I have not yet completed
the second part, it takes a lot of time). The review of the first part
will be posted as a follow-up to this article, the review of the second
part will not be posted, but is available through the following link:
<http://www.cwi.nl/~dik/english/mathematics/mueck/book2.html>
It will be updated while I read more.
My initial conclusion is that chapters 1 to 8 can serve quite well in
a course on the history of set theory. The historical context is set
out extremely well. Also the outlining of current set theory appears
to be adequate. The second part (at least what I have read until now)
is nonsense. It consists entirely of misunderstandings, bad logic,
ill-defined objects and whatever you can think. Moreover, I fail to
see why the authors needs over ten examples to show that actual infinity
leads to an inconsistency. Only one would be sufficient, if it were
valid. The lack of mathematical content in chapter 9 is shocking. And
I think it is even worse in chapter 10 (which appears to be more
philosophical than mathematical).
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
Dik T. Winter
These chapters provide an exceptional well written showing about how
things developed through the ages. I will note what each chapter
contains, and also note some of the errors in chapters 7 and 8.
Chapter 1. Natürlich unendlich
Is about the natural numbers and some properties (including primes).
They are introduced using the Peano axioms. Also induction is
explained. What I am missing here a bit is a remark that the Peano
axioms can be used to define arithmetic that conforms with the common
arithmetic. And I think that the remark that Euclid used a proof by
contradiction is wrong (actually the theorem and proof are quite
Cantoresque):
THEOREM
Given any collection of primes, there is a prime that is not in
that collection (at that time collections were still finite).
PROOF
The well-known construction, that leads to either a new prime or
a natural number that is the product of two or more primes not in
the collection.
COROLLARY (current)
There are infinitely many primes.
And, although the sizes of various numbers are explained, in this
chapter there is not yet much about infinity.
Chapter 2. Gegen Unendlich
This chapter introduces sequences and series and silently the rational
numbers are introduced. Properly is explained that oo in limits and
infinite sums is just a notational convenience (or in his terms a
potential infinity that is never reached). Also is explained
something about the historical development. There is further an
introduction to infinite products and continued fractions, but both
are not explained further (e.g. with infinite products it is not
explained that the product diverges if it is 0).
Chapter 3. Alogos
Here we see the history of the founding of irrational numbers,
including some irrationality proofs. The Fundamental Theorem of
Algebra is introduced, and proof by infinite descent. However,
the author apparently takes the position that the symbol
sequence 'sqrt(2)' does not prove the existence of that number,
because it is nothing more than the question to find the number
whose square is 2. With this following Cantor. I wonder however
why a notation like 3/7 does not fall under the same verdict, as
it is nothing more than the question to find the number that,
when multiplied by 7 yields 3. So we can say that the number
sequence '3/7' does not prove existence of that number. And,
indeed, when operating in the natural numbers that question makes
no sense. It does not exist in the natural numbers, so you have
to extend the natural numbers (with rationals) to have existence.
In the same way 'sqrt(2)' does not exist in the rational numbers,
you have to extend the set of numbers to find a solution.
We see here a bias from the author that extension to rational
numbers is natural, but extension to algebraic numbers is not.
But the point is moot. In both cases, given some number system,
both or neither or one of them does exist. In Z_p (the integers
mod p) 3/7 does exist when p > 7 and sqrt(2) does exist when
p = 4.k-1 for some integer k. And neither exist in the natural
numbers.
Chapter 4. Tranzendent
This chapter explains transcendental numbers and how they are
developed. Also the theorem by Liouville is given (with proof),
and with it the explicit showing of Liouville's number (together
with a few others). Here again, the existence of numbers is too
much connected with 'actual infinity', because the basis is
decimal representation.
Chapter 5. Infinitesimal
This is mostly historical, showing an overview about the struggle
with the infinitesimal, where finally Cauchy and Weierstraß gave
the solution to the problems (differentiating and integrating) without
using infinitesimals at all, but using limits. It is acknowledged
that infinitesimals can be given a proper basis using Abraham
Robinson's non-standard analysis. Alas, other approaches are not
even mentioned (John Conway, Anders Kock).
Chapter 6. Paradoxien des Unendlichen
Bolzano's musings are set out here, especially for those cases where
intuition does not work. For instance, he does not like bijections
to prove equality of "size" of sets as it does not conform with
intuition of infinite sets (the set of squares of natural numbers is
intuitively less than the set of all natural numbers). He fails to
see that set inclusion does not give a trichonomy of sets, and it
is difficult to obtain that when set inclusion should always be
decisive when present.
Chapter 7. Transfinit
Set theory is described, both in the older (Cantor time) form as in
the modern form. However, I think there is an error on p. 88. It
is stated that (translated): "and it is even provable that a
well-ordering [of the reals] can not be defined at all..."
My understanding is that a definable well-order is not inconsistent
with ZF, and there are indeed models were a well-order can be defined.
See: <a href="http://arxiv.org/pdf/math.LO/9812115">Uri Abraham and
Saharon Shelah, A delta^2_2 well-order of the reals and incompactness
of L(Q^MM)</a>.
Chapter 8. Potentiell versus aktual
Here an overview is given about the historical struggle between
potential and actual infinity. From the quoted text it is clear
that the issue is not mathematical, but philosophical, and many
times even religious.
Also a few of the more esoteric findings of modern set theory are
indicated, but clearly the author is wrong in some of the cases.
The first such case is after the incompleteness theorem by Gödel
(page 106):
"any sufficient complex consistent system contains theorems that
are true but that can not be proven."
After this follows what appears like a simple conclusion by the author,
because no reference is given:
"The consistency of such an arithmetic is not provable within
the system - because otherwise such a system would be inconsistent
as, as is known, within an inconsistent system everything can
be proven."
But that is simply the second incompleteness theorem in disguise:
"For any formal theory ..., the theory includes a statement
of its own consistency if and only if the theory is inconsistent."
The inconsistent system proves the "if" part, but from this the
"only if" part can not be concluded. Also the "only if" part does
not follow trivially from the first incompleteness theorem.
The second case is more serious, because it tends to show
misunderstandings by the author (more of this in the reviews of
later chapters). Page 107 deals with Skolem's results that there
is a countable model of ZF. And the question is, how is that possible
when there are uncountable sets? The authors quotes three ways to
explain it. The first argument is contending the existence of N within
the model, the third argument is about it being impossible to look too
close at the countable model. I tend to agree with the author that the
first and third do not hold. However, he has apparently serious
problems with the second. To quote (translated):
"It is assumed that the axiom of the power set does not give the
complete powerset, but only a subset that is countable from the
outside, but not so in the model, because there is no mapping
from N to P'(N)."
He counters this with:
"That is very much a dubious explanation, because a mapping is
a set itself. A map from N to a countable set P'(N) is a
countable set. Why should it fail to be in a model that
contains N?"
But that is wrong, because when a map 'f' is viewed as a set, it is
defined as:
{ (n, s) | n in N, s in P(N), f(n) = s }
So when 'f' can not be defined in a model, neither can the set
representing 'f'. He further states:
"Moreover, as by the axiom of the power set P(N) contains each
subset of N. So which subset fails to be in P(N)?"
The answer is simply: those that can not be defined in the model.
The model has serious influences on the properties that can be used
in the axiom of subsets, and so also on the power set as defined by
the axiom of the power set.
Aatu Koskensilta
> However, I think there is an error on p. 88. It is stated that
> (translated): "and it is even provable that a well-ordering [of
> the reals] can not be defined at all..."
> My understanding is that a definable well-order is not inconsistent
> with ZF, and there are indeed models were a well-order can be defined.
> See: <a href="http://arxiv.org/pdf/math.LO/9812115">Uri Abraham and
> Saharon Shelah, A delta^2_2 well-order of the reals and incompactness
> of L(Q^MM)</a>.
There is no need to refer to such a recent paper. V=L already implies
that there is a definable well-ordering of the reals of ridiculously low
complexity. Of course, in reality there is no definable well-ordering of
the reals.
--
Aatu Koskensilta (aatu.kos...@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
James Waldby
> Chapter 6. Paradoxien des Unendlichen
>
> Bolzano's musings are set out here, especially for those cases where
> intuition does not work. For instance, he does not like bijections to
> prove equality of "size" of sets as it does not conform with intuition
> of infinite sets (the set of squares of natural numbers is intuitively
> less than the set of all natural numbers). He fails to see that set
> inclusion does not give a trichonomy of sets, and it is difficult to
> obtain that when set inclusion should always be decisive when present.
You should indicate who "he" is, ie Bolzano or Mueckenheim.
-jiw
Dik T. Winter
> Dik T. Winter wrote:
> > However, I think there is an error on p. 88. It is stated that
> > (translated): "and it is even provable that a well-ordering [of
> > the reals] can not be defined at all..."
> > My understanding is that a definable well-order is not inconsistent
> > with ZF, and there are indeed models were a well-order can be defined.
> > See: <a href="http://arxiv.org/pdf/math.LO/9812115">Uri Abraham and
> > Saharon Shelah, A delta^2_2 well-order of the reals and incompactness
> > of L(Q^MM)</a>.
>
> There is no need to refer to such a recent paper. V=L already implies
> that there is a definable well-ordering of the reals of ridiculously low
> complexity. Of course, in reality there is no definable well-ordering of
> the reals.
Yes, I know, and I already have discussed that with WM in this newsgroup.
But that was the only quickly available reference I found that contains
a proof.
muec...@rz.fh-augsburg.de
Hahaha! Are you joking, Mynheer Winter?
Remember Burali-Forti, Russell, Skolem, Banach and Tarski: They all
had a proof that set theory is wrong. What happened? Pasting of new
axioms, superseding, psychological repression (see your assertion of a
mistake in my chapter 8).
And if there are many errors, why not bring them all to light?
Further, there are different kinds of contradictions in my book. One
sort refutes the trichotomy of naturals and alephs, another shows that
the cardinality of the continuum, 2^aleph_0, is not larger than
aleph_0. A third sort of proofs (by MatheRealism, cp.chapter 10) shows
that there is no infinitset at all.
Please distinguish carefully these different parts.
Regards, WM
muec...@rz.fh-augsburg.de
> Chapters 1 to 8:
>
> These chapters provide an exceptional well written showing about how
> things developed through the ages.
Thank you very much, Dik. I will use this as a cover text for the
second or third edition.
> Chapter 1. Natürlich unendlich
>
> Is about the natural numbers and some properties (including primes).
> They are introduced using the Peano axioms. Also induction is
> explained. What I am missing here a bit is a remark that the Peano
> axioms can be used to define arithmetic that conforms with the common
> arithmetic. And I think that the remark that Euclid used a proof by
> contradiction is wrong
??? It is said (I didn't read the original text) that Euclid assumed
that n primes exist and then he contradicted this assumption by
proving the existence of prime number n+1. Many authors call this a
proof by contradiction.
> (actually the theorem and proof are quite
> Cantoresque):
Yes. Cantor assumed a complete list of reals and showed the existence
of another real. Many authors call this a proof by contradiction.
>
> THEOREM
> Given any collection of primes, there is a prime that is not in
> that collection (at that time collections were still finite).
> PROOF
> The well-known construction, that leads to either a new prime or
> a natural number that is the product of two or more primes not in
> the collection.
> COROLLARY (current)
> There are infinitely many primes.
That however is not the wording used by Euclid.
>
> And, although the sizes of various numbers are explained, in this
> chapter there is not yet much about infinity.
>
> Chapter 3. Alogos
>
> Here we see the history of the founding of irrational numbers,
> including some irrationality proofs. The Fundamental Theorem of
> Algebra is introduced, and proof by infinite descent. However,
> the author apparently takes the position that the symbol
> sequence 'sqrt(2)' does not prove the existence of that number,
> because it is nothing more than the question to find the number
> whose square is 2. With this following Cantor.
Who was correct in this point.
> I wonder however
> why a notation like 3/7 does not fall under the same verdict, as
> it is nothing more than the question to find the number that,
> when multiplied by 7 yields 3. So we can say that the number
> sequence '3/7' does not prove existence of that number.
I think I told you already: 3/7 is 0.3 in base 7 and this can be
compared with any existing nunmber n/B (in natural base B) by 7n/7B <
3B/7B or 7n/7B = 3B/7B or 7n/7B > 3B/7B. So we need only find the
natural numbers 7n and 3B which exist for every natural numbers n and
B we can use. (In case all bits of our bit reservoir were required to
establish n and B, then the reservoir of bits must be extended by 7 or
3 which should be possible in all practicable cases.)
> Chapter 5. Infinitesimal
>
> This is mostly historical, showing an overview about the struggle
> with the infinitesimal, where finally Cauchy and Weierstraß gave
> the solution to the problems (differentiating and integrating) without
> using infinitesimals at all, but using limits. It is acknowledged
> that infinitesimals can be given a proper basis using Abraham
> Robinson's non-standard analysis. Alas, other approaches are not
> even mentioned (John Conway, Anders Kock).
And Robinson is only mentioned to support his authority when quoting
that there are no infinite sets at all. The reader can see, that
Robinson knows about what he talks.
In fact I never lectured in that depth as only to mention hyperreal
numbers. The audience is not mathematicians. And as I state on the
cover, my book is for everbody with a good knowledge of school
mathematics.
>
> Chapter 6. Paradoxien des Unendlichen
>
> Bolzano's musings are set out here, especially for those cases where
> intuition does not work. For instance, he does not like bijections
> to prove equality of "size" of sets as it does not conform with
> intuition of infinite sets
It does not conform with facts. If points (reals) do exist, then there
are more points in a long distance than in a short distance as becomes
evident when considering the short one as a part of the longer one.
_________
_____________
Two identical distances can be interchanged and have same number of
points; they cannot be distinguished. In the difference of longer and
shorter distance there are points too. This is a very simple proof
that bijections can yield false results (whehn interpreted as yielding
numbers of elements). The longer distance "hat mehr Realtät" as Cantor
would say. And mathematics is about reality, at least about geometry.
(the set of squares of natural numbers is
> intuitively less than the set of all natural numbers). He fails to
> see that set inclusion does not give a trichonomy of sets,
Better: He fails to fail to see that it gives trichotomy of sets.
>
> Chapter 7. Transfinit
>
> Set theory is described, both in the older (Cantor time) form as in
> the modern form. However, I think there is an error on p. 88. It
> is stated that (translated): "and it is even provable that a
> well-ordering [of the reals] can not be defined at all..."
> My understanding is that a definable well-order is not inconsistent
> with ZF, and there are indeed models were a well-order can be defined.
> See: <a href="http://arxiv.org/pdf/math.LO/9812115">Uri Abraham and
> Saharon Shelah, A delta^2_2 well-order of the reals and incompactness
> of L(Q^MM)</a>.
You have seen my list of ZFC axioms. It has been proved that this list
does not yield a definable well ordering of the reals, i.e., without
introducing additional axioms. I did not write that it would be in
contradiction with ZFC; this would have been silly because ZFC proves
the existence of a well ordering.
I wrote:
Es ist bisher niemandem gelungen, eine Wohlordnung für R anzuge¬ben,
und es läßt sich sogar beweisen, daß eine Wohlordnung gar nicht
definiert, also angegeben oder gefunden werden kann - auch bei
Gültigkeit des Auswahl¬axioms (you see the reference to ZFC by AC)
Of course you can add axioms to ZFC. But then you have no longer ZFC.
And why then use V=L? Why not use the axiom: There is a definable well
ordering of the reals? V=L is "Augenwischerei" (as well as AC).
Further, all your arguing is in vain unless you can present a well
ordering working in reality. And you cannot.
This part of the book is absolutely correct.
>
> Chapter 8. Potentiell versus aktual
>
> Here an overview is given about the historical struggle between
> potential and actual infinity. From the quoted text it is clear
> that the issue is not mathematical, but philosophical, and many
> times even religious.
>
> Also a few of the more esoteric findings of modern set theory are
> indicated, but clearly the author is wrong in some of the cases.
>
> The first such case is after the incompleteness theorem by Gödel
> (page 106):
> "any sufficient complex consistent system contains theorems that
> are true but that can not be proven."
> After this follows what appears like a simple conclusion by the author,
> because no reference is given:
Ah yes, you are right. I forgot it. Credit to Gödel.
>
> The second case is more serious, because it tends to show
> misunderstandings by the author (more of this in the reviews of
> later chapters). Page 107 deals with Skolem's results that there
> is a countable model of ZF. And the question is, how is that possible
> when there are uncountable sets? The authors quotes three ways to
> explain it. The first argument is contending the existence of N within
> the model, the third argument is about it being impossible to look too
> close at the countable model. I tend to agree with the author that the
> first and third do not hold. However, he has apparently serious
> problems with the second. To quote (translated):
> "It is assumed that the axiom of the power set does not give the
> complete powerset, but only a subset that is countable from the
> outside, but not so in the model, because there is no mapping
> from N to P'(N)."
> He counters this with:
> "That is very much a dubious explanation, because a mapping is
> a set itself. A map from N to a countable set P'(N) is a
> countable set. Why should it fail to be in a model that
> contains N?"
> But that is wrong, because when a map 'f' is viewed as a set, it is
> defined as:
> { (n, s) | n in N, s in P(N), f(n) = s }
> So when 'f' can not be defined in a model, neither can the set
> representing 'f'.
When P(N) exists in the model, then at least a function f from P(N)
to ... exists there. It is the similar to the nodes and edges of the
tree. Every existing node is the end of an edge. There are not less
edges than nodes. If all elements of P(N) exist, then all elements of
f exist. And the existence of P(N) is guaranteed by the power set
axiom.
> He further states:
> "Moreover, as by the axiom of the power set P(N) contains each
> subset of N. So which subset fails to be in P(N)?"
> The answer is simply: those that can not be defined in the model.
> The model has serious influences on the properties that can be used
> in the axiom of subsets, and so also on the power set as defined by
> the axiom of the power set.
No. The existence of N and of the power set axiom guarantee the
existence of P(N) and of at least a mapping
(P(N))-->N. Your explanation is ungereimt, as Bolzano would have
said. And Skolem himself did not believe in his own explanations, as
his attitude towards set theory clearly shows.
Nevertheless, a very good review. Thank you, Dik. As no part will
follow in public, it should be mentioned that in the book about 170
references are given, that the index contains over 550 subjects, and
that the customer buys a magnificent hard cover issue.
Regards, WM
muec...@rz.fh-augsburg.de
wrote:
> Dik T. Winter wrote:
> > However, I think there is an error on p. 88. It is stated that
> > (translated): "and it is even provable that a well-ordering [of
> > the reals] can not be defined at all..."
> > My understanding is that a definable well-order is not inconsistent
> > with ZF, and there are indeed models were a well-order can be defined.
> > See: <a href="http://arxiv.org/pdf/math.LO/9812115">Uri Abraham and
> > Saharon Shelah, A delta^2_2 well-order of the reals and incompactness
> > of L(Q^MM)</a>.
>
> There is no need to refer to such a recent paper. V=L already implies
> that there is a definable well-ordering of the reals of ridiculously low
> complexity. Of course, in reality there is no definable well-ordering of
> the reals.
Sic!
Regards, WM
muec...@rz.fh-augsburg.de
Both!
Both fail to fail to see that set inclusion gives trichotomy of sets -
in reality.
Regards, WM
muec...@rz.fh-augsburg.de
Chapter 9. Infinit
In this chapter the author tries to explain how the existence of a
cardinal number for the set of natural numbers leads to an
inconsistency, or how some proofs in set theory are wrong. He
does
that by examples and counter-reasoning, but in all those examples
he
either uses bad logic, misunderstanding, circular reasoning or
ill-defined terms. You may look at a few of them and skip the
others,
or possibly look at all of them, but it becomes a bit tedious. I
have tried to comment on all of them, but I may have missed some.
The first is based on a quote from Cantor (parafrased):
"omega is the first whole number that follows all numbers nu,
i.e. it is larger than every number nu."
[ Here nu is a natural number, Cantor uses the term "whole number"
for what is currently called "ordinal number". ] The author
continues:
"The last sentence shows an inherent contradiction of the
notion
omega: omega in no way follows all natural numbers, because
each
is followed by a single next larger number. So omega follows
the
last natural number, which does not exist. Because omega has
no
immediate predecessor, it does not follow anything."
The author obviously thinks that Cantor means with "follows",
"follows
immediately". If that were so the statement is indeed obviously
false.
WM:
Cantor said it. He wrote (Collected Works, p. 324, p. 326, p. 330)
about the "der Größe nach zunächst folgende Zahl", i.e. the limit
ordinal number which immediately follows.
We can safely assume that Cantor did not think that.
WM:
Of course, no reason to mention that. The conclusion is mine.
Moreover it is
interesting that the author does himself use the term "immediate
predecessor". Why the distinction "predecessor" vs. "immediate
predecessor", but not the distinction "follows" vs. "follows
immediately"?
And, moreover, Cantor actually does *write* what he meant with
that
terminology!
The second is an example that tries to refute the possibility that
there
is an aleph-0 that is the cardinal number of the set of natural
numbers.
He states that every natural number has in its decimal expansion
with
leading zeros omitted only digits at places that are indexed by
natural
numbers (true). Moreover, he states that also the reverse holds,
every
decimal expansion (again with leading zeros omitted) that has only
digits at positions that are indexed by natural numbers is a
natural
numbers (given without proof, and false, it is only true when
there
are finitely many such positions).
WM:
As long as there are natural indexes, the positions are finitely many,
because every natural index marks a finite segment.
From the (false) equivalence of the
two statements the author concludes, correctly, that if there was
an
actual infinity there would exist a decimal expansion 111...,
that,
because of the equivalence would be a natural number, which is a
contradiction. That is right, but the premissa is unproven, a
clear
case of circular reasoning. He moreover adstruates it with the
decimal expansion 0.111... and starts to reflect. A common
misunder-
standing. When you reflect an infinite decimal expansion of a
rational
you do not necessarily get a natural number.
WM:
I is easy to prove.
0.000...0001 reflected is the natural number 1000...000. You get it
for every finite index n, i.e., for any natural number of zeros.
And although the author
wishes to remove convergence from this discussion, the value of an
infinite decimal expansion is *defined* using convergence, so it
*can*
not be omitted. So his conclusion that the existence of aleph-0
has
different effects before and after the decimal point is nonsense.
In
both cases you have to use limits and convergence to obtain a
number.
When there are infinitely many decimals in front the limit does
not
converge, so is not a natural number.
WM:
Every position idexed by a natural number is a finite position. As
only natural numbers are used to index, all positons are finite (have
a finite distance from the decimal point).
When there are such behind, the
limit does converge, and you have a real number. There are
however
decimal systems where it is the other way around, infinitely
decimals
in front and only finitely many decimals behind the decimal point
(the 10-adics, a non-prime version of the p-adics).
The third will be well known to readers that follow the author on
sci.math. The author forms a list of numbers 'a' of the form
0.11...1.
In the list a natural number matches with a number 'a' with that
number
of ones in the expansion. Moreover, it matches (at the end) omega
with the decimal expansion of 1/9. He continues with the height
and the width of the table, without actually defining what those
*are*.
WM:
It is sobvious. The number of digits (or bits) 1.
I think (from what is written) that he intends the triangular
portion
of the table that contains only ones, and the height and width as
a
natural (or ordinal, or cardinal) number of the maximal number of
lines
and the maximal number of ones in a line. And in the complete
table
there are omega+1 lines and omega ones in the expansion. But even
when we omit the last line, the width and height are both omega,
at
least if we define the height and the width as limits, and define
the
limits in the appropriate way. You get at the last line *only*
when
you are using limits on the entries. And those are quite
differently
defined, so there is no reason why width and height should be
equal.
I will skip his ludicrous conclusions from this.
WM:
There is no possibility that width an height could be different.
Proof: One-to-one mapping of the vertical digits onto the horizontal
digits, by means of the diagonal elements a_ii.
The fourth case is a bit strange, in my opinion. As Bolzano
states
(quoted) that some counter-intuitive results can be resolved when
one
realises that there is no largest natural number. The author
maintains
that that is not the basic question, the basic question is whether
all natural numbers are finite. It is a bit unclear *why* he
questions
that, as it follows immediately from the definition.
WM:
But this definition is not compatible with the actual existence of all
natural numbers.
But he states
that for a set that contains *all* natural numbers, there is
always
a natural number that is not in the set. A strange statement, to
say
the least, moreover, it is not based on anything.
WM:
As you have read, Cantor enjoys the splitting of numbers into ordinal
and cardinal number. This splitting cannot happen as long as there are
only finite numbers. It may happen for infinite numbers like omega or
aleph_0. But if it has happened, then there is the number aleph_0
already among the numbers which it counts.
Regards, WM
William Hughes
> As long as there are natural indexes, the positions are finitely many,
> because every natural index marks a finite segment.
[All statements are statements that are true in Wolkenmueckenheim.
(Some, like "a potentially infinite set is finite", only make sense
in Wolkenmueckenheim)]
You cannot use the fact that every index
is finite to show that the set of indexes is not potentially infinite.
Yes, a potentially infinite set is finite,
but a potentially infinite set does not have a cardinality
nor does it correspond to a natural number.
- William Hughes
MoeBlee
> Remember Burali-Forti, Russell, Skolem, Banach and Tarski: They all
> had a proof that set theory is wrong.
Unless you give a mathematical definition of 'wrong', I don't know
what you mean by 'a proof that set theory is wrong'.
Burali-Forti showed that there is no greatest ordinal. In Z set
theories, it is not a contradiction that there is no greatest ordinal.
Russell showed that Frege's system is inconsistent. That does not
impinge on Z set theories.
Skolem showed that a theory can have a countable model and also a
theorem that there are sets that are uncountable. But it is not a
contradiction that a theory has a countable model and also a theorem
that there are sets that are uncountable.
Tarkski and Banach showed that ZFC has a certain result that is
puzzling at first glance. But it requires the axiom of choice; it is
not a contradiction; and (as far as I understand) would only
contradict a physical theory if matter had infinite density.
MoeBlee
MoeBlee
> On 14 Feb., 03:57, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> And Robinson is only mentioned to support his authority when quoting
> that there are no infinite sets at all. The reader can see, that
> Robinson knows about what he talks.
Robinson's own formalist philosophy includes his claim that the notion
of an infinite set is meaningless. However, Robinson states why
infinite sets may be used formally. Robinson's mathematics are
developed on the usual basis of mathematical logic and set theory,
including infinite sets, uncountability, and the axiom of choice.
MoeBlee
G. Frege
>>
>> And Robinson is only mentioned to support his authority when quoting
>> that there are no infinite sets at all. The reader can see, that
>> Robinson knows about what he talks.
>>
> Robinson's own formalist philosophy includes his claim that the notion
> of an infinite set is meaningless. However, Robinson states why
> infinite sets may be used formally. Robinson's mathematics are
> developed on the usual basis of mathematical logic and set theory,
> including infinite sets, uncountability, and the axiom of choice.
>
(I) Infinite totalities do not exist in any sense of the
word (i.e., either really or ideally). More precisely, any
mention, or purported mention, of infinite totalities is,
literally, meaningless.
(II) Nevertheless, we should continue the business of
mathematics 'as usual', i.e., we should act as if infinite
totalities really existed.
(Abraham Robinson)
F.
--
E-mail: info<at>simple-line<dot>de
David Marcus
> Tarkski and Banach showed that ZFC has a certain result that is
> puzzling at first glance. But it requires the axiom of choice; it is
> not a contradiction; and (as far as I understand) would only
> contradict a physical theory if matter had infinite density.
"Density" probably isn't the correct word. But, I don't think Banach-
Tarski poses any problems for physics.
--
David Marcus
Ralf Bader
On p. 324 I read:
"Somit ist beta die auf alle a_nu der Größe nach nächstfolgende
Ordnungszahl;" and, a couple of lines later,
"Zu jeder Fundamentalreihe {a_nu} von Ordnungszahlen gehört eine
Ordnungszahl Lim_nu a_nu, welche auf alle a_nu der Größe nach zunächst
folgt;"
It seems that Cantor is introducing the general notion of limit ordinal at
this point. The a_nu form a "fundamental sequence", and beta=Lim_nu a_nu
is the smallest ordinal larger than all ordinals in that sequence.
Similar on p. 326, 330.
Mückenheim distorted Cantor's clear statements by pulling the words "der
Größe nach zunächst folgende Zahl" ("following, in size, next after all
a_nu") out of their context. I'll keep for myself what I think about
Mückenheim's style of quoting, of which this is another example. Just this:
I don't believe in the favourable review of the first 8 chapters of his
book. I wouldn't trust Mückenheim in anything without
cross-and-double-checking.
Ross A. Finlayson
There are several mistakes.
I don't really care because Muckenheim's potential finitism is from
the last century ago already. It is not that there's something wrong
with that, only that denial doesn't lead to knowledge. I don't
disapprove of his discussion of potential infinity vis-a-vis the
finite, yet would find him hard-pressed to deny infinity in the
system.
Hopefully, it's a reasonable scholarly article.
Cantor's writings are perhaps not as you'd expect given their modern
interpretation. It is refreshing to read translations, generally
English readers read the Jourdain translation, Mueckenheim offers
sometimes alternate transcriptions, translations. For example, I use
them both.
What is this supposed to be? I don't see room for the infinitesimal
calculus in that system, which is obviously widely used.
Ross
--
Ross Finlayson
Finlayson Consulting
Virgil
muec...@rz.fh-augsburg.de wrote:
They may all have had proofs that some particular form of set theory was
'wrong', but where did any of them ever have a proof that all forms of
set theory must be wrong?
The calculus of Leibitz and Newton, being based on undefined
infinitesimals, was "wrong" in a way, too, but that has since been fixed.
So why does WM continue to claim ( without proof, or even evidence) that
whatever was once "wrong" with set theory is incapable of being fixed?
> Further, there are different kinds of contradictions in my book.
All based on assumptions contradicting the axiom systems which WM is
attempting to criticize.
muec...@rz.fh-augsburg.de
Now the fifth. The author states that *every* set of natural
numbers
contains a number that is larger than the cardinal number of the
set.
WM:
Are you sure, Dik? If so, then it would be a big error. But I could
not find it. I wrote "Eine nicht leere, endliche Menge positiver
gerader Zahlen, enthält stets Elemente, welche die Kardinalzahl der
Men¬ge übertref¬fen."
(Oviously this is wrong. There is a number that is larger than or
equal to the cardinal number of the set, consider {1}. But he
continues
with sets of even numbers, and there it is indeed true.) He
states
that it can be proven by induction for each natural number, and
that
is true, but concludes that so it is true for the "potential
infinite"
set of whole numbers (natural numbers is meant here). Upto this
point
of the book the "potential infinite" set of natural numbers has
not yet
been defined, so the meaning is pretty unclear.
WM: Didn't you read chapter 8?
But he asks how it is
possible that in the finite case:
| { 2, 4, 6, ..., 2n } | = n
and how in the infinite case 2n becomes equal to n. He states
that if
| { 2, 4, 6, ... } | = aleph-0
we have:
lim{n -> oo} | { 2, 4, 6, ..., 2n } | = aleph-0
(the author does not define that limit, but under some reasonable
definition that is true), but concludes:
2n / aleph-0 <= 1
which is true for finite n and so also in the limit. How the
author
comes to the first (finite n) and to the second (limit), and how
that
division is defined is left to the imagination of the reader.
WM: 2n < aleph_0 by definition for every natural number.
But for
finite n, I would assume that that quotient is 0 with a reasonable
definition, and so the limit is 0. But from this the conclusion
is
formed that:
lim{n -> oo} ( 2n / | { 2, 4, 6, ..., 2n } | ) <= 1,
a clear contradiction, because it is 2. The conclusion is wrong
because it assumes that also in this case:
lim{n -> oo} ( f(n) / g(n) } = lim{m -> oo} f(m) / lim{n ->
oo} g(n).
Clearly false. This is only true when both limits do exist (as
finite
results).
WM:
No. It is nonsense to speak of the infinite set of finite numbers. If
we do it, then we find literally:
n is finite in every case including n --> oo, because there is no
infinite natural number.
The limit | { 2, 4, 6, ... } | can be defined by the union of all
initial segments. It is aleph_0, also by definition. Clearly these two
results do not fit together, as shown by letting
lim{n -> oo} | { 2, 4, 6, ..., 2n } | / 2n = 1/2
Both limits are infinite, and (again) with a reasonable
definition of limit both are aleph-0. But there is no definition
of
aleph-0/aleph-0. And, indeed, we can get any number from 0 to oo
when we increase m in f(m) and n in g(n) independently (under
fairly
reasonably definitions of both limit and quotient). This is quite
similar to what we can do with conditional convergent series.
The sixth one is the following:
consider |{ 0, 1, 2, ..., n-1 }| = n < aleph-0
Obviously for every finite n, there are fewer than aleph-0 natural
numbers less than n. Let's make n a variable and assume that it
goes through all natural numbers. Because for each finite n there
are fewer than aleph-0 natural numbers less than n, there are few
natural numbers less than aleph-0, while the cardinality is
aleph-0.
The fallacy is of course the assumption that what is true for each
finite segment is also true for the complete set.
WM: The "fallacy" is to conclude a =< z from a<b<c<...<z even in case
of infinitely many terms.
Number seven. The author quotes Cantor as saying (parafrased):
"It is even permitted to think about the newly created number
omega as limit, which the numbers nu try to reach, when we
mean with that, that omega is the first whole number that
follows all numbers nu, that is, it is larger than all
numbers nu."
And the author states that it is not permitted. Why not?
Because a number omega that follows all natural numbers and also
gives the number of them does not exist. And attempts an analogy
with negative unit fractions. Zero is not a unit fraction, so it
is the first number that follows the unit fractions. He notices
two differences. For the unit fractions, the distance to 0
becomes smaller and smaller, for omega, the distance to omega
remains the same. The (in the authors eyes) more serious
difference,
so serious that it would lead to a collapse of Cantors analogy, is
that with the unit fractions, 0 is not their number, while it
should
be omega for the natural numbers. I am really impressed. For the
first difference: it depends entirely how we measure distances,
and
the measurement of distances is not part of set theory.
WM: But it can be applied to set theoretic theorems.
With other
measuring devices it would be the other way around. The second is
also nonsense. While with the natural numbers an initial set of
them has cardinality that is closely related to the smallest
number
following them all, but that it not true with sets of unit
fractions.
Further he states that the cardinality of an initial set of
natural
numbers is one more than the sum of the distances (but this
depends on
the way measuring is done)
WM: It is the measure of set theory. The distances are in one-to-one
correspondence with all the numbers except the first one.
, and this sum is finite as long as all
numbers involved are finite. And the last is obviously false when
there are infitely many finite numbers, in that case the sum does
not exist.
WM: Neither do the numbers.
On to the eighth one. Here the author considers a new concept:
"Interzession", which I would translate as "mildly interleaved",
but
there are perhaps better terms.
WM: Intercession, for instance.
Two subsets of a set are mildly
interleaved if between each two elements of the first set there is
at least one element of the other, and the reverse. Two sets can
be mildly interleaved when they can be put as subsets in a set
that
are mildly interleaved. Clearly if there is a bijection between
two ordered sets they can be mildly interleaved (put a particular
destination element just after the corresponding source element).
He states that "can be mildly interleaved" is an equivalence
relation
on sets. But he does not prove transitivity, he only shows it by
example, but perhaps it holds, and it is an equivalence relation,
and
so equivalence classes can be formed. Obviously all ordered sets
with
the same cardinality fall in the same class. But the author fails
to
mention that the concept makes sense only for sets that *are*
ordered.
So can all sets be ordered?
WM:
All sets of finite numbers can be ordered. They need not be well
ordered for intercession. For applicaton of Cantor's aephs they must
even be well ordered.
Obviously, under CH they can, they can
even be well-ordered, but I do not know whether the same is true
when
CH does not hold.
You mean AC.
I would challenge the author to give an ordering
(not necessarily a well-ordering) of the subsets of R.
WM: I stated on p. 117: For all finite numbers: "Alle unendlichen
Zahlenmengen (aus endlichen Zahlen)". Most subsets of R are not finite
and, therefore, do not belong to species "finite numbers".
So his claim
that it is an equivalence relation on sets, and as such provides a
better measure for the size of a set is unfounded.
WM: It is an equivalence relation on sets of finite numbers, i.e. all
numbers < omega.
Number nine. Also well-known to readers of sci.math: the binary
tree.
But with a twist. He gives an infinite binary tree with nodes and
edges and path-bundles. The root node represents "0," each
sibling
node represents either a following "0" or "1". So going through
the edges and nodes we accumulate a binary number. A path-bundle
is an edge together with all the edges leading to the root, with a
twist. He states that given a path-bundle that splits in two
sibling
path-bundles, each of the siblings receives one half of the edges
of
the parent path-bundles. (What *is* one half of an edge?)
What is half a cake?
With this
he finds that each path-bundle contains more than one edge (the
level
1 path-bundles contain one edge, the level-2 path-bundles contain
3/2 edges, etc.). And so each path-bundle contains at least one
edge.
Yes, indeed, even with this construction, it is the last one. So
he
concludes:
"This way no path-bundle (and of course also no path) can
split from
another one without an edge being associated with it."
"it" meaning the path or path-bundle. Which is, eh, quite
strange.
WM: What is strange about this obvious fact? Simply follow a path like
0.000... or 0,010101... and you will see it. Therefore there can be
not more paths than edges. It is really quite strange that you seem to
believe that "in the infinite", the paths spring off the edges and
exist without any foundation, the usual form of existence in set
theory.
For path-bundles the strange construction is not needed at all.
WM: Paths are bunches too: singletons, aren't they? Or are they
nothing at all?
There
is a clear bijection between edges and path-bundles (map a path-
bundle
to its last edge).
For paths the situation is different. You can not
map a particular edge to a path because the paths are not
terminating.
WM: Paths are path bunches too, namely such with one path only, unless
paths do not exist.
So there is no clear reason why the cardinality of the set of
paths
should be less than the cardinality of the set of path-bundles.
WM: It is not less but it canot be larger. I find it highly
(HIGHLY!!!) strange how an intelligent being can claim (and, perhaps,
even believe) that there are more bunches of paths (including all
singletons, i.e., paths) in the tree than nodes where they can split.
So there are more results of spits than splits, although we know that
every split creates exactly one result. Strange.
Regards, WM
Virgil
muec...@rz.fh-augsburg.de wrote:
Bolzano is long since dead, and WM is much like him in that respect.
David Marcus
> REPLY TO THE SECRET REVIEW OF CHAPTER 9 (PART 2 of the Reply)
Secret? Since when is something posted on the Web a secret?
--
David Marcus
Dik T. Winter
> On 14 Feb., 03:44, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> > Book:
> > Die Mathematik des Unendlichen
> > Shaker Verlag, Aachen, Germany, 2006, ISBN 3-8322-5587-7.
> > part will not be posted, but is available through the following link:
> > <http://www.cwi.nl/~dik/english/mathematics/mueck/book2.html>
> > It will be updated while I read more.
I have completed reading chapter 9 (it took more time than I actually
wished to spend), and the file at the above link is updated. Still
chapter 10 to go.
> > My initial conclusion is that chapters 1 to 8 can serve quite well in
> > a course on the history of set theory. The historical context is set
> > out extremely well. Also the outlining of current set theory appears
> > to be adequate. The second part (at least what I have read until now)
> > is nonsense. It consists entirely of misunderstandings, bad logic,
> > ill-defined objects and whatever you can think. Moreover, I fail to
> > see why the authors needs over ten examples to show that actual infinity
> > leads to an inconsistency. Only one would be sufficient, if it were
> > valid.
>
> Hahaha! Are you joking, Mynheer Winter?
No.
> Remember Burali-Forti, Russell, Skolem, Banach and Tarski: They all
> had a proof that set theory is wrong. What happened? Pasting of new
> axioms, superseding, psychological repression (see your assertion of a
> mistake in my chapter 8).
I will look at it.
> And if there are many errors, why not bring them all to light?
> Further, there are different kinds of contradictions in my book. One
> sort refutes the trichotomy of naturals and alephs, another shows that
> the cardinality of the continuum, 2^aleph_0, is not larger than
> aleph_0. A third sort of proofs (by MatheRealism, cp.chapter 10) shows
> that there is no infinitset at all.
I have gone over chapter 9 now, read my review.
Dik T. Winter
> On 14 Feb., 03:57, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> > Chapters 1 to 8:
> >
> > These chapters provide an exceptional well written showing about how
> > things developed through the ages.
>
> Thank you very much, Dik. I will use this as a cover text for the
> second or third edition.
>
> > contradiction is wrong
>
> ??? It is said (I didn't read the original text) that Euclid assumed
> that n primes exist and then he contradicted this assumption by
> proving the existence of prime number n+1. Many authors call this a
> proof by contradiction.
It would be a proof by contradiction, if there is an initial assumption
that is falsified by the reasoning. But see below.
> > (actually the theorem and proof are quite
> > Cantoresque):
>
> Yes. Cantor assumed a complete list of reals and showed the existence
> of another real. Many authors call this a proof by contradiction.
Wrong. Cantor does not assume that the given list is complete. He shows
that given *any* list of reals there is a real not on the list. There is
*no* initial assumption of completeness.
> > THEOREM
> > Given any collection of primes, there is a prime that is not in
> > that collection (at that time collections were still finite).
> > PROOF
> > The well-known construction, that leads to either a new prime or
> > a natural number that is the product of two or more primes not in
> > the collection.
> > COROLLARY (current)
> > There are infinitely many primes.
>
> That however is not the wording used by Euclid.
What is the wording used by Euclid? (I may not that the Corollary was indeed
*not* by Euclid, but that is the modern corollary.)
> > However,
> > the author apparently takes the position that the symbol
> > sequence 'sqrt(2)' does not prove the existence of that number,
> > because it is nothing more than the question to find the number
> > whose square is 2. With this following Cantor.
>
> Who was correct in this point.
Maybe, maybe.
> > I wonder however
> > why a notation like 3/7 does not fall under the same verdict, as
> > it is nothing more than the question to find the number that,
> > when multiplied by 7 yields 3. So we can say that the number
> > sequence '3/7' does not prove existence of that number. =20
>
> I think I told you already: 3/7 is 0.3 in base 7
But in my opinion 0.3 in base 7 is still a formula telling how the number
is calculated. It tells me that it is 3 * 1/7. And before Simon Stevin's
"De Thiende" from 1585, decimal numbers (and base 'n' notated rationals)
did not actually exist. The Babylonians used a base 60 notation, also
for fractions, but there was no true zero and there was no fraction
indicator. And what I have seen from Indian mathematics does not contain
decimal fractions either. If you look at the Surya Siddhanta (a classic
work to calculate the calendar) you will find that it contains a sine
table, but the values are represented as integers, to be divided by 1000.
The angles themselves are represented in degrees, sixtieth of degrees,
and so on. But as Simon Stevin writes (parafrased):
"The importance of the proposal is not so much mathematical profoundness,
but more broad applicability."
And it was merely used as an aid to ease calculations.
> I think I told you already: 3/7 is 0.3 in base 7 and this can be
> compared with any existing nunmber n/B (in natural base B) by 7n/7B <
> 3B/7B or 7n/7B = 3B/7B or 7n/7B > 3B/7B.
So for the comparison you do not use "0.3" but the original "3/7"?
Well, I can compare sqrt(2) with any rational number. Square the two.
> 3B/7B or 7n/7B = 3B/7B or 7n/7B > 3B/7B. So we need only find the
> natural numbers 7n and 3B which exist for every natural numbers n and
> B we can use. (In case all bits of our bit reservoir were required to
> establish n and B, then the reservoir of bits must be extended by 7 or
> 3 which should be possible in all practicable cases.)
For squaring not much more is needed. But whether you can compare easily
or not does *not* depend on whether the notation actually is an algorithm
or not.
> > It is acknowledged
> > that infinitesimals can be given a proper basis using Abraham
> > Robinson's non-standard analysis. Alas, other approaches are not
> > even mentioned (John Conway, Anders Kock).
>
> And Robinson is only mentioned to support his authority when quoting
> that there are no infinite sets at all. The reader can see, that
> Robinson knows about what he talks.
What quote of Robinson are you talking about? I do not find any such
quote. Pray state page number in your book.
> > Bolzano's musings are set out here, especially for those cases where
> > intuition does not work. For instance, he does not like bijections
> > to prove equality of "size" of sets as it does not conform with
> > intuition of infinite sets
>
> It does not conform with facts.
What facts? Facts are subjective.
> If points (reals) do exist, then there
> are more points in a long distance than in a short distance as becomes
> evident when considering the short one as a part of the longer one.
Yes, and that kind of comparisons lead to lack of trichotomy.
> Two identical distances can be interchanged and have same number of
> points; they cannot be distinguished. In the difference of longer and
> shorter distance there are points too. This is a very simple proof
> that bijections can yield false results (whehn interpreted as yielding
> numbers of elements). The longer distance "hat mehr Realtä4t" as Cantor
> would say. And mathematics is about reality, at least about geometry.
Well, when I did learn geometry the reality was that you could actually
not state that a longer distance had more points. Consider the construction
to divide a particular line in five pieces of equal length using compass
and straight-edge. All such constructions are actually bijections in
disguise.
> > (the set of squares of natural numbers is
> > intuitively less than the set of all natural numbers). He fails to
> > see that set inclusion does not give a trichonomy of sets,
>
> Better: He fails to fail to see that it gives trichotomy of sets.
Pray elaborate, I do not understand this.
> > is an error on p. 88. It
> > is stated that (translated): "and it is even provable that a
> > well-ordering [of the reals] can not be defined at all..."
...
> You have seen my list of ZFC axioms. It has been proved that this list
> does not yield a definable well ordering of the reals, i.e., without
> introducing additional axioms. I did not write that it would be in
> contradiction with ZFC; this would have been silly because ZFC proves
> the existence of a well ordering.
> I wrote:
> Es ist bisher niemandem gelungen, eine Wohlordnung für R anzugeben,
> und es läßt sich sogar beweisen, daß eine Wohlordnung gar nicht
> definiert, also angegeben oder gefunden werden kann
And you are stating here that a well-ordering can not be defined in ZFC.
So, what is it. Can you show the proof that a well-ordering can not be
defined? Or at least a source for such a proof? I state that ZFC is
*not* inconsistent with a definable well-ordering of R.
> Further, all your arguing is in vain unless you can present a well
> ordering working in reality. And you cannot.
What reality? Did you read the reference I presented? Strange enough,
you object to V=L as axiom. As far as I know, V=L provides only the
computable reals in the set of reals.
> > But that is wrong, because when a map 'f' is viewed as a set, it is
> > defined as:
> > { (n, s) | n in N, s in P(N), f(n) =3D s }
> > So when 'f' can not be defined in a model, neither can the set
> > representing 'f'.
>
> When P(N) exists in the model,
Of course it exists if it is a model of ZF: the axiom of the powerset.
> then at least a function f from P(N)
> to ... exists there.
To what?
> It is the similar to the nodes and edges of the
> tree. Every existing node is the end of an edge. There are not less
> edges than nodes. If all elements of P(N) exist, then all elements of
> f exist. And the existence of P(N) is guaranteed by the power set
> axiom.
No argument here, it does not contradict what I stated.
>
> > He further states:
> > "Moreover, as by the axiom of the power set P(N) contains each
> > subset of N. So which subset fails to be in P(N)?"
> > The answer is simply: those that can not be defined in the model.
> > The model has serious influences on the properties that can be used
> > in the axiom of subsets, and so also on the power set as defined by
> > the axiom of the power set.
>
> No. The existence of N and of the power set axiom guarantee the
> existence of P(N) and of at least a mapping
> (P(N))-->N. Your explanation is ungereimt, as Bolzano would have
> said. And Skolem himself did not believe in his own explanations, as
> his attitude towards set theory clearly shows.
Sorry. (Going back to the ZF definitions), loosely speaking, the
axiom of the power set states that there is a set that contains all
subsets of a given set. The axiom of specification states what are
subsets and connects them with properties phi that can be possessed by
elements. The model states what properties are valid properties in
the model. So whether something is a subset or not of a particular
set depends on the model. So when looking from outside the model you
may find a subset that is not a subset in the model.
MoeBlee
> In article <1171972752.151611.207...@q2g2000cwa.googlegroups.com> mueck...@rz.fh-augsburg.de writes:
> > And Robinson is only mentioned to support his authority when quoting
> > that there are no infinite sets at all. The reader can see, that
> > Robinson knows about what he talks.
>
> What quote of Robinson are you talking about? I do not find any such
> quote. Pray state page number in your book.
See Robinson's essay 'Formalism 64'. Robinson says that the notion of
an infinite set is meaningless. But he also endorses working with
infinite sets in formal theories.
MoeBlee
Dik T. Winter
> REPLY TO THE SECRET REVIEW OF CHAPTER 9 (PART 1 of the Reply)
Eh? What is secret about it? I have published widely enough how it could
be found. <http://www.cwi.nl/~dik/english/mathematics/mueck/book2.html>,
nothing secret about it. And within a few days you can find it by Google.
> The first is based on a quote from Cantor (parafrased):
> "omega is the first whole number that follows all numbers nu,
> i.e. it is larger than every number nu."
> [ Here nu is a natural number, Cantor uses the term "whole number"
> for what is currently called "ordinal number". The author
> continues:
> "The last sentence shows an inherent contradiction of the notion
> omega: omega in no way follows all natural numbers, because each
> is followed by a single next larger number. So omega follows the
> last natural number, which does not exist. Because omega has no
> immediate predecessor, it does not follow anything."
> The author obviously thinks that Cantor means with "follows", "follows
> immediately". If that were so the statement is indeed obviously false.
WM:
> Cantor said it. He wrote (Collected Works, p. 324, p. 326, p. 330)
> about the "=EF=80=A0der Größe nach zunächst folgende Zahl",
> i.e. the limit ordinal number which immediately follows.
I have no idea what =EF=80=A0 is, it is no valid character in UTF-8 encoded
Unicode. But, which immediately follows what? Not a predecessor, I think.
And in the quote above he *explicitly* states what he means with the term
(i.e. it is larger than every number nu).
> The second is an example that tries to refute the possibility that there
> is an aleph-0 that is the cardinal number of the set of natural numbers.
> He states that every natural number has in its decimal expansion with
> leading zeros omitted only digits at places that are indexed by natural
> numbers (true). Moreover, he states that also the reverse holds, every
> decimal expansion (again with leading zeros omitted) that has only
> digits at positions that are indexed by natural numbers is a natural
> numbers (given without proof, and false, it is only true when there
> are finitely many such positions).
WM:
> As long as there are natural indexes, the positions are finitely many,
> because every natural index marks a finite segment.
Wrong. You are assuming what you are trying to prove.
> From the (false) equivalence of the
> two statements the author concludes, correctly, that if there was an
> actual infinity there would exist a decimal expansion 111..., that,
> because of the equivalence would be a natural number, which is a
> contradiction. That is right, but the premissa is unproven, a clear
> case of circular reasoning. He moreover adstruates it with the
> decimal expansion 0.111... and starts to reflect. A common misunder-
> standing. When you reflect an infinite decimal expansion of a rational
> you do not necessarily get a natural number.
WM:
> I is easy to prove.
> 0.000...0001 reflected is the natural number 1000...000. You get it
> for every finite index n, i.e., for any natural number of zeros.
So you are *not* reflecting an infinite decimal expansion.
> And although the author
> wishes to remove convergence from this discussion, the value of an
> infinite decimal expansion is *defined* using convergence, so it *can*
> not be omitted. So his conclusion that the existence of aleph-0 has
> different effects before and after the decimal point is nonsense. In
> both cases you have to use limits and convergence to obtain a number.
> When there are infinitely many decimals in front the limit does not
> converge, so is not a natural number.
WM:
> Every position idexed by a natural number is a finite position. As
> only natural numbers are used to index, all positons are finite (have
> a finite distance from the decimal point).
Yes, so what? All positions are finite, but there are infinitely many of
them.
> The third will be well known to readers that follow the author on
> sci.math. The author forms a list of numbers 'a' of the form 0.11...1.
> In the list a natural number matches with a number 'a' with that number
> of ones in the expansion. Moreover, it matches (at the end) omega
> with the decimal expansion of 1/9. He continues with the height
> and the width of the table, without actually defining what those *are*.
WM:
> It is sobvious. The number of digits (or bits) 1.
Eh? What is the height of the unending table? What is the width of it? How
do you define that?
> I think (from what is written) that he intends the triangular portion
> of the table that contains only ones, and the height and width as a
> natural (or ordinal, or cardinal) number of the maximal number of lines
> and the maximal number of ones in a line. And in the complete table
> there are omega+1 lines and omega ones in the expansion. But even
> when we omit the last line, the width and height are both omega, at
> least if we define the height and the width as limits, and define the
> limits in the appropriate way. You get at the last line *only* when
> you are using limits on the entries. And those are quite differently
> defined, so there is no reason why width and height should be equal.
> I will skip his ludicrous conclusions from this.
WM:
> There is no possibility that width an height could be different.
> Proof: One-to-one mapping of the vertical digits onto the horizontal
> digits, by means of the diagonal elements a_ii.
That works perfectly well for the triangle without the last line.
> The fourth case is a bit strange, in my opinion. As Bolzano states
> (quoted) that some counter-intuitive results can be resolved when one
> realises that there is no largest natural number. The author maintains
> that that is not the basic question, the basic question is whether
> all natural numbers are finite. It is a bit unclear *why* he questions
> that, as it follows immediately from the definition.
WM:
> But this definition is not compatible with the actual existence of all
> natural numbers.
What actual existence? Until this point you have not even stated what that
*means*.
> But he states
> that for a set that contains *all* natural numbers, there is always
> a natural number that is not in the set. A strange statement, to say
> the least, moreover, it is not based on anything.
WM:
> As you have read, Cantor enjoys the splitting of numbers into ordinal
> and cardinal number. This splitting cannot happen as long as there are
> only finite numbers. It may happen for infinite numbers like omega or
> aleph_0. But if it has happened, then there is the number aleph_0
> already among the numbers which it counts.
In what way is that a response to what I wrote?
Dik T. Winter
...
> I don't believe in the favourable review of the first 8 chapters of his
> book.
You can trust it, I wrote it. The first eight chapters give good insight
in the history (but there are a few flaws I noticed). Chapter 9 however
is clear nonsense (and I still have to review chapter 10).
Dik T. Winter
> REPLY TO THE SECRET REVIEW OF CHAPTER 9 (PART 2 of the Reply)
There is nothing secret about it.
> Now the fifth. The author states that *every* set of natural numbers
> contains a number that is larger than the cardinal number of the set.
WM:
> Are you sure, Dik?
No, I was wrong. The review has been corrected. (But it was irrelevant to
the remainder.)
> Upto this point
> of the book the "potential infinite" set of natural numbers has not yet
> been defined, so the meaning is pretty unclear.
>
WM: Didn't you read chapter 8?
I thought so. Can you point me to the page where you define it?
> But he asks how it is
> possible that in the finite case:
> | { 2, 4, 6, ..., 2n } | =3D n
> and how in the infinite case 2n becomes equal to n. He states that if
> | { 2, 4, 6, ... } | =3D aleph-0
> we have:
> lim{n -> oo} | { 2, 4, 6, ..., 2n } | =3D aleph-0
> (the author does not define that limit, but under some reasonable
> definition that is true), but concludes:
> 2n / aleph-0 <= 1
> which is true for finite n and so also in the limit. How the author
> comes to the first (finite n) and to the second (limit), and how that
> division is defined is left to the imagination of the reader.
>
WM: 2n < aleph_0 by definition for every natural number.
Does not explain either.
> But for
> finite n, I would assume that that quotient is 0 with a reasonable
> definition, and so the limit is 0. But from this the conclusion is
> formed that:
> lim{n -> oo} ( 2n / | { 2, 4, 6, ..., 2n } | ) < 1,
> a clear contradiction, because it is 2. The conclusion is wrong
> because it assumes that also in this case:
> lim{n -> oo} ( f(n) / g(n) } = lim{m -> oo} f(m) / lim{n -> oo} g(n).
> Clearly false. This is only true when both limits do exist (as finite
> results).
WM:
> No. It is nonsense to speak of the infinite set of finite numbers. If
> we do it, then we find literally:
> n is finite in every case including n --> oo, because there is no
> infinite natural number.
Eh? But perhaps yes. But I do not see in what way that contradicts what I
wrote.
> The limit | { 2, 4, 6, ... } | can be defined by the union of all
> initial segments. It is aleph_0, also by definition. Clearly these two
> results do not fit together, as shown by letting
> lim{n -> oo} | { 2, 4, 6, ..., 2n } | / 2n = 1/2
I have no idea what this shows, and how it contradicts what I did write.
The above limit is valid. So what? What does it contradict? You refrained
from any comment on what did follow. So you do agree with that analysis?
> The sixth one is the following:
> consider |{ 0, 1, 2, ..., n-1 }| = n < aleph-0
> Obviously for every finite n, there are fewer than aleph-0 natural
> numbers less than n. Let's make n a variable and assume that it
> goes through all natural numbers. Because for each finite n there
> are fewer than aleph-0 natural numbers less than n, there are few
> natural numbers less than aleph-0, while the cardinality is aleph-0.
> The fallacy is of course the assumption that what is true for each
> finite segment is also true for the complete set.
>
WM: The "fallacy" is to conclude a =< z from a<b<c<...<z even in case
> of infinitely many terms.
Where do I conclude that?
> I am really impressed. For the
> first difference: it depends entirely how we measure distances, and
> the measurement of distances is not part of set theory.
>
> WM: But it can be applied to set theoretic theorems.
Perhaps.
> With other
> measuring devices it would be the other way around. The second is
> also nonsense. While with the natural numbers an initial set of
> them has cardinality that is closely related to the smallest number
> following them all, but that it not true with sets of unit fractions.
> Further he states that the cardinality of an initial set of natural
> numbers is one more than the sum of the distances (but this depends on
> the way measuring is done)
>
WM: It is the measure of set theory. The distances are in one-to-one
> correspondence with all the numbers except the first one.
What measure are you talking about? What are distances in set theory?
How do you define them?
> , and this sum is finite as long as all
> numbers involved are finite. And the last is obviously false when
> there are infitely many finite numbers, in that case the sum does
> not exist.
>
WM: Neither do the numbers.
So numbers do not exist?
> On to the eighth one. Here the author considers a new concept:
> "Interzession", which I would translate as "mildly interleaved", but
> there are perhaps better terms.
>
WM: Intercession, for instance.
An English speaker would not understand that for what you are meaning.
Have a look at Merriam-Webster. And, yes, I did look.
> But the author fails to
> mention that the concept makes sense only for sets that *are* ordered.
> So can all sets be ordered?
WM:
> All sets of finite numbers can be ordered. They need not be well
> ordered for intercession. For applicaton of Cantor's aephs they must
> even be well ordered.
So it only works for sets of finite numbers, and does not provide an
equivalence relation amongst sets.
> Obviously, under CH they can, they can
> even be well-ordered, but I do not know whether the same is true when
> CH does not hold.
>
> You mean AC.
Indeed. Corrected.
> I would challenge the author to give an ordering
> (not necessarily a well-ordering) of the subsets of R.
>
WM: I stated on p. 117: For all finite numbers: "Alle unendlichen
> Zahlenmengen (aus endlichen Zahlen)". Most subsets of R are not finite
> and, therefore, do not belong to species "finite numbers".
So it is not an equivalence relation amongst sets.
> So his claim
> that it is an equivalence relation on sets, and as such provides a
> better measure for the size of a set is unfounded.
>
WM: It is an equivalence relation on sets of finite numbers, i.e. all
> numbers < omega.
Yes, and in that case all infinite sets of finite numbers fall in the same
class. But that says nothing about other sets. So, in what way is it
better than bijection?
> Number nine. Also well-known to readers of sci.math: the binary tree.
> But with a twist. He gives an infinite binary tree with nodes and
> edges and path-bundles. The root node represents "0," each sibling
> node represents either a following "0" or "1". So going through
> the edges and nodes we accumulate a binary number. A path-bundle
> is an edge together with all the edges leading to the root, with a
> twist. He states that given a path-bundle that splits in two sibling
> path-bundles, each of the siblings receives one half of the edges of
> the parent path-bundles. (What *is* one half of an edge?)
>
> What is half a cake?
For a cake I know the answer, but I ask what it is for an edge. Is half
an edge the part of an edge that starts at a node and stops half-way to
the next node?
> Yes, indeed, even with this construction, it is the last one. So he
> concludes:
> "This way no path-bundle (and of course also no path) can split from
> another one without an edge being associated with it."
> "it" meaning the path or path-bundle. Which is, eh, quite strange.
>
WM: What is strange about this obvious fact?
The strangeness is that it is not an abvious fact. I state, quite clearly,
that each path-bundle can be associated with an edge, namely the last one
in the path-bundle. How you come from that that also each path can be
associated with an edge is completely unfounded.
WM: What is strange about this obvious fact? Simply follow a path like
> 0.000... or 0,010101... and you will see it. Therefore there can be
> not more paths than edges. It is really quite strange that you seem to
> believe that "in the infinite", the paths spring off the edges and
> exist without any foundation, the usual form of existence in set
> theory.
Clear as mud.
> For path-bundles the strange construction is not needed at all.
>
WM: Paths are bunches too: singletons, aren't they? Or are they
> nothing at all?
A path-bundle has a last edge, a path does not have a last edge. Quite
some difference.
> There
> is a clear bijection between edges and path-bundles (map a path-bundle
> to its last edge).
>
> For paths the situation is different. You can not
> map a particular edge to a path because the paths are not terminating.
>
WM: Paths are path bunches too, namely such with one path only, unless
> paths do not exist.
But they do not have a last edge.
> So there is no clear reason why the cardinality of the set of paths
> should be less than the cardinality of the set of path-bundles.
>
WM: It is not less but it canot be larger. I find it highly
> (HIGHLY!!!) strange how an intelligent being can claim (and, perhaps,
> even believe) that there are more bunches of paths (including all
> singletons, i.e., paths) in the tree than nodes where they can split.
> So there are more results of spits than splits, although we know that
> every split creates exactly one result. Strange.
You may think it strange. But I see simply that there is no bijection
possible between the elements of the set of paths and the set of nodes.
I see only an injection from the set of nodes to the set of paths, and
from that I conclude that there are fewere nodes than paths.
G. Frege
>
> See Robinson's essay 'Formalism 64'. Robinson says that the notion of
> an infinite set is meaningless. But he also endorses working with
> infinite sets in formal theories.
>
(I) Infinite totalities do not exist in any sense of the
word (i.e., either really or ideally). More precisely, any
mention, or purported mention, of infinite totalities is,
literally, meaningless.
(II) Nevertheless, we should continue the business of
mathematics 'as usual', i.e., we should act as if infinite
totalities really existed.
(Abraham Robinson)
But the claim (as stated by Robinson) is certainly wrong. It may be
the case that there are no denotations for certain terms (names)
seemingly refer to, say, infinite sets. B u t certainly those terms
do have senses (i.e. a meaning). (It seems that Robinson wasn't
familiar with Frege's theory and/or terminology concerning sense and
reference, or meaning and denotation, etc.) Otherwise Robinson's own
advice (II) wouldn't make any sense.
muec...@rz.fh-augsburg.de
> You cannot use the fact that every index
> is finite to show that the set of indexes is not potentially infinite.
> Yes, a potentially infinite set is finite,
That is what I say above.
> but a potentially infinite set does not have a cardinality
OK
> nor does it correspond to a natural number.
But it can be many natural numbers.
Regards, WM
muec...@rz.fh-augsburg.de
> On Feb 20, 3:44 am, mueck...@rz.fh-augsburg.de wrote:
>
> > Remember Burali-Forti, Russell, Skolem, Banach and Tarski: They all
> > had a proof that set theory is wrong.
>
> Unless you give a mathematical definition of 'wrong', I don't know
> what you mean by 'a proof that set theory is wrong'.
>
> Burali-Forti showed that there is no greatest ordinal. In Z set
> theories, it is not a contradiction that there is no greatest ordinal.
>
> Russell showed that Frege's system is inconsistent. That does not
> impinge on Z set theories.
>
> Skolem showed that a theory can have a countable model and also a
> theorem that there are sets that are uncountable. But it is not a
> contradiction that a theory has a countable model and also a theorem
> that there are sets that are uncountable.
In any model of ZFC the ZFC axioms are valid. With them there exists
the empty set (usually by axiom). With the empty set there exists the
set omega (by axiom f infinity) and with omega there exists the power
set of omega (by axiom of power set). Why should there exist all these
elements but not a mapping, not even the identity mapping?
>
> Tarkski and Banach showed that ZFC has a certain result that is
> puzzling at first glance. But it requires the axiom of choice; it is
> not a contradiction; and (as far as I understand) would only
> contradict a physical theory if matter had infinite density.
Put a marble in your living room and pray that ZFC may be valid
tonight. Perhaps your prayer is heard and our universe is full of
marbles tomorrow. Then I will believe in ZFC. Otherwise you should
begin to suspect it.
Of course this all would not affect any physical theory. --- And as
infinite density of matter cannot happen in physics it would be
nonsense to have it in a physical theory other than an idealization.
Regards, WM
muec...@rz.fh-augsburg.de
> On Feb 20, 3:59 am, mueck...@rz.fh-augsburg.de wrote:
>
> > On 14 Feb., 03:57, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> > And Robinson is only mentioned to support his authority when quoting
> > that there are no infinite sets at all. The reader can see, that
> > Robinson knows about what he talks.
>
> Robinson's own formalist philosophy includes his claim
his deep insight
> that the notion
> of an infinite set is meaningless.
Regards, WM
muec...@rz.fh-augsburg.de
Yes, because physics is run by phsicists and not by cranks.
Regards, WM
muec...@rz.fh-augsburg.de
That is a necessity when quoting. Cantor's collected works cover 450
pages. I could not post all of them. But in an attempt to obtain your
approval, here are some paragraphs I did not yet quote. (Unreadable
symbols are mostly greek letters.)
=======================A remark by Zeremelo, p. 208 - 209
Zu S. 200. Daß es in jeder Menge (') transfiniter Zahlen immer eine
kleinste gibt, läßt sich folgendermaßen einsehen. Es sei () die
Gesamtheit aller (endlichen und unendlichen) Zahlen , welche kleiner
sind als alle Zahlen '; solche Zahlen muß es geben, z. B. die Zahl 1,
sofern sie nicht selbst zu ' gehört und dann natürlich die kleinste
der Menge ist. Unter den Zahlen gibt es nun entweder eine größte 1,
so daß die unmittelbar folgende '1 nicht zu () gehört, aber ' ist
für jedes ', dann gehört '1 selbst zu (') und ist ihre kleinste.
Oder aber die Zahlen enthalten keine größte, dann besitzen sie (nach
dem zweiten Erzeugungsprinzip) eine "Grenze" ', welche auf alle
zunächst folgt, also wieder jedem ' ist, und diese Zahl ' muß
dann wieder notwendig zu (') gehören und die kleinste aller '
darstellen.
================================= p 3.12
§ 12.
Die wohlgeordneten Mengen.
Unter den einfach geordneten Mengen gebührt den wohlgeordneten Mengen
eine ausgezeichnete Stelle; ihre Ordnungstypen, die wir
"Ordnungszahlen" nennen, bilden das natürliche Material für eine
genaue Definition der höheren transfiniten Kardinalzahlen oder
Mächtigkeiten, eine Definition, die durchaus konform ist derjenigen,
welche uns für die kleinste transfinite Kardinalzahl Alef-null durch
das System aller endlichen Zahlen geliefert worden ist (§ 6).
"Wohlgeordnet" nennen wir eine einfach geordnete Menge F (§ 7), wenn
ihre Elemente f von einem niedersten f1 an in bestimmter Sukzession
aufsteigen, so daß folgende zwei Bedingungen erfüllt sind:
I. "Es gibt in F ein dem Range nach niederstes Element f1".
II. "Ist F' irgendeine Teilmenge von F und besitzt F ein oder mehrere
Elemente höheren Ranges als alle Elemente von F', so existiert ein
Element f' von F, welches auf die Gesamtheit F' zunächst folgt, so daß
keine Elemente in F vorkommen, die ihrem Range nach zwischen F' und f'
fallen 1"[16].
======================================== p. 313
A. "Jede Teilmenge F1 einer wohlgeordneten Menge F hat ein niederstes
Element."
Beweis. Gehört das niederste Elemente f1 von F zu F1, so ist es
zugleich das niederste Elemente von F1.
Andernfalls sei F' die Gesamtheit aller Elemente von F, welche
niederen Rang haben als alle Elemente von F1, so ist eben deshalb kein
Element von F zwischen F' und F1 gelegen.
Folgt also f' (nach II) zunächst auf F', so gehört es notwendig zu F1
und nimmt hier den niedersten Rang ein.
B. "Ist eine einfach geordnete Menge F so beschaffen, daß sowohl F
wie auch jede ihrer Teilmengen ein niederstes Element haben, so ist F
eine wohlgeordnete Menge."
Beweis. Da F ein niederstes Element hat, so ist die Bedingung I
erfüllt.
Sei F' eine Teilmenge von F derart, daß es in F ein oder mehrere
Elemente F' gibt; F1 sei die Gesamtheit aller dieser Elemente und f'
das niederste Element von F1, so ist offenbar f' das auf F' zunächst
folgende Element von F. Somit ist auch die Bedingung II erfüllt und es
ist daher F eine wohlgeordnete Menge.
C. "Jede Teilmenge F' einer wohlgeordneten Menge F ist gleichfalls
eine wohlgeordnete Menge."
Beweis. Nach Satz A hat F' sowohl wie jede Teilmenge F'' von F' (da
sie zugleich Teilmenge von F ist) ein niederstes Element; daher ist
nach Satz B F' eine wohlgeordnete Menge.
========================================= p. 331
Sei J irgendeine Teilmenge von {} derart, daß es Zahlen in {} gibt,
die größer sind als alle Zahlen von J. Sei etwa 0 eine dieser Zahlen.
Dann ist auch J eine Teilmenge von A0+1, und zwar eine solche, daß
mindestens die Zahl 0 von A0+1 größer ist als alle Zahlen von J.
DaA0+1 eine wohlgeordnete Menge ist, so muß (§ 12) eine Zahl ' von
A0+1, die daher auch eine Zahl von {} ist, auf alle Zahlen von J
zunächst folgen.
===========================================
I'll keep for myself what I think about
> Mückenheim's style of quoting,
Taken a course in good behaviour and self-disciplin?
> of which this is another example. Just this:
> I don't believe in the favourable review of the first 8 chapters of his
> book.
Buy it, read it. In contrast to set theory there is no necessity to
believe in anything with my book.
Regards, WM
muec...@rz.fh-augsburg.de
> I don't really care because Muckenheim's potential finitism is from
> the last century ago already.
That would resemble set theory. No, it has been approved by millenia.
> It is not that there's something wrong
> with that,
Correct!
>only that denial doesn't lead to knowledge.
Not to putative knowledge.
> I don't
> disapprove of his discussion of potential infinity vis-a-vis the
> finite, yet would find him hard-pressed to deny infinity in the
> system.
The social system of contemporary set theorists? Yes.
But that doesn't matter.
Regards, WM
muec...@rz.fh-augsburg.de
> In article <1171971859.370048.181...@l53g2000cwa.googlegroups.com>,
That, admittedly, is new. It is the binary tree to which you, as far
as I remember, ascribed different sets of paths according to how one
is looking at it. This is obviously a contradiction in every set
theory.
> The calculus of Leibitz and Newton, being based on undefined
> infinitesimals, was "wrong" in a way, too, but that has since been fixed.
>
> So why does WM continue to claim ( without proof, or even evidence) that
> whatever was once "wrong" with set theory is incapable of being fixed?
Everything will be fixed with no doubt. You fixed it by stating that a
unique set of nodes in a tree with unique structure yields different
sets of paths. Others will fix it in their way, perhaps with some more
conjuring ticks like unaccessible cardinals and hyper unaccessible
ordinals to avoid the immediate impression of naiveté.
Regards, WM
muec...@rz.fh-augsburg.de
In number ten the author attempts to show that the first proof by
Cantor
of the uncountability is invalid. That proof is given on page 77
of the
book, but let me recap. Given a list of real numbers, it is
possible
to find for any arbitrary interval (a, b) a number in that
interval that
is not in the list. The proof is by a sequence of nested
intervals in
(a, b), where the first is found by the first two numbers in the
list
that are larger than a and smaller than b. We replace a by the
lower
number and b by the larger number and repeat the process, Going
further
through the list. This way a list of nested intervals is found
that is
either finite, or is infinite. If infinite, the limits of the
lower
bounds and the upper bounds can be equal or not. If equal, that
limit
is the required number, if not equal, each number in the interval
given
by the limits can be used. If the list of intervals is finite, we
take
the last one. That one can contain at most one more number from
the
list, so we can take every number from that interval, except the
additional number, if it is in there. In this section it is clear
that the author does not understand the basics about how that
shows
that the reals are not countable.
WM:
How could this simple stuff be not understandable? Of course it is
easy ti understand, but it seems not so easy to understand my critics,
Because the proof above is valid
for *every* list of reals, a list of reals does not exist (every
list
is doomed to be incomplete), and so the reals are not countable.
The proof must be valid for every list of reals. Yes. Therefore it
must also work for a very artificial list.
Now the author tries to use the proof to show that even countable
sets are uncountable. He starts with the sequence of rationals:
-1, 1/2, -1/3, 1/4, ...
and concludes from that that the (single) limit is 0, not in the
set
used, and attempts to show by this that the set of numbers in the
sequence with 0 added is uncountable. The author fails to see one
important difference: in the set of reals, each open interval
contains
reals, in the set of rationals from the sequence above that is not
true. But that *is* necessary to make it work.
Each open interval of rational contains rationals. They can be treated
like the sequence above by a simplke coordinate transformation.
But what I said in the book is that from th result of Cantor's first
proof alone one cannot distinguish between the cardinal numbers of
transcendental and rational numbers.
In eleven the author attempts to do the same with the second
proof.
(Although the author disagrees with me, the second proof is
actually
not about the reals, but about sequences of binary digits without
interpretation, somebody has later modified it to a proof about
the
reals. Was that Zermelo (see below under twenty)? In the
original
version it proves the theorem that there are uncountable sets.)
But
in this case it is in the modified, decimal, version (who came
first
with the decimal version?). Here he does not even show a
contradiction,
or whatever, but only asks how it is possible that a complete list
indeed *does* exist, and asks whether that is the right way to
look at
the infinite. But that is not a mathematical but a philosophical
question. Here he shows only a contradiction with his
sensibilities
(remember Bolzano's musings in chapter 6?).
In twelve he attempts to find a contradiction, and builds a list
(ternary I think),
Doesn't matter. The idea was decimal, but the digits 3 to 9 are not
used.
where in the building of the diagonal every 0 and 1
is replaced by a 2, and concludes that the diagonal found at every
step
is in the list. Strange that he sees that as a contradiction,
because
in the building of the diagonal *each* digit has to be changed.
But it cannot be done other than step by step.
I hesitate to name the next one thirteen, because it is strongly
connected with the previous, but the authors now gets some quite
different conclusions. This example is also repeatedly mentioned
on
sci.math, it is about the list 0.000..., 0.1000..., 0.11000...,
0.11000..., etc. The 0 in the diagonal is replaced by a 1 and he
shows
that the diagonal as built up to the n-th row is equal to the (n
+1)-th
row. As he rightly states, the limit (when seen as decimal: 1/9)
is not
in the list. The proper conclusion from this is that the list is
incomplete. However, the author comes to a quite different
conclusion,
for this purpose he constructs a new list, where the first 0
(everything
is happening after the decimal point) is replaced by 1. He states
that
the diagonal of that new list does not exist because is should
have
aleph-0 1's, and that would require that there is a list element
with
aleph-0 1's. He tries to show that with finite segments. But
this
fails, obviously, as the (complete) diagonal is constructed with
limits
in mind.
There are limits in set theory?
Using the tools of set theory we can define a bijection. An existing
diagonal with X elements guarantees the existence of as many lines and
columns.
Fourteen is also closely connected. Obviously the author is
thinking
that the creation of the diagonal is an iterative process, which
is
*not* the case.
Which *is* the case. You cannot start at the end because it is not
there. So you cannot start with anaction including the end. So you can
only start at the first or at least a finite position n.
Given a list, you can refer to the n-th digit of the
created diagonal without ever referring to any other digit of the
created diagonal.
Yes, but with the n-th digit you have not all of them. And you cannot
start at he non existing end. So you must start at a finite oposition.
The creation of the digits are independent of each
other. This is probably because the author has not read much
beyond
Cantor, who indeed saw everything as sequential processes.
Of course the natural numbers are a sequence. In Cantor's case, the
sequential construction leads to contradictions. Some advocates of set
theory have recognized this and, therefore, postulated that everything
happens at once, but that does not mean that this position is correct
or sensible. One cannot do infinitely many transactions at once.
So the
author is thinking that if a sequential process (creating the
diagonal)
can be done in one step, so can every sequential process. A deep
misunderstanding.
LOL.
If operations do not depend on each other, you can
do them all at once, if they do depend on each other you can not.
And the existence of the number digit n+1 depends on the existence of
the digit number n.
Regards, WM
William Hughes
> On 20 Feb., 17:59, "William Hughes" <wpihug...@hotmail.com> wrote:
>
> > You cannot use the fact that every index
> > is finite to show that the set of indexes is not potentially infinite.
> > Yes, a potentially infinite set is finite,
>
> That is what I say above.
And this is only true using the Wolkenmueckenheim concept of
finite. Outside of Wolkenmuekenheim, only sets with
a fixed maximum are called finite.
The fact that you call a potentially infinite set
finite does not change its properties. It does
not give it properties that sets with a fixed maximum have.
>
> > but a potentially infinite set does not have a cardinality
>
> OK
>
> > nor does it correspond to a natural number.
>
> But it can be many natural numbers.
>
This is statement ii.
Statement i (all natural numbers have
property X) and statement ii (a potentiall
infinite set contains only natural numbers) do
not imply property iii (a potentially infinite
set has property X).
- William Hughes
Aatu Koskensilta
> See Robinson's essay 'Formalism 64'. Robinson says that the notion of
> an infinite set is meaningless. But he also endorses working with
> infinite sets in formal theories.
Right, and as an antidote to Robinson's (and Cohen's) "formalism" one
could well use Kreisel's _Observations on Popular Discussions of
Foundations_.
--
Aatu Koskensilta (aatu.kos...@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Dik T. Winter
Stating that the notion of an infinite set is meaningless is not stating
that there are no infinite sets at all.
G. Frege
wrote:
>
> Stating that the notion of an infinite set is meaningless is not stating
> that there are no infinite sets at all.
>
"(I) Infinite totalities do not exist in any sense of the
word (i.e., either really or ideally). More precisely, any
mention, or purported mention, of infinite totalities is,
literally, meaningless.
(II) Nevertheless, we should continue the business of
mathematics 'as usual', i.e., we should act as if infinite
totalities really existed."
(Abraham Robinson)
Though certainly the the notion of /infinite sets/ is NOT meaningless.
David Marcus
And, mathematics is run by mathematicians, and not by cranks.
--
David Marcus
David Marcus
> REPLY TO THE SECRET REVIEW OF CHAPTER 9 (PART 3 of the Reply)
This is rather strange, even for WM: did he really think that Dik would
think his book was correct? I guess narcissism is very strong.
--
David Marcus
Carsten Schultz
It doesn't matter much to him. The review gives WM an opportunity to
repeat all of his usual boring fallacies.
Note: Putting something in a book does not make it correct.
Best,
Carsten
--
Carsten Schultz (2:38, 33:47)
http://carsten.codimi.de/
PGP/GPG key on the pgp.net key servers,
fingerprint on my home page.
Virgil
muec...@rz.fh-augsburg.de wrote:
Singular is plural? No wonder WM stays so confused.
Virgil
muec...@rz.fh-augsburg.de wrote:
> In any model of ZFC the ZFC axioms are valid. With them there exists
> the empty set (usually by axiom). With the empty set there exists the
> set omega (by axiom f infinity) and with omega there exists the power
> set of omega (by axiom of power set). Why should there exist all these
> elements but not a mapping, not even the identity mapping?
Who says there aren't any mappings?
> >
> > Tarkski and Banach showed that ZFC has a certain result that is
> > puzzling at first glance. But it requires the axiom of choice; it is
> > not a contradiction; and (as far as I understand) would only
> > contradict a physical theory if matter had infinite density.
>
> Put a marble in your living room and pray that ZFC may be valid
> tonight. Perhaps your prayer is heard and our universe is full of
> marbles tomorrow. Then I will believe in ZFC. Otherwise you should
> begin to suspect it.
WM seems to have lost his marbles. When marbles are made of up only of
points, as spheres are, only then can they, a la banach-Tarski, breed.
That WM does not yet understand the difference between physical marbles
and geometric spheres is a measure of his ignorance.
Virgil
muec...@rz.fh-augsburg.de wrote:
But Robinson imbues infinite sets with his own meanings anyway.
Virgil
muec...@rz.fh-augsburg.de wrote:
Actually, physics is not run by "phsicists" [sic], but largely by
physicists, although they rely heavily on the work of mathematicians.
Virgil
muec...@rz.fh-augsburg.de wrote:
> > I don't believe in the favourable review of the first 8 chapters of his
> > book.
>
>
> Buy it, read it. In contrast to set theory there is no necessity to
> believe in anything with my book.
>
> Regards, WM
In the first 8 chapters, there is apparently nothing of WM, and in the
ramaining chapters there is apparently nothing of mathematics, so why
would anyone want to buy it.
Virgil
muec...@rz.fh-augsburg.de wrote:
> On 20 Feb., 22:18, Virgil <vir...@comcast.net> wrote:
> > In article <1171971859.370048.181...@l53g2000cwa.googlegroups.com>,
> >
> > > Remember Burali-Forti, Russell, Skolem, Banach and Tarski: They all
> > > had a proof that set theory is wrong.
> >
> > They may all have had proofs that some particular form of set theory was
> > 'wrong', but where did any of them ever have a proof that all forms of
> > set theory must be wrong?
>
> That, admittedly, is new.
It is so new that it does not even exist.
> It is the binary tree to which you, as far
> as I remember, ascribed different sets of paths according to how one
> is looking at it. This is obviously a contradiction in every set
> theory.
It was not one binary tree, but a variety of differently defined binary
trees that had different properties in accord with the differences in
their definitions.
>
> > The calculus of Leibitz and Newton, being based on undefined
> > infinitesimals, was "wrong" in a way, too, but that has since been fixed.
> >
> > So why does WM continue to claim ( without proof, or even evidence) that
> > whatever was once "wrong" with set theory is incapable of being fixed?
>
> Everything will be fixed with no doubt. You fixed it by stating that a
> unique set of nodes in a tree with unique structure yields different
> sets of paths.
I said that a binary tree can as correctly be defined by its set of
paths as by its set of nodes.
> Others will fix it in their way, perhaps with some more
> conjuring ticks like unaccessible cardinals and hyper unaccessible
> ordinals to avoid the immediate impression of naiveté.
WM's conjuring tricks, however, are not sufficient to destroy any part
of mathematics, including mathematical set theories, since WM is
incapable of providing any of the essential proofs of his claims.
Virgil
David Marcus <David...@alumdotmit.edu> wrote:
And what is so "secret" about review whose URL has been published?
MoeBlee
> On 20 Feb., 20:09, "MoeBlee" <jazzm...@hotmail.com> wrote:
> >
> > On Feb 20, 3:44 am, mueck...@rz.fh-augsburg.de wrote:
>
> > > Remember Burali-Forti, Russell, Skolem, Banach and Tarski: They all
> > > had a proof that set theory is wrong.
>
> > Unless you give a mathematical definition of 'wrong', I don't know
> > what you mean by 'a proof that set theory is wrong'.
>
> > Burali-Forti showed that there is no greatest ordinal. In Z set
> > theories, it is not a contradiction that there is no greatest ordinal.
>
> > Russell showed that Frege's system is inconsistent. That does not
> > impinge on Z set theories.
>
> > Skolem showed that a theory can have a countable model and also a
> > theorem that there are sets that are uncountable. But it is not a
> > contradiction that a theory has a countable model and also a theorem
> > that there are sets that are uncountable.
>
> In any model of ZFC the ZFC axioms are valid.
Some writers do use the word 'valid' in the sense of 'true'. But it
helps to keep the distinction. In any model of ZFC, the ZFC axioms are
true. A formula is valid iff it is true in every model for the
language.
> With them there exists
> the empty set (usually by axiom).
With what? The models are not objects of the theory but rather of the
meta-theory (well, there are inner models too, but that's a bit more
(or less?) complicated). Also, I already told you that with the schema
of separation as as an axiom schema or theorem schema, we don't need
an empty set axiom.
> With the empty set there exists the
> set omega (by axiom f infinity) and with omega there exists the power
> set of omega (by axiom of power set). Why should there exist all these
> elements but not a mapping, not even the identity mapping?
Are you talking about Skolem's paradox now? First, what identity
mapping are you referring to? There always exists a bijection of a set
onto itself. As to other mappings, they exist or they don't as
provided by the axioms of the theory. And we have to be careful to be
specific as to what requirements we place as to in WHICH sets a
mapping does or does not exist. If you (generic 'you') just blur such
important distinctions, then you get crank claims of contradictions
that are not contradictions.
> > Tarkski and Banach showed that ZFC has a certain result that is
> > puzzling at first glance. But it requires the axiom of choice; it is
> > not a contradiction; and (as far as I understand) would only
> > contradict a physical theory if matter had infinite density.
>
> Put a marble in your living room and pray that ZFC may be valid
> tonight. Perhaps your prayer is heard and our universe is full of
> marbles tomorrow. Then I will believe in ZFC. Otherwise you should
> begin to suspect it.
ZFC is not a theory of physics. I don't know much about the Tarski-
Banach theorem, but I have not read that it entails anything contrary
to physics. As I understand, the theorem does not render that a
PHYSICAL object may be perform in the way the ABSTRACT object in the
theorem performs.
Also, I notice that you did not respond to my other points in reply to
you. And I notice that is a habit you have of brushing off from among
the more obviously solid points of rebuttal to you. Of course, I
understand that no one can be expected to answer every line of every
post. But I do note that I've written quite a bit to you, with a lot
of tissue of explanation, yet you demur from following up on those
substantive matters as, what seems to me, to be a way of avoiding an
engagement of ideas that would lead you into considerations beyond
your program of denunciations of set theory. My point is not to go
back over all those posts, but rather to suggest that we discuss these
points for all they are worth and not just as fodder for your
continued harrumping that set theory is "wrong".
> Of course this all would not affect any physical theory. --- And as
> infinite density of matter cannot happen in physics it would be
> nonsense to have it in a physical theory other than an idealization.
I was told by another poster that my use of 'density' is not correct
here. Nevertheless, I see that we agree on this point (at least to the
extent that it is a given that matter does not have whatever property
it is allows the Tarski-Banach behavior).
The point you keep missing is that set theory does NEED to be intended
as a theory in which we can read off each of its sentences as a
statement about physical objects.
MoeBlee
MoeBlee
Notice how you snipped what else I reported - which is that Robinson
still endorses infinitistic set theory and that he uses the full array
of mathematical logic, including infinite sets, uncountable sets, and
the axiom of choice.
Why don't you think about Robinson's mathematics and philosophy a bit
more so that you won't be so inclined to miss and snip the TOTALITY of
what he's saying rather than use him (as you've used other
mathematicians) as mere fodder for quotes (taken out of context and
misunderstood by you) for your continual harrumping that set theory is
"wrong".
Such distortions of context and misrpresentation of what the writers
really are saying is indicative of someone whose interest is not in
really understand the subject, even as a controversy, but rather in
just demanding that he is right and all his interlocuters are wrong,
and motivated to do so by his conceptual inability (or at least
disinclination turned to stone) even to understand the basics of the
subject. That is, to be a crank.
MoeBlee
David Marcus
http://en.wikipedia.org/wiki/Banach-Tarski_Paradox
The first paragraph of the Wikipedia article is a good summary.
--
David Marcus
MoeBlee
> Of course the natural numbers are a sequence. In Cantor's case, the
> sequential construction leads to contradictions. Some advocates of set
> theory have recognized this and, therefore, postulated that everything
> happens at once, but that does not mean that this position is correct
> or sensible.
It would be better to understand that IF the mathematics is described
with a temporal ANALOGY, then the best fit is that "it all happens at
once". But we are NOT obligated to regarding mathematics as being
describable with a temproral analogy. Any state of affairs that is a
model of set theory is an ABSTRACT state of affairs in which time is
not even expressed as a feature. So surely we do NOT have to describe
the state of affairs as a chronology, not even as a singular
"happening all at once". Set theory does axiomatize the mathematics
that is used for theories in which time is a feature, but that does
not entail that a model of set theory is itself or must be itself a
state of affairs in which time is a feature.
> One cannot do infinitely many transactions at once.
And set theory is not a theory about what transactions a finite
intelligence can perform. However, set theory does axiomatize the
theorems about computability which can be taken as about what
transactions a finite intelligence and/or also a computing machine can
perform. And meanwhile, you are welcome to advance a mathematical
theory that limits itself to such computability.
The question for you is: Are you interested in understanding that or
are you vastly more interested in continuing to harrump that set
theory is "wrong"?
MoeBlee
MoeBlee
wrote:
> On 2007-02-21, MoeBlee <jazzm...@hotmail.com> wrote:
>
> > See Robinson's essay 'Formalism 64'. Robinson says that the notion of
> > an infinite set is meaningless. But he also endorses working with
> > infinite sets in formal theories.
>
> Right, and as an antidote to Robinson's (and Cohen's) "formalism" one
> could well use Kreisel's _Observations on Popular Discussions of
> Foundations_.
I know. I think that is the essay of Kreisel's that I read. I had
difficulty following his erudite polemics. I will consult it again at
various junctures as I learn more the subject.
MoeBlee
MoeBlee
> In article <1172023295.358280.62...@p10g2000cwp.googlegroups.com> "MoeBlee" <jazzm...@hotmail.com> writes:
> > On Feb 20, 5:46 pm, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> > > In article <1171972752.151611.207...@q2g2000cwa.googlegroups.com> mueck...@rz.fh-augsburg.de writes:
> >
> > > > And Robinson is only mentioned to support his authority when quoting
> > > > that there are no infinite sets at all. The reader can see, that
> > > > Robinson knows about what he talks.
> > >
> > > What quote of Robinson are you talking about? I do not find any such
> > > quote. Pray state page number in your book.
> >
> > See Robinson's essay 'Formalism 64'. Robinson says that the notion of
> > an infinite set is meaningless. But he also endorses working with
> > infinite sets in formal theories.
>
> Stating that the notion of an infinite set is meaningless is not stating
> that there are no infinite sets at all.
On that point, vis-a-vis Robinson's own views, I would defer to
whatever he said about that as the best representation of his view. I
would think that (from what I recall in the essay) he would
distinguish between existence literally and existence formally, though
I have no cause to insist that that is the best rendering of his
explanation.
MoeBlee
MoeBlee
> Notice how you snipped what else I reported - which is that Robinson
> still endorses infinitistic set theory
To be more clear, I should say, he doesn't endores *platonistic*
infinitisitc notions, so perhaps 'infinitistic set theory' is not a
good choice of words. In any case, he does endorse and works in set
theory with infinite sets.
MoeBlee
MoeBlee
> http://en.wikipedia.org/wiki/Banach-Tarski_Paradox
>
> The first paragraph of the Wikipedia article is a good summary.
>From that paragraph: "Banach and Tarski intended this proof to be
evidence in favor of rejecting the axiom of choice [...]"
Source?
MoeBlee
MoeBlee
Formatting of that came out wrong. Should be:
Ross A. Finlayson
Yeah, where's the truth?
You know, the truth, that which is not false.
Got any of that?
Ross
--
Ross Finlayson
Finlayson Consulting
Ross A. Finlayson
Some abstract says Jourdain proved AC is a theorem. Actually, it just
implies that.
Not having read the article, I feel free to make statements about it
having read the abstract.
Ross
--
Finlayson Consulting
Dik T. Winter
> REPLY TO THE SECRET REVIEW OF CHAPTER 9 (PART 3 of the Reply)
What is so secret about it?
[ About Cantor's first uncountability proof.]
> Because the proof above is valid
> for *every* list of reals, a list of reals does not exist (every list
> is doomed to be incomplete), and so the reals are not countable.
>
> The proof must be valid for every list of reals. Yes. Therefore it
> must also work for a very artificial list.
>
> Now the author tries to use the proof to show that even countable
> sets are uncountable. He starts with the sequence of rationals:
> -1, 1/2, -1/3, 1/4, ...
> and concludes from that that the (single) limit is 0, not in the set
> used, and attempts to show by this that the set of numbers in the
> sequence with 0 added is uncountable. The author fails to see one
> important difference: in the set of reals, each open interval contains
> reals, in the set of rationals from the sequence above that is not
> true. But that *is* necessary to make it work.
>
> Each open interval of rational contains rationals. They can be treated
> like the sequence above by a simplke coordinate transformation.
> But what I said in the book is that from th result of Cantor's first
> proof alone one cannot distinguish between the cardinal numbers of
> transcendental and rational numbers.
But you *did* use the above reasoning to show that even countable sets
could be proven to be uncountable. But that you can not distingsuish
this way the cardinal numbers of the transcendental and rational numbers
is correct (that is why I did not comment on that), but is no problem at
all.
> In twelve he attempts to find a contradiction, and builds a list
> (ternary I think),
>
> Doesn't matter. The idea was decimal, but the digits 3 to 9 are not
> used.
>
> where in the building of the diagonal every 0 and 1
> is replaced by a 2, and concludes that the diagonal found at every step
> is in the list. Strange that he sees that as a contradiction, because
> in the building of the diagonal *each* digit has to be changed.
>
> But it cannot be done other than step by step.
That is what you state. But as each replacement is independent of all
other replacements, it can be done in parallel. No step by step needed.
[ About the infinite triangle.]
> Using the tools of set theory we can define a bijection. An existing
> diagonal with X elements guarantees the existence of as many lines and
> columns.
Yes, and there are. Both are aleph-0, and neither does have an aleph-0-th
element.
> Fourteen is also closely connected. Obviously the author is thinking
> that the creation of the diagonal is an iterative process, which is
> *not* the case.
>
> Which *is* the case. You cannot start at the end because it is not
> there. So you cannot start with anaction including the end. So you can
> only start at the first or at least a finite position n.
All replacements can be done in parallel.
> Given a list, you can refer to the n-th digit of the
> created diagonal without ever referring to any other digit of the
> created diagonal.
>
> Yes, but with the n-th digit you have not all of them. And you cannot
> start at he non existing end. So you must start at a finite oposition.
You need not start anywhere, you can start with all of them at once.
> The creation of the digits are independent of each
> other. This is probably because the author has not read much beyond
> Cantor, who indeed saw everything as sequential processes.
>
> Of course the natural numbers are a sequence. In Cantor's case, the
> sequential construction leads to contradictions.
That is what you think but can not prove.
> Some advocates of set
> theory have recognized this and, therefore, postulated that everything
> happens at once, but that does not mean that this position is correct
> or sensible. One cannot do infinitely many transactions at once.
As time and size plays no role in set theory, this makes not much sense.
> So the
> author is thinking that if a sequential process (creating the diagonal)
> can be done in one step, so can every sequential process. A deep
> misunderstanding.
>
> LOL.
Oh.
> If operations do not depend on each other, you can
> do them all at once, if they do depend on each other you can not.
>
> And the existence of the number digit n+1 depends on the existence of
> the digit number n.
Wrong. The existence of the n+1-st digit depends only on the n+1-st number
in the list which is guaranteed by the mapping given.
Jesse F. Hughes
> Put a marble in your living room and pray that ZFC may be valid
> tonight. Perhaps your prayer is heard and our universe is full of
> marbles tomorrow. Then I will believe in ZFC. Otherwise you should
> begin to suspect it.
Wow! What a clear and concise proposal! Keen.
I will let you know my findings in the morn, I tell you what!
But I'm a bit unclear on one little point. I'm sure it's not
important, but still.... How does ZFC prove that prayer will result in
an overflowing universe of marbles again? You were a bit sketchy on
that li'l detail.
--
Jesse F. Hughes
"Doesn't pay to lie if you aren't good at it."
-- Captain Friday, /City of the Dead/
Adventures by Morse radio show
Dik T. Winter
> In article <1172058191.6...@v45g2000cwv.googlegroups.com> muec...@rz.fh-augsburg.de writes:
> > REPLY TO THE SECRET REVIEW OF CHAPTER 9 (PART 3 of the Reply)
>
> What is so secret about it?
Wolfgang apparently does not answer this. Should we talk about the
secret papers by Wolfgang?
muec...@rz.fh-augsburg.de
> >
> > > And I think that the remark that Euclid used a proof by
> > > contradiction is wrong
> >
> > ??? It is said (I didn't read the original text) that Euclid assumed
> > that n primes exist and then he contradicted this assumption by
> > proving the existence of prime number n+1. Many authors call this a
> > proof by contradiction.
>
> It would be a proof by contradiction, if there is an initial assumption
> that is falsified by the reasoning. But see below.
Whether this is the case depends on the person. Euclid is said to have
assumed that.
>
> > > (actually the theorem and proof are quite
> > > Cantoresque):
> >
> > Yes. Cantor assumed a complete list of reals and showed the existence
> > of another real. Many authors call this a proof by contradiction.
>
> Wrong. Cantor does not assume that the given list is complete. He shows
> that given *any* list of reals there is a real not on the list. There is
> *no* initial assumption of completeness.
So Cantor constructs one or the other list without attempting to
include all the reals? Then he finds another real and takes it as
evidence for what? That he personally is not able or was too lazy to
construct a complete list?
> >
> > That however is not the wording used by Euclid.
>
> What is the wording used by Euclid? (I may note that the Corollary was indeed
> *not* by Euclid, but that is the modern corollary.)
Euclid is reported to have said (of course in Greek and slightly
differing in the thousands of copies of his Elements made by hand
before the 1500 printed editionsappeared): "There are more prime
numbers than any given set of primes contains" or "in any given set
of primes, there is prime number missing".
This is of course correct and can be generalized to natural numbers:
For any given set of natural numbers, there is a natural number not in
that set.
> > I think I told you already: 3/7 is 0.3 in base 7
>
> But in my opinion 0.3 in base 7 is still a formula telling how the number
> is calculated. It tells me that it is 3 * 1/7. And before Simon Stevin's
> "De Thiende" from 1585, decimal numbers (and base 'n' notated rationals)
> did not actually exist.
Once upon a time there were only unary numbers existing. Everything
else is abbreviation which has to be introduced by someone.
But as Simon Stevin writes (parafrased):
> "The importance of the proposal is not so much mathematical profoundness,
> but more broad applicability."
> And it was merely used as an aid to ease calculations.
Yes. Also decimal representation is only to ease calculation (and
recognition - if that is different).
>
> > I think I told you already: 3/7 is 0.3 in base 7 and this can be
> > compared with any existing nunmber n/B (in natural base B) by 7n/7B <
> > 3B/7B or 7n/7B = 3B/7B or 7n/7B > 3B/7B.
>
> So for the comparison you do not use "0.3" but the original "3/7"?
> Well, I can compare sqrt(2) with any rational number. Square the two.
How can a number be squared which does not exist? First it must exist.
Then it can be handled.
>
> > 3B/7B or 7n/7B = 3B/7B or 7n/7B > 3B/7B. So we need only find the
> > natural numbers 7n and 3B which exist for every natural numbers n and
> > B we can use. (In case all bits of our bit reservoir were required to
> > establish n and B, then the reservoir of bits must be extended by 7 or
> > 3 which should be possible in all practicable cases.)
>
> For squaring not much more is needed.
Except the number to be squared.
> What quote of Robinson are you talking about? I do not find any such
> quote. Pray state page number in your book.
quote. Pray state page number in your book.
Robinson says on page 110 (last lines) "Infinite totalities do not
exist in any sense of the word (i.e., either really or ideally). More
precisely, any mention, or purported mention, of infinite totalities
is, literally, meaningless" [ROB64].
By the way: There is a comprehensive index at the end of my book,
including the data of about 170 scholars and the page numbers where
they are mentioned.
>
> Well, when I did learn geometry the reality was that you could actually
> not state that a longer distance had more points.
No, you cannot, unless you believe that all real numbers do actually
exist. Then you must think that the points are all there.
> >
> > Better: He fails to fail to see that it gives trichotomy of sets.
>
> Pray elaborate, I do not understand this.
Bolzano was correct when he said that set inclusion gives trichotomy.
A long line has more points than a short line, notwithstanding
bijection. (Of course, all this is only correct, if reals actually
exist.)
>
> And you are stating here that a well-ordering can not be defined in ZFC.
Yes, not without additional axioms. Easiest by the axiom that a well
order has been defined.
> So, what is it. Can you show the proof that a well-ordering can not be
> defined? Or at least a source for such a proof? I state that ZFC is
> *not* inconsistent with a definable well-ordering of R.
It has been proven by forcing, AFAIK, but being in vacations, I have
no access to literarture presently.
>
> > Further, all your arguing is in vain unless you can present a well
> > ordering working in reality. And you cannot.
>
> What reality? Did you read the reference I presented?
I would like to read the well-ordering.
>
> > > But that is wrong, because when a map 'f' is viewed as a set, it is
> > > defined as:
> > > { (n, s) | n in N, s in P(N), f(n) =3D s }
> > > So when 'f' can not be defined in a model, neither can the set
> > > representing 'f'.
> >
> > When P(N) exists in the model,
>
> Of course it exists if it is a model of ZF: the axiom of the powerset.
>
> > then at least a function f from P(N)
> > to ... exists there.
>
> To what?
The identity mapping of P(N) exists when P(N) exist. This is including
the mapping from the singletons of N to N to P(N).
>
> > It is the similar to the nodes and edges of the
> > tree. Every existing node is the end of an edge. There are not less
> > edges than nodes. If all elements of P(N) exist, then all elements of
> > f exist. And the existence of P(N) is guaranteed by the power set
> > axiom.
>
> No argument here, it does not contradict what I stated.
So you think that a set may exist but no identity mapping? I agree.
There is no mapping from N or R. But most set theorists deny that (In
fact, I never met a set theorist who missed the identiy mapping R -->
R in real mathematics. How could it get lost in some model which obeys
the same set of axioms?)
>
>
> >
> > > He further states:
> > > "Moreover, as by the axiom of the power set P(N) contains each
> > > subset of N. So which subset fails to be in P(N)?"
> > > The answer is simply: those that can not be defined in the model.
> > > The model has serious influences on the properties that can be used
> > > in the axiom of subsets, and so also on the power set as defined by
> > > the axiom of the power set.
> >
> > No. The existence of N and of the power set axiom guarantee the
> > existence of P(N) and of at least a mapping
> > (P(N))-->N. Your explanation is ungereimt, as Bolzano would have
> > said. And Skolem himself did not believe in his own explanations, as
> > his attitude towards set theory clearly shows.
>
> Sorry. (Going back to the ZF definitions), loosely speaking, the
> axiom of the power set states that there is a set that contains all
> subsets of a given set. The axiom of specification states what are
> subsets and connects them with properties phi that can be possessed by
> elements. The model states what properties are valid properties in
> the model. So whether something is a subset or not of a particular
> set depends on the model. So when looking from outside the model you
> may find a subset that is not a subset in the model.
> --
> dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
> home: bovenover 215, 1025 jn amsterdam, nederland;http://www.cwi.nl/~dik/
23. Dik T. Winter Profil anzeigen
Weitere Optionen 21 Feb., 02:46
Newsgroups: sci.math
Von: "Dik T. Winter" <Dik.Win...@cwi.nl>
Datum: Wed, 21 Feb 2007 01:46:30 GMT
Lokal: Mi 21 Feb. 2007 02:46
Betreff: Re: Review of Mueckenheims book.
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In article <1171972752.151611.207...@q2g2000cwa.googlegroups.com>
mueck...@rz.fh-augsburg.de writes:
> On 14 Feb., 03:57, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> > Chapters 1 to 8:
> >
> > These chapters provide an exceptional well written showing about
how
> > things developed through the ages.
>
> Thank you very much, Dik. I will use this as a cover text for the
> second or third edition.
>
> > And I think that the remark that Euclid used a
proof by
> > contradiction is wrong
>
> ??? It is said (I didn't read the original text) that Euclid
assumed
> that n primes exist and then he contradicted this assumption by
> proving the existence of prime number n+1. Many authors call this
a
> proof by contradiction.
It would be a proof by contradiction, if there is an initial
assumption
that is falsified by the reasoning. But see below.
> > (actually the theorem and proof are quite
> > Cantoresque):
>
> Yes. Cantor assumed a complete list of reals and showed the
existence
> of another real. Many authors call this a proof by contradiction.
Wrong. Cantor does not assume that the given list is complete. He
shows
that given *any* list of reals there is a real not on the list. There
is
*no* initial assumption of completeness.
So Cantor constructs one or the other list without atempting to
include all the reals? Then he finds another real and takes it a
sevidence for what? That he personally is not able or was too lazy to
construct a complete list?
> > THEOREM
> > Given any collection of primes, there is a prime that is
not in
> > that collection (at that time collections were still
finite).
> > PROOF
> > The well-known construction, that leads to either a new
prime or
> > a natural number that is the product of two or more
primes not in
> > the collection.
> > COROLLARY (current)
> > There are infinitely many primes.
>
> That however is not the wording used by Euclid.
What is the wording used by Euclid? (I may not that the Corollary was
indeed
*not* by Euclid, but that is the modern corollary.)
Euclid is reported to have said (of course in Greek and slightly
differing in the thousands of copies of his Elements made by hand
before the 1500 printed editions): "There are more prime numbers than
any given set of primes contains" or "for any given set of primes,
there is prim number missing". This is of course correct and can be
generalized to natural numbers: For any given set of natural numbers,
there is a natural number not in that set.
> >
However,
> > the author apparently takes the position that the symbol
> > sequence 'sqrt(2)' does not prove the existence of that
number,
> > because it is nothing more than the question to find the
number
> > whose square is 2. With this following Cantor.
>
> Who was correct in this point.
Maybe, maybe.
> > I wonder
however
> > why a notation like 3/7 does not fall under the same verdict,
as
> > it is nothing more than the question to find the number
that,
> > when multiplied by 7 yields 3. So we can say that the
number
> > sequence '3/7' does not prove existence of that number. =20
>
> I think I told you already: 3/7 is 0.3 in base 7
But in my opinion 0.3 in base 7 is still a formula telling how the
number
is calculated. It tells me that it is 3 * 1/7. And before Simon
Stevin's
"De Thiende" from 1585, decimal numbers (and base 'n' notated
rationals)
did not actually exist.
Once upon a time there were only unary numbers existing. Everything
else is abbreviation which has to be introduced by someone.
The Babylonians used a base 60 notation, also
for fractions, but there was no true zero and there was no fraction
indicator. And what I have seen from Indian mathematics does not
contain
decimal fractions either. If you look at the Surya Siddhanta (a
classic
work to calculate the calendar) you will find that it contains a sine
table, but the values are represented as integers, to be divided by
1000.
The angles themselves are represented in degrees, sixtieth of
degrees,
and so on. But as Simon Stevin writes (parafrased):
"The importance of the proposal is not so much mathematical
profoundness,
but more broad applicability."
And it was merely used as an aid to ease calculations.
Yes. Also decimal representation is only to ease calculation.
> I think I told you already: 3/7 is 0.3 in base 7 and this can be
> compared with any existing nunmber n/B (in natural base B) by 7n/7B
<
> 3B/7B or 7n/7B = 3B/7B or 7n/7B > 3B/7B.
So for the comparison you do not use "0.3" but the original "3/7"?
Well, I can compare sqrt(2) with any rational number. Square the
two.
How can a number be squared which does not exist?
> 3B/7B or 7n/7B = 3B/7B or 7n/7B > 3B/7B. So we need only find the
> natural numbers 7n and 3B which exist for every natural numbers n
and
> B we can use. (In case all bits of our bit reservoir were required
to
> establish n and B, then the reservoir of bits must be extended by 7
or
> 3 which should be possible in all practicable cases.)
For squaring not much more is needed.
Except the number to be squared.
But whether you can compare easily
or not does *not* depend on whether the notation actually is an
algorithm
or not.
> > It is
acknowledged
> > that infinitesimals can be given a proper basis using
Abraham
> > Robinson's non-standard analysis. Alas, other approaches are
not
> > even mentioned (John Conway, Anders Kock).
>
> And Robinson is only mentioned to support his authority when
quoting
> that there are no infinite sets at all. The reader can see, that
> Robinson knows about what he talks.
What quote of Robinson are you talking about? I do not find any such
quote. Pray state page number in your book.
Robinson says on page 110 (last lines) "Infinite totalities do not
exist in any sense of the word (i.e., either really or ideally). More
precisely, any mention, or purported mention, of infinite totalities
is, literally, meaningless" [ROB64].
By the way: There is a comprehensive index at the end of my book,
including the data of about 170 scholars.
> > Bolzano's musings are set out here, especially for those
cases where
> > intuition does not work. For instance, he does not like
bijections
> > to prove equality of "size" of sets as it does not conform
with
> > intuition of infinite sets
>
> It does not conform with facts.
What facts? Facts are subjective.
> If points (reals) do exist, then
there
> are more points in a long distance than in a short distance as
becomes
> evident when considering the short one as a part of the longer
one.
Yes, and that kind of comparisons lead to lack of trichotomy.
> Two identical distances can be interchanged and have same number
of
> points; they cannot be distinguished. In the difference of longer
and
> shorter distance there are points too. This is a very simple proof
> that bijections can yield false results (whehn interpreted as
yielding
> numbers of elements). The longer distance "hat mehr Realtä4t" as
Cantor
> would say. And mathematics is about reality, at least about
geometry.
Well, when I did learn geometry the reality was that you could
actually
not state that a longer distance had more points.
No, you cannot, unless you believe that all real numbers do actually
exist. Then you must think that the points are all there.
Consider the construction
to divide a particular line in five pieces of equal length using
compass
and straight-edge. All such constructions are actually bijections in
disguise.
> > (the set of squares of natural
numbers is
> > intuitively less than the set of all natural numbers). He
fails to
> > see that set inclusion does not give a trichonomy of sets,
>
> Better: He fails to fail to see that it gives trichotomy of sets.
Pray elaborate, I do not understand this.
Bolzano was correct when he said that set inclusion gives trichotomy.
A long line has more points than a short line, notwithstanding
bijection. (All this is only correct, if reals actually exist, of
course.)
> > is an error on p.
88. It
> > is stated that (translated): "and it is even provable that a
> > well-ordering [of the reals] can not be defined at all..."
...
> You have seen my list of ZFC axioms. It has been proved that this
list
> does not yield a definable well ordering of the reals, i.e.,
without
> introducing additional axioms. I did not write that it would be in
> contradiction with ZFC; this would have been silly because ZFC
proves
> the existence of a well ordering.
> I wrote:
> Es ist bisher niemandem gelungen, eine Wohlordnung für R
anzugeben,
> und es läßt sich sogar beweisen, daß eine Wohlordnung gar nicht
> definiert, also angegeben oder gefunden werden kann
And you are stating here that a well-ordering can not be defined in
ZFC.
So, what is it. Can you show the proof that a well-ordering can not
be
defined? Or at least a source for such a proof? I state that ZFC is
*not* inconsistent with a definable well-ordering of R.
It has been proven by forcing, AFAIK, but being in vacations, I have
no access to literarture presently.
> Further, all your arguing is in vain unless you can present a well
> ordering working in reality. And you cannot.
What reality? Did you read the reference I presented? Strange
enough,
you object to V=L as axiom. As far as I know, V=L provides only the
computable reals in the set of reals.
> > But that is wrong, because when a map 'f' is viewed as a set,
it is
> > defined as:
> > { (n, s) | n in N, s in P(N), f(n) =3D s }
> > So when 'f' can not be defined in a model, neither can the
set
> > representing 'f'.
>
> When P(N) exists in the model,
Of course it exists if it is a model of ZF: the axiom of the
powerset.
> then at least a function f from
P(N)
> to ... exists there.
To what?
The identity mapping of P(N) exists with P(N). This is including the
mapping N --> P(N).
> It is the similar to the nodes and edges of
the
> tree. Every existing node is the end of an edge. There are not
less
> edges than nodes. If all elements of P(N) exist, then all elements
of
> f exist. And the existence of P(N) is guaranteed by the power set
> axiom.
No argument here, it does not contradict what I stated.
So you think that a set may exist but no identity mapping? I agree.
There is no mapping from N or R. But most set theorists deny that (In
fact, I never met a set theorist who missed the identiy mapping R -->
R in real mathematics. How could it get lost in some model which obeys
the same set of axioms?)
>
> > He further states:
> > "Moreover, as by the axiom of the power set P(N) contains
each
> > subset of N. So which subset fails to be in P(N)?"
> > The answer is simply: those that can not be defined in the
model.
> > The model has serious influences on the properties that can
be used
> > in the axiom of subsets, and so also on the power set as
defined by
> > the axiom of the power set.
>
> No. The existence of N and of the power set axiom guarantee the
> existence of P(N) and of at least a mapping
> (P(N))-->N. Your explanation is ungereimt, as Bolzano would have
> said. And Skolem himself did not believe in his own explanations,
as
> his attitude towards set theory clearly shows.
Sorry. (Going back to the ZF definitions), loosely speaking, the
axiom of the power set states that there is a set that contains all
subsets of a given set. The axiom of specification states what are
subsets
What are the subsets of N in any model containing N? Every combination
of natural numbers is a subset of N. Your "axiom of specification"
seems to be introduced to have a tool to exclude some subsets?
Regards, WM
muec...@rz.fh-augsburg.de
> > As long as there are natural indexes, the positions are finitely many,
> > because every natural index marks a finite segment.
>
> Wrong. You are assuming what you are trying to prove.
I assume: every natural index marks a finite segment. Not more. But
that is enough.
>
> > I is easy to prove.
> > 0.000...0001 reflected is the natural number 1000...000. You get it
> > for every finite index n, i.e., for any natural number of zeros.
>
> So you are *not* reflecting an infinite decimal expansion.
I am reflecting every existing position. Which position cannot be
reflected?
> > Every position idexed by a natural number is a finite position. As
> > only natural numbers are used to index, all positons are finite (have
> > a finite distance from the decimal point).
>
> Yes, so what? All positions are finite, but there are infinitely many of
> them.
I do not map the *number of positions* but *every position*. Everyone
is finite and, therefore, belongs to a natural reflected number.
Every natural number n can be mapped on its finite initial segment
(i.e. that one where n is the largest number). Therefore, there are as
many finite initial segments of natural numbers as are natural numbers
(one-to-one).
Conclusion: Should there be an infinite initial segment, we had one
more initial segments than numbers. How could the infinite segment be
distinguished from the finite ones? Remember, in set theory sets are
to be distinguished by elements.
>
> Eh? What is the height of the unending table? What is the width of it? How
> do you define that?
If it exists, then it need not be defined. If it does not exist, then
a definition cannot help either.
But we can define a bijection between lines and columns by the
diagonal: Bot lines and columns must have the same number n from the
diagonal element a_nn, be it a finite number or a transfinite number.
That is "the height = width of the table".
> > There is no possibility that width an height could be different.
> > Proof: One-to-one mapping of the vertical digits onto the horizontal
> > digits, by means of the diagonal elements a_ii.
>
> That works perfectly well for the triangle without the last line.
No, you are in error. And you know it.
>
> What actual existence? Until this point you have not even stated what that
> *means*.
See Cantor on p. 99 of my book.
Regards, WM
muec...@rz.fh-augsburg.de
> > Upto this point
> > of the book the "potential infinite" set of natural numbers has not yet
> > been defined, so the meaning is pretty unclear.
> >
> WM: Didn't you read chapter 8?
>
> I thought so. Can you point me to the page where you define it?
Cantor definies it at p. 99.
>
> > But he asks how it is
> > possible that in the finite case:
> > | { 2, 4, 6, ..., 2n } | =3D n
> > and how in the infinite case 2n becomes equal to n. He states that if
> > | { 2, 4, 6, ... } | =3D aleph-0
> > we have:
> > lim{n -> oo} | { 2, 4, 6, ..., 2n } | =3D aleph-0
> > (the author does not define that limit, but under some reasonable
> > definition that is true), but concludes:
> > 2n / aleph-0 <= 1
> > which is true for finite n and so also in the limit. How the author
> > comes to the first (finite n) and to the second (limit), and how that
> > division is defined is left to the imagination of the reader.
> >
> WM: 2n < aleph_0 by definition for every natural number.
>
> Does not explain either.
Every natural number is finite. Therefore every natural number is less
than aleph_0 --- whether you use a limit or not. The number of
numbers, however, is not restricted to finite values --- allegedly.
>
> > No. It is nonsense to speak of the infinite set of finite numbers. If
> > we do it, then we find literally:
> > n is finite in every case including n --> oo, because there is no
> > infinite natural number.
>
> Eh? But perhaps yes. But I do not see in what way that contradicts what I
> wrote.
The limit lim{n -> oo} 2n remains finite while the limit
lim{n -> oo} | { 2, 4, 6, ..., 2n } | is aleph_0.
This is the mathematical version of the talk of infinitely many finite
numbers.
>
> > The limit | { 2, 4, 6, ... } | can be defined by the union of all
> > initial segments. It is aleph_0, also by definition. Clearly these two
> > results do not fit together, as shown by letting
> > lim{n -> oo} | { 2, 4, 6, ..., 2n } | / 2n = 1/2
>
> I have no idea what this shows, and how it contradicts what I did write.
This shows that the talking about infinitely many finite numbers is
nonsense.
> The above limit is valid. So what? What does it contradict? You refrained
> from any comment on what did follow. So you do agree with that analysis?
>
> > The sixth one is the following:
> > consider |{ 0, 1, 2, ..., n-1 }| = n < aleph-0
> > Obviously for every finite n, there are fewer than aleph-0 natural
> > numbers less than n. Let's make n a variable and assume that it
> > goes through all natural numbers. Because for each finite n there
> > are fewer than aleph-0 natural numbers less than n, there are few
> > natural numbers less than aleph-0, while the cardinality is aleph-0.
> > The fallacy is of course the assumption that what is true for each
> > finite segment is also true for the complete set.
> >
> WM: The "fallacy" is to conclude a =< z from a<b<c<...<z even in case
> > of infinitely many terms.
>
> Where do I conclude that?
When you see that 2n/|{2,4,6,...,2n} with growing n gets larger and
larger but "in infinity" is 0.
> >
> What measure are you talking about? What are distances in set theory?
> How do you define them?
The distance between numbers is their difference.
>
> > , and this sum is finite as long as all
> > numbers involved are finite. And the last is obviously false when
> > there are infitely many finite numbers, in that case the sum does
> > not exist.
> >
> WM: Neither do the numbers.
>
> So numbers do not exist?
Those which were required for N to have infinitely many finite
numbers. No, those do not exist.
>
> > On to the eighth one. Here the author considers a new concept:
> > "Interzession", which I would translate as "mildly interleaved", but
> > there are perhaps better terms.
> >
> WM: Intercession, for instance.
>
> An English speaker would not understand that for what you are meaning.
> Have a look at Merriam-Webster. And, yes, I did look.
It is a new expression coined for a new discovery. Also in Germany
Interzession has a different meaning (used in jurisprudence), but it
is not yet occupied in mathematics. And its Latin meaning fits
extremely well with the situation.
> > I would challenge the author to give an ordering
> > (not necessarily a well-ordering) of the subsets of R.
> >
> WM: I stated on p. 117: For all finite numbers: "Alle unendlichen
> > Zahlenmengen (aus endlichen Zahlen)". Most subsets of R are not finite
> > and, therefore, do not belong to species "finite numbers".
>
> So it is not an equivalence relation amongst sets.
It is an equivalence relation amongst sets with elements which are
finite numbers.
>
> > So his claim
> > that it is an equivalence relation on sets, and as such provides a
> > better measure for the size of a set is unfounded.
> >
> WM: It is an equivalence relation on sets of finite numbers, i.e. all
> > numbers < omega.
>
> Yes, and in that case all infinite sets of finite numbers fall in the same
> class. But that says nothing about other sets. So, in what way is it
> better than bijection?
It does not raise the wrong impression that there were infinite
numbers.
> > What is half a cake?
>
> For a cake I know the answer, but I ask what it is for an edge. Is half
> an edge the part of an edge that starts at a node and stops half-way to
> the next node?
Half an edge is that part of an edge which together with another half
gives a full edge. There is no topology on edges.
>
> > Yes, indeed, even with this construction, it is the last one. So he
> > concludes:
> > "This way no path-bundle (and of course also no path) can split from
> > another one without an edge being associated with it."
> > "it" meaning the path or path-bundle. Which is, eh, quite strange.
> >
> WM: What is strange about this obvious fact?
>
> The strangeness is that it is not an abvious fact. I state, quite clearly,
> that each path-bundle can be associated with an edge, namely the last one
> in the path-bundle. How you come from that that also each path can be
> associated with an edge is completely unfounded.
Try to construct a path which does not splitt off by an edge.
>
> WM: What is strange about this obvious fact? Simply follow a path like
> > 0.000... or 0,010101... and you will see it. Therefore there can be
> > not more paths than edges. It is really quite strange that you seem to
> > believe that "in the infinite", the paths spring off the edges and
> > exist without any foundation, the usual form of existence in set
> > theory.
>
> Clear as mud.
>
> > For path-bundles the strange construction is not needed at all.
> >
> WM: Paths are bunches too: singletons, aren't they? Or are they
> > nothing at all?
>
> A path-bundle has a last edge, a path does not have a last edge. Quite
> some difference.
How many paths must a bundle have to be considered a bundle?
But you believe in the empty set, don't you?
> > For paths the situation is different. You can not
> > map a particular edge to a path because the paths are not terminating.
> >
> WM: Paths are path bunches too, namely such with one path only, unless
> > paths do not exist.
>
> But they do not have a last edge.
Perhaps this is so because they do not have existence at all?
>
> > So there is no clear reason why the cardinality of the set of paths
> > should be less than the cardinality of the set of path-bundles.
> >
> WM: It is not less but it canot be larger. I find it highly
> > (HIGHLY!!!) strange how an intelligent being can claim (and, perhaps,
> > even believe) that there are more bunches of paths (including all
> > singletons, i.e., paths) in the tree than nodes where they can split.
> > So there are more results of spits than splits, although we know that
> > every split creates exactly one result. Strange.
>
> You may think it strange. But I see simply that there is no bijection
> possible between the elements of the set of paths and the set of nodes.
> I see only an injection from the set of nodes to the set of paths, and
> from that I conclude that there are fewere nodes than paths.
Couldn't it be that the bijection is not the answer to everything and
that aleph_0 < 2^aleph_0 is not always valid?
It is impossible that there are more splitting results than splitting
positions. Do you agree???
Not even by tunnel effect the tree can create or contain more paths
than nodes.
Regards, WM
Virgil
muec...@rz.fh-augsburg.de wrote:
> On 21 Feb., 02:46, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> > >
> > > > And I think that the remark that Euclid used a proof
> > > > by
> > > > contradiction is wrong
> > >
> > > ??? It is said (I didn't read the original text) that Euclid assumed
> > > that n primes exist and then he contradicted this assumption by
> > > proving the existence of prime number n+1. Many authors call this a
> > > proof by contradiction.
> >
> > It would be a proof by contradiction, if there is an initial assumption
> > that is falsified by the reasoning. But see below.
>
> Whether this is the case depends on the person. Euclid is said to have
> assumed that.
Not by anyone familiar with Euclid's proof.
> >
> > > > (actually the theorem and proof are quite
> > > > Cantoresque):
> > >
> > > Yes. Cantor assumed a complete list of reals and showed the existence
> > > of another real.
False! Cantor merely assumed a list of reals ( a mapping from N to R)
which he then showed necessarily omitted some real number.
That WM should misrepresent both Euclid and Cantor as as having proved
by contradiction what they proved directly, marks WM again as being
mathematically incompetent.
> Many authors call this a proof by contradiction.
> >
> > Wrong. Cantor does not assume that the given list is complete. He shows
> > that given *any* list of reals there is a real not on the list. There is
> > *no* initial assumption of completeness.
>
>
> So Cantor constructs one or the other list without attempting to
> include all the reals? Then he finds another real and takes it as
> evidence for what? That he personally is not able or was too lazy to
> construct a complete list?
WM again shows his gross incompetence at anything mathematical by his
failure to understand the conseqeuences of showing that any listing of
reals must be incomplete.
> > >
> > > That however is not the wording used by Euclid.
> >
> > What is the wording used by Euclid? (I may note that the Corollary was
> > indeed
> > *not* by Euclid, but that is the modern corollary.)
>
>
> Euclid is reported to have said (of course in Greek and slightly
> differing in the thousands of copies of his Elements made by hand
> before the 1500 printed editionsappeared): "There are more prime
> numbers than any given set of primes contains" or "in any given set
> of primes, there is prime number missing".
>
> This is of course correct and can be generalized to natural numbers:
> For any given set of natural numbers, there is a natural number not in
> that set.
What natural number is not in the set of all natural numbers?
> > Well, I can compare sqrt(2) with any rational number. Square the two.
>
> How can a number be squared which does not exist? First it must exist.
> Then it can be handled.
Then 2 cannot exist in Wolkenmuekenheim, as it must be the square of
something, as each positive real is.
> > Well, when I did learn geometry the reality was that you could actually
> > not state that a longer distance had more points.
>
> No, you cannot, unless you believe that all real numbers do actually
> exist. Then you must think that the points are all there.
But it is just those who believe that real numbers actually exist who
deny that longer lines have more points.
> There is no mapping from N or R.
There is in mathematics. A natural injection.
Whatever sort of world WM lives in that has no such mapping must be very
peculiar.
Virgil
muec...@rz.fh-augsburg.de wrote:
> On 21 Feb., 03:25, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
>
> > > As long as there are natural indexes, the positions are finitely many,
> > > because every natural index marks a finite segment.
> >
> > Wrong. You are assuming what you are trying to prove.
>
> I assume: every natural index marks a finite segment. Not more. But
> that is enough.
You also assume that there is an end to the indices, which for natural
numbers indices is false.
>
> I am reflecting every existing position. Which position cannot be
> reflected?
It is not single positions but sets of positions which determine the e
meaning of a number.
The set of positions required for the decimal expansion of 1/9 cannot be
reflected.
In fact for every rational whose denominator has a prime factor other
than 2 or 5, its decimal representation cannot be reversed.
WM's ignorance again shows itself.
>
> I do not map the *number of positions* but *every position*. Everyone
> is finite and, therefore, belongs to a natural reflected number.
Reflect the decimal representation of 1/9.
>
> Every natural number n can be mapped on its finite initial segment
> (i.e. that one where n is the largest number). Therefore, there are as
> many finite initial segments of natural numbers as are natural numbers
> (one-to-one).
But neither set (that of finite initial segments or that of natural
numbers) can be numbered by a natural number.
>
> Conclusion: Should there be an infinite initial segment, we had one
> more initial segments than numbers.
It is the 'number' of finite initial segments one need worry about
pairing off with naturals, unless WM claims to have an infinite natural.
> How could the infinite segment be
> distinguished from the finite ones? Remember, in set theory sets are
> to be distinguished by elements.
By having for each finite segment an element not in that segment.
>
> See Cantor on p. 99 of my book.
As it is the many errors in your book that are the issue here, citing
your book does not get you off the hook.
Virgil
muec...@rz.fh-augsburg.de wrote:
> Every natural number is finite. Therefore every natural number is less
> than aleph_0 --- whether you use a limit or not. The number of
> numbers, however, is not restricted to finite values --- allegedly.
And is also not a natural.
>
> The limit lim{n -> oo} 2n remains finite while the limit
> lim{n -> oo} | { 2, 4, 6, ..., 2n } | is aleph_0.
What are WM's deltas and epsilons justifying those limit claims?
>
> This is the mathematical version of the talk of infinitely many finite
> numbers.
It may be WM's version of mathematics, but we all know how unreliable
that is. Absent some reasonable definitions, which WM has yet to
provide, for evaluating his claimed limits, they do not exist.
> This shows that the talking about infinitely many finite numbers is
> nonsense.
When WM talks it is almost always nonsense, even when talking about what
from others is quite sensible.
> >
> > So numbers do not exist?
>
> Those which were required for N to have infinitely many finite
> numbers. No, those do not exist.
Then there must be infinitely many finite naturals missing from WM's
numerology.
> Couldn't it be that the bijection is not the answer to everything and
> that aleph_0 < 2^aleph_0 is not always valid?
NO!
> It is impossible that there are more splitting results than splitting
> positions. Do you agree???
Not at all. In a tree in which every path is of length n, there are at
most n "splitting positions", in at least one sense, but there are 2^n
paths.
> Not even by tunnel effect the tree can create or contain more paths
> than nodes.
That only can hold for finite binary trees, as maximal infinite binary
trees can be shown to have uncountably many paths
muec...@rz.fh-augsburg.de
> mueck...@rz.fh-augsburg.de writes:
> > Put a marble in your living room and pray that ZFC may be valid
> > tonight. Perhaps your prayer is heard and our universe is full of
> > marbles tomorrow. Then I will believe in ZFC. Otherwise you should
> > begin to suspect it.
>
> Wow! What a clear and concise proposal! Keen.
>
> I will let you know my findings in the morn, I tell you what!
>
> But I'm a bit unclear on one little point. I'm sure it's not
> important, but still.... How does ZFC prove that prayer will result in
> an overflowing universe of marbles again?
Shit happens.
Regards, WM
muec...@rz.fh-augsburg.de
> In article <JDuD88....@cwi.nl> "Dik T. Winter" <Dik.Win...@cwi.nl> writes:
> > In article <1172058191.620157.204...@v45g2000cwv.googlegroups.com> mueck...@rz.fh-augsburg.de writes:
> > > REPLY TO THE SECRET REVIEW OF CHAPTER 9 (PART 3 of the Reply)
> >
> > What is so secret about it?
>
> Wolfgang apparently does not answer this.
You said "the review of the second part will not be posted". So I
called it secret.
Secret somethings are always attracting.
Regards, WM
muec...@rz.fh-augsburg.de
> In article <1172052400.509592.146...@m58g2000cwm.googlegroups.com>,
>
> mueck...@rz.fh-augsburg.de wrote:
> > > I don't believe in the favourable review of the first 8 chapters of his
> > > book.
>
> > Buy it, read it. In contrast to set theory there is no necessity to
> > believe in anything with my book.
>
> > Regards, WM
>
> In the first 8 chapters, there is apparently nothing of WM, and in the
> ramaining chapters there is apparently nothing of mathematics, so why
> would anyone want to buy it.
Amusing. When Sigund Freud introduced his theory, it was criticised
for containng many correct ideas and many new ideas, alas, the correct
ideas were not new and the new ideas were not correct.
Regards, WM
muec...@rz.fh-augsburg.de
>
> > It is the binary tree to which you, as far
> > as I remember, ascribed different sets of paths according to how one
> > is looking at it. This is obviously a contradiction in every set
> > theory.
>
> It was not one binary tree, but a variety of differently defined binary
> trees that had different properties in accord with the differences in
> their definitions.
All these trees have in common the same set of nodes and edges.
>
>
> > > The calculus of Leibitz and Newton, being based on undefined
> > > infinitesimals, was "wrong" in a way, too, but that has since been fixed.
>
> > > So why does WM continue to claim ( without proof, or even evidence) that
> > > whatever was once "wrong" with set theory is incapable of being fixed?
>
> > Everything will be fixed with no doubt. You fixed it by stating that a
> > unique set of nodes in a tree with unique structure yields different
> > sets of paths.
>
> I said that a binary tree can as correctly be defined by its set of
> paths as by its set of nodes.
You said you had discovered something new, about different sets of
paths ...?
Regards, WM
Regards, WM
Alois Steindl
>
> In the first 8 chapters, there is apparently nothing of WM, and in the
> ramaining chapters there is apparently nothing of mathematics, so why
> would anyone want to buy it.
Hello,
I think the book - and the review! - should be carefully read by those
people, who hired WM to teach mathematics at the University of Applied
Sciences Augsburg.
Alois
Dik T. Winter
> On 22 Feb., 05:12, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> > In article <JDuD88....@cwi.nl> "Dik T. Winter" <Dik.Win...@cwi.nl> writes:
> > > In article <1172058191.620157.204...@v45g2000cwv.googlegroups.com> mueck...@rz.fh-augsburg.de writes:
> > > > REPLY TO THE SECRET REVIEW OF CHAPTER 9 (PART 3 of the Reply)
> > >
> > > What is so secret about it?
> >
> > Wolfgang apparently does not answer this.
>
> You said "the review of the second part will not be posted".
And in the same article I posted an URL where it can be read.
> So I
> called it secret.
> Secret somethings are always attracting.
So something that is not posted in this newsgroup is secret? In that
case all your papers are secret, but I do not think they are
attracting...
muec...@rz.fh-augsburg.de
I said: Nur aus der zusätzlichen Information, wonach die Menge
abzählbar ist, und dem Satz, daß die Vereinigung von zwei abzählbaren
Mengen stets wieder eine abzählbare Menge ergibt, kann auf die
Überabzählbarkeit der Menge geschlossen werden.
Andererseits könnte man aus diesem Überabzählbarkeitsbeweis auf die
Überabzähl¬barkeit abzähl¬barer Mengen schließen.
Then comes the example.
so finden wir eine unendliche Folge (-1, 1/2), (-1/3, 1/4), (-1/5,
1/6), ... von Inter¬vallen ((), ()) und den nicht zur Folge
gehörenden Grenzwert = = = 0, obwohl die Menge der Glieder
einschließlich Grenzwert gewiß abzählbar ist. Nur durch diese
Zusatzinformation wird das Beweis¬ergebnis korrigiert und die
Abzähl¬barkeit der Folge einschließlich Grenzwert fest¬gestellt. Dies
gilt für jede um den Grenzwert alternierende konver¬gente Folge wie
( + (-1)n/n). Wie können wir aber sicherstellen, daß nicht auch
andere Ergebnisse durch (bislang möglicherweise noch unbekannte)
Zusatz¬informationen richtiggestellt werden müs¬sen?
But that you can not distingsuish
> this way the cardinal numbers of the transcendental and rational numbers
> is correct (that is why I did not comment on that), but is no problem at
> all.
>
> > where in the building of the diagonal every 0 and 1
> > is replaced by a 2, and concludes that the diagonal found at every step
> > is in the list. Strange that he sees that as a contradiction, because
> > in the building of the diagonal *each* digit has to be changed.
> >
> > But it cannot be done other than step by step.
>
> That is what you state. But as each replacement is independent of all
> other replacements, it can be done in parallel. No step by step needed.
The natural numbers are created step by step by the Peano axioms,
i.e., by induction. And every application has to take this into
account.
>
> [ About the infinite triangle.]
>
> > Using the tools of set theory we can define a bijection. An existing
> > diagonal with X elements guarantees the existence of as many lines and
> > columns.
>
> Yes, and there are. Both are aleph-0, and neither does have an aleph-0-th
> element.
Nothing but wrong belief. Counter proof by bijection by means of the
diagonal element a_nn.
>
> > Fourteen is also closely connected. Obviously the author is thinking
> > that the creation of the diagonal is an iterative process, which is
> > *not* the case.
> >
> > Which *is* the case. You cannot start at the end because it is not
> > there. So you cannot start with an action including the end. So you can
> > only start at the first or at least a finite position n.
>
> All replacements can be done in parallel.
Nothing but wrong belief. Absolutely unjustified.
>
> > Given a list, you can refer to the n-th digit of the
> > created diagonal without ever referring to any other digit of the
> > created diagonal.
> >
> > Yes, but with the n-th digit you have not all of them. And you cannot
> > start at he non existing end. So you must start at a finite oposition.
>
> You need not start anywhere, you can start with all of them at once.
If you could, you would see that there are not infinitely many. But
you cannot. Couterproof: What comes before [pi*10^10^100]?
Regards, WM
muec...@rz.fh-augsburg.de
> Are you talking about Skolem's paradox now? First, what identity
> mapping are you referring to? There always exists a bijection of a set
> onto itself. As to other mappings, they exist or they don't as
> provided by the axioms of the theory. And we have to be careful to be
> specific as to what requirements we place as to in WHICH sets a
> mapping does or does not exist.
In any model of ZFC: With the empty set there exists the set omega and
with it is power set, and with it the identity mapping of the power
set onto itself
>
> > > Tarkski and Banach showed that ZFC has a certain result that is
> > > puzzling at first glance. But it requires the axiom of choice; it is
> > > not a contradiction; and (as far as I understand) would only
> > > contradict a physical theory if matter had infinite density.
>
> > Put a marble in your living room and pray that ZFC may be valid
> > tonight. Perhaps your prayer is heard and our universe is full of
> > marbles tomorrow. Then I will believe in ZFC. Otherwise you should
> > begin to suspect it.
>
> ZFC is not a theory of physics. I don't know much about the Tarski-
> Banach theorem, but I have not read that it entails anything contrary
> to physics. As I understand, the theorem does not render that a
> PHYSICAL object may be perform in the way the ABSTRACT object in the
> theorem performs.
It is about geometry. Geometry is about physics. If the dissection as
proven possible by Banach and Tarski (but, of course, not concretely
defined --- see well ordering of the reals) could be applied in
geometry, then it could be applied in physics. Obviously this result
of set theory contradicts set theory (more precisely: the axiom of
choice) for anybody who is not a strong believer without a critical
sense or the wish to accept the truth.
>
> Also, I notice that you did not respond to my other points in reply to
> you.
Sorry, I am neither able nor willing to answer every contribution in
detail. Therefore I must snip much. But if you have a serious request,
please repeat it. As far as I remember you wanted to do some proofs
concerning the binary tree, which from my diskussion with Dik should
be clear to anybody.
> And I notice that is a habit you have of brushing off from among
> the more obviously solid points of rebuttal to you. Of course, I
> understand that no one can be expected to answer every line of every
> post.
In particular if he is in a position as I am here.
> But I do note that I've written quite a bit to you, with a lot
> of tissue of explanation, yet you demur from following up on those
> substantive matters as, what seems to me, to be a way of avoiding an
> engagement of ideas that would lead you into considerations beyond
> your program of denunciations of set theory.
In ordinary mathematics it is sufficient to show one contradiction.
There are many. To say that, is not a denunciation.
>My point is not to go
> back over all those posts, but rather to suggest that we discuss these
> points for all they are worth .
We were discussing the binary tree. Please continue. Or whatever you
want to discuss in detail.
Regards, WM
muec...@rz.fh-augsburg.de
> On Feb 21, 2:03 am, mueck...@rz.fh-augsburg.de wrote:
>
> > On 20 Feb., 20:17, "MoeBlee" <jazzm...@hotmail.com> wrote:
>
> > > On Feb 20, 3:59 am, mueck...@rz.fh-augsburg.de wrote:
>
> > > > On 14 Feb., 03:57, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> > > > And Robinson is only mentioned to support his authority when quoting
> > > > that there are no infinite sets at all. The reader can see, that
> > > > Robinson knows about what he talks.
>
>
> > his deep insight
>
> > > that the notion
> > > of an infinite set is meaningless.
>
> of mathematical logic, including infinite sets, uncountable sets, and
> the axiom of choice.
That is all well known.
Look, when contributions become too long, then they are not fully
displayed in my news reader and I have to load them separaely. In
order to avoid this as much as possible, I snip as much as possible.
>
> Why don't you think about Robinson's mathematics and philosophy a bit
> more so that you won't be so inclined to miss and snip the TOTALITY of
> what he's saying rather than use him (as you've used other
> mathematicians) as mere fodder for quotes (taken out of context and
> misunderstood by you) for your continual harrumping that set theory is
> "wrong".
>
> Such distortions of context and misrpresentation of what the writers
> really are saying is indicative of someone whose interest is not in
> really understand the subject,
Be sure that I read your text and that I understand what Robinson was
doing (not all of his work, of course, but the most famous part). Bu
why repeat paragraphs which are not in question?
Regards, WM
muec...@rz.fh-augsburg.de
> On Feb 21, 3:43 am, mueck...@rz.fh-augsburg.de wrote:
>
> > Of course the natural numbers are a sequence. In Cantor's case, the
> > theory have recognized this and, therefore, postulated that everything
> > happens at once, but that does not mean that this position is correct
> > or sensible.
>
> with a temporal ANALOGY, then the best fit is that "it all happens at
> once". But we are NOT obligated to regarding mathematics as being
> describable with a temproral analogy. Any state of affairs that is a
> model of set theory is an ABSTRACT state of affairs in which time is
> not even expressed as a feature. So surely we do NOT have to describe
> the state of affairs as a chronology, not even as a singular
> "happening all at once". Set theory does axiomatize the mathematics
> that is used for theories in which time is a feature, but that does
> not entail that a model of set theory is itself or must be itself a
> state of affairs in which time is a feature.
>
> > One cannot do infinitely many transactions at once.
>
> intelligence can perform. However, set theory does axiomatize the
> theorems about computability which can be taken as about what
> transactions a finite intelligence and/or also a computing machine can
> perform. And meanwhile, you are welcome to advance a mathematical
> theory that limits itself to such computability.
I will not build a new mathematical theory. I would not be able to do
so, so I don't even start.
>
> The question for you is: Are you interested in understanding that or
> are you vastly more interested in continuing to harrump that set
> theory is "wrong"?
Read chapter 7 of my book in oreder to judge about what I understand
by set theory and how I understand it. I am sure to know what you
understand by it, but I am not sure that that is the correct way to
think.
Regards, WM
Fuckwit
>
> Robinson says on page 110 (last lines) "[i] Infinite totalities do not
> exist in any sense of the word (i.e., either really or ideally). More
> precisely, any mention, or purported mention, of infinite totalities
> is, literally, meaningless." [ROB64].
>
"(ii) Nevertheless, we should continue the business of
mathematics as usual', i.e., we should act as if infinite
totalities really existed."
(Abraham Robinson)
F.
muec...@rz.fh-augsburg.de
> Why don't you think about Robinson's mathematics and philosophy a bit
> more so that you won't be so inclined to miss and snip the TOTALITY of
> what he's saying rather than use him (as you've used other
> mathematicians) as mere fodder for quotes (taken out of context and
> misunderstood by you)
That is what I mostly like and what is typical for set theorists. One
can express ones opinions as precisely as possible, they will say that
the opposite is meant.
Skolem held set theory as wrog.
Banch and Taski tried to dismiss AC.
Robinson stated as cannot be statet more clearly:
Indeed, I think that there is a real need, in formalism and elsewhere,
to link our understanding of mathematics with our understanding of the
physical world.
(i) Infinite totalities do not exist in any sense of the word (i.e.,
either really or ideally). More precisely, any mention, or purported
mention, of infinite totalities is, literally, meaningless. (ii)
Nevertheless, we should continue the business of Mathematics 'as
usual', i.e., we should act as if infinite totalities really
existed."
Hey, dreamer, wake up! "any mention, or purported mention, of infinite
totalities is, literally, meaningless." Can you understand the meaning
of these words, literally?
I did not misunderstand them. You are sleepwalking.
Regards, WM
muec...@rz.fh-augsburg.de
Number fifteen is again closely connected. The author quotes
Cantor,
I think from a letter to Dedekind, as stating (translated):
"If from a well-ordered set any two elements change their place
in
the ordering, does that not change the order type, so also not
the
ordinal number. From this it follows that such order changes
of
a well-ordered set leave the ordinal number unchanged when
they
are based on a finite or infinite series of transpositions."
Interpreting this sentence is in itself already quite difficult.
Anyhow, the conclusion of the author that it means that any finite
or
infinite series of transpositions leave the order type unchanged
is
incorrect, because there are many infinite series of
transpositions
that do not even give a well-defined set (and it is my opinion
that
Cantor was wrong, but that depends on interpretation).
WM:
Cantor was wrong, as I indicated by "Trotzdem könnte mit Hilfe des in
Kasten 9.2 be¬schriebenen "Beweises" jede Liste in eine vorgegebene
Ordnung gebracht werden."
So the conclusion
by the author is false.
WM:
Again I find you reporting an error which is not in the book. In this
way you get a fine collection. Did you not see "Beweis" in quotation
marks?
So his conclusion that some countable number
of transpositions starting with a well-ordering of the rationals,
leading to the natural order is false. The resulting ordered set
is
not well-defined.
WM: Of course, but if the set really existed, as Cantor believed, then
Cantor was correct. tThen we could get the smallest positive rational
number as the first. That shows again: Set theory is incorrect.
In sixteen he looks at the extension Cantor gave to the diagonal
proof
by showing that the powerset of the reals was strictly larger than
the
set of the reals (and said this in the same way holds for every
set).
However, the author talks about a proof that any interval can not
be bijected to the set of real-valued functions on that interval,
whether
continuous or not. This is actually less than what Cantor did
proof,
in his proof the functions used are those functions that have only
0 and 1 as function values (indicator functions), so these
function
correspond to subsets, and that set is a subset of what the author
thinks
it is about. Cantor's proof in essence states that if 'f' is a
map
from the reals in the interval to the indicator functions, we can
define the indicator function 'f(x)(x)' (remember that 'f(x)' is
an
indicator function and so 'f(x)(x)' is either 0 or 1). And he
shows
in essence that the indicator function 'G(x) = 1 - f(x)(x)' is not
in
the map (pretty similar to Hessenbergs argument). But even if it
was
a proof about all real valued functions, the authors
counterargument
is invalid, because in his counterargument the map is constructed
during a process of finding functions.
WM: As Cantor's argument must hold for every set, it must also hold
for an artificially constructed set. And here it does not.
And in seventeen he states that the diagonal argument is wrong
because
it requires a matrix of equal height and width, both aleph-0. See
also above. More serious is that he states (explicitly) that
adding
one or more reals to the list should increase the "number" of
elements
in the list.
WM: In analysis we have limits where the later digits get less and
less weight (10^-n). In the limit the weight is zero.You fail to see
that Cantor's argument requires a one-to-one mapping, because thre
every digit has the same weight. That means that there are *exactly*
as many lines as columns.And if this is accomplished, then by adding
one singe line, the diagonal does no longer cross every line.
It is incredible: You conclude from the construction of *one* diagonal
number to the uncountability of the reals, but you do not see that the
addition of one single line to the list (let alone doubling of its
lines) makes this proof impossible.
Eighteen. Rational numbers. Again a strange formulation because
again he argues that the list of rational numbers (whatever way
that
is created) does not contain a diagonal, because there are far
more
lines than digit positions.
WM: This is only a slight remark questioning the assumption that
Cantor's list must be a square. This assumption, however, must
necessarily be fulfilled --- not in the manner of a limit but one-to-
ne.
And gives as example the list of rationals
in [0, 1) with finite binary expansion and argues that there are
2^n
such numbers with n binary digits after the decimal point.
Obviously
correct. He fails to see that to create a diagonal you need
infinite
expansions for all rational numbers, i.e. augment them with an
infinite
number of trailing zeroes.
WM: You may need them, but do you have them?
But apparently the author has the impression
that, because 2^n > n for finite n, lim{n -> oo} 2^n > lim{n ->
oo} n,
or something like that. Well, even when we define limits on
cardinal
numbers so that they work, that is not the case. If we do that we
find
that both limits are aleph-0.
WM: You define and believe it so, but erroneously.
In nineteen the author goes a bit deeper. König tried to prove
that
the reals (the continuum) could not be well-ordered, however, he
retracted
his first proof, and adjusted it later with the remark:
"It is easy to prove that the finitely defined elements of the
continuum form a subset of the continuum with cardinality
aleph-0".
As is known, quite true. It is a pity that the author has
restricted
his readings, a look at Allan Turing, "On Computable numbers, with
an
application to the Entscheidungsproble", proc. London M. S.,
series
2, 42(1936), pp 230-265, would have shown the author that there
has
been much thought about that already, and in later papers it has
been
completely formalised. But for some reason the author ignores all
papers after, say, 1930.
WM: Please read the list of references. It is given at the end of the
book.
Well, the author tries to find a paradox
(following König): suppose there is a well-ordering of the
continuum.
The well-ordering corresponds to the sequence of ordinals without
gaps.
In that ordering, there is a first ordinal that corresponds to a
finitely
defined number. But that ordinal is finitely defined: a paradox.
It is
not. The error is here that, while that ordinal can be finitely
defined,
that is not necessarily true for the number it corresponds with.
The
fatal error is the assumption that if there is a well-order, that
it
can be finitely defined.
WM: Unless it can be finitely defined, it is not defined at all.
Further Koenig was completely correct: Man beachte ferner, daß nach
den jetzt gültigen Annahmen das Kontinuum, wie jede wohlgeordnete
Menge, eine lückenlose Folge bestimmter Ordnungszahlen definiert; und
zwar in der Weise, daß jedem Elemente des Kontinuums eine und nur eine
solche Ordnungszahl entspricht, wie auch umgekehrt. Es ist demnach
"die einem endlich definierten Elemente des Kontinuums entsprechende
Ordnungszahl", sowie auch "das einer endlich definierten solchen
Ordnungszahl entsprechende Element des Kontinuums" endlich definiert.
Twenty shows a common fallacy. The computable numbers are
countable,
so they can be put in a list, we can compute the diagonal, so that
is
a computable number not in the list, showing that there is a
paradox.
The problem here is that it can be shown that
WM:
that the reviewer coflates computable and constructable.
Leaving alone that it conflates constructable with computable, the
answer is no. It is only yes when all numbers in the list are
computable.
Please do not conflate constructable with computable. A constructable
number is a number which can be constructed like pi or Cantor's
diagonal. The set of all constructable numbers is countable. So Cantor
shows the uncountability of a set by constructing another number of a
countable set.
Alle "endlich definierbaren Zahlen", alle konstru¬ierbaren, alle
individualisierbaren Zahlen sind abzählbar. Es gilt dasselbe wie für
die Zugehörigkeit eines Elementes zu einer Menge: "Im allge¬meinen
werden die betref¬fenden Entscheidun¬gen nicht mit den zu Gebote
stehenden Methoden oder Fähigkeiten in Wirklichkeit sicher und genau
ausführbar sein; darauf kommt es aber hier durchaus nicht an, sondern
allein auf die interne Determination welche in konkreten Fällen, wo es
die Zwecke fordern, durch Vervollkommnung der Hilfsmittel zu einer
aktuellen (externen) Determination aus¬zubilden ist" [CAN32, p. 150].
Trotzdem ist es unmöglich, mehr als abzählbar viele individuell
unterscheidbare Zahlen zu konstruieren. Alle durch kein einziges
Merkmal individualisierbaren oder determinierbaren Zahlen (WEYLs Brei
des Kontinu¬ums) sind als Elemente einer Menge unzulässig.
Did you read anyhing about computable in my book?
Regards, WM
Carsten Schultz
(quoting Dik, I think)
> REPLY TO THE NONSECRET REVIEW OF CHAPTER 9 (PART 4 of the Reply)
>
> Number fifteen is again closely connected. The author quotes
> Cantor,
> I think from a letter to Dedekind, as stating (translated):
> "If from a well-ordered set any two elements change their place
> in
> the ordering, does that not change the order type, so also not
> the
> ordinal number. From this it follows that such order changes
> of
> a well-ordered set leave the ordinal number unchanged when
> they
> are based on a finite or infinite series of transpositions."
> Interpreting this sentence is in itself already quite difficult.
I do not know the context, but I would think that this just means that
isomorphic well-orderings yield the same ordinal number. Of course,
this is obvious to us, but I do not know how well established the notion
of an isomorphism was back then. I think "infinite series of
transpositions" might just be a crude way of saying "permutation", i.e.
bijection of the set onto itself.
Of course, I may be wrong.
Just my 2 cents,
Carsten
--
Carsten Schultz (2:38, 33:47)
http://carsten.codimi.de/
PGP/GPG key on the pgp.net key servers,
fingerprint on my home page.
MoeBlee
> On 21 Feb., 21:50, "MoeBlee" <jazzm...@hotmail.com> wrote:
>
> > Are you talking about Skolem's paradox now? First, what identity
> > mapping are you referring to? There always exists a bijection of a set
> > onto itself. As to other mappings, they exist or they don't as
> > provided by the axioms of the theory. And we have to be careful to be
> > specific as to what requirements we place as to in WHICH sets a
> > mapping does or does not exist.
>
> In any model of ZFC: With the empty set there exists the set omega and
> with it is power set, and with it the identity mapping of the power
> set onto itself
It is not excluded "a priori" that a model of ZFC might not map '0' to
the empty set, 'w' to omega, or even 'e' to the membership relation.
Anyway, the identity function on the power set of omega is a bijection
from the power set of omega onto the power set of omega, yes, we know
that.
> > ZFC is not a theory of physics. I don't know much about the Tarski-
> > Banach theorem, but I have not read that it entails anything contrary
> > to physics. As I understand, the theorem does not render that a
> > PHYSICAL object may be perform in the way the ABSTRACT object in the
> > theorem performs.
>
> It is about geometry. Geometry is about physics.
Geometry is USED for physics. That does not entail that all geometry
has a direct object-by-object correlation with physics.
> If the dissection as
> proven possible by Banach and Tarski (but, of course, not concretely
> defined --- see well ordering of the reals) could be applied in
> geometry, then it could be applied in physics.
Only you could say how YOU would apply it. Meanwhile, other people are
free not to apply it the way YOU apply it.
> Obviously this result
> of set theory contradicts set theory (more precisely: the axiom of
> choice) for anybody who is not a strong believer without a critical
> sense or the wish to accept the truth.
More empty rhetoric by you. No P and ~P in set theory do you adduce.
> > Also, I notice that you did not respond to my other points in reply to
> > you.
>
> Sorry, I am neither able nor willing to answer every contribution in
> detail.
I myself mentioned that I don't expect you to answer every post or
every argument in every post, let alone every detail in every post.
> Therefore I must snip much.
I don't blame you for merely snipping. I snip too to get to the gist
of some point. But your snipping is often to snip CRUCIAL context, as
I mentioned elsewhere, in particular, your reply to my reply about
Robinson.
Moreover, you are now responding to my compaint that you skip
responding to certain hard arguments put to you. That is not a matter
of snipping.
> But if you have a serious request,
> please repeat it.
Again, you ignore that I said that my point is not to go over previous
discussions but to suggest a better pattern in future discussions.
> As far as I remember you wanted to do some proofs
> concerning the binary tree, which from my diskussion with Dik should
> be clear to anybody.
There've been a lot of posts I've made to you regarding things other
than your tree. I wasn't talking about the tree. But as to the tree,
you completely miss my point about that, even as I BELABORED it: I was
willing to discuss your tree argument with you IF you would give me
the courtesy of FIRST telling me what the ground rules are,
specifically whether you were claiming to work strictly in a Z set
theory (or whether you were working in some combination of set theory
and your own personal mathematical ideation) and whether your claimed
contradiction was indeed an actual P and ~P in the language of set
theory. You never gave me a straight answer to that, so I told you I'm
not interested in investing my time on a project that is completely
undetermined as to such ground rules.
> > And I notice that is a habit you have of brushing off from among
> > the more obviously solid points of rebuttal to you. Of course, I
> > understand that no one can be expected to answer every line of every
> > post.
>
> In particular if he is in a position as I am here.
>
> > But I do note that I've written quite a bit to you, with a lot
> > of tissue of explanation, yet you demur from following up on those
> > substantive matters as, what seems to me, to be a way of avoiding an
> > engagement of ideas that would lead you into considerations beyond
> > your program of denunciations of set theory.
>
> In ordinary mathematics it is sufficient to show one contradiction.
> There are many. To say that, is not a denunciation.
You're not even TRYING to understand anything I'm saying here.
> >My point is not to go
> > back over all those posts, but rather to suggest that we discuss these
> > points for all they are worth .
>
> We were discussing the binary tree. Please continue. Or whatever you
> want to discuss in detail.
No, I was referring to matters other than the binary tree. And AGAIN,
I said my point is not to go back to discuss what I consider you to
have elided, but rather to suggest a better conversation on a going
forward basis. But it seems that a better conversation with you is
unlikely; It seems to me that you simply do not have the willingness
to LISTEN to what other people are trying to convey to you.
MoeBlee
MoeBlee
> On 21 Feb., 22:07, "MoeBlee" <jazzm...@hotmail.com> wrote:
>
> > On Feb 21, 2:03 am, mueck...@rz.fh-augsburg.de wrote:
>
> > > On 20 Feb., 20:17, "MoeBlee" <jazzm...@hotmail.com> wrote:
>
> > > > On Feb 20, 3:59 am, mueck...@rz.fh-augsburg.de wrote:
>
> > > > > On 14 Feb., 03:57, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> > > > > And Robinson is only mentioned to support his authority when quoting
> > > > > that there are no infinite sets at all. The reader can see, that
> > > > > Robinson knows about what he talks.
>
> > > > Robinson's own formalist philosophy includes his claim
>
> > > his deep insight
>
> > > > that the notion
> > > > of an infinite set is meaningless.
>
> > Notice how you snipped what else I reported - which is that Robinson
> > still endorses infinitistic set theory and that he uses the full array
> > of mathematical logic, including infinite sets, uncountable sets, and
> > the axiom of choice.
>
> That is all well known.
Not to everybody. In fact, I spent multiple posts trying to get
another poster to recognize that Robinson's non-standard analysis
takes place in a greater context of classical mathematical logic and
ZFC.
> Look, when contributions become too long, then they are not fully
> displayed in my news reader and I have to load them separaely. In
> order to avoid this as much as possible, I snip as much as possible.
I don't begrudge snipping for concision and emphasis. But your
presenting only the first half of Robinson's view of infinity distorts
the important point that Robinson works with infinite sets in the same
way as many of the people in this thread who you are claiming to be
wrong about working with infinite sets.
> > Why don't you think about Robinson's mathematics and philosophy a bit
> > more so that you won't be so inclined to miss and snip the TOTALITY of
> > what he's saying rather than use him (as you've used other
> > mathematicians) as mere fodder for quotes (taken out of context and
> > misunderstood by you) for your continual harrumping that set theory is
> > "wrong".
>
> > Such distortions of context and misrpresentation of what the writers
> > really are saying is indicative of someone whose interest is not in
> > really understand the subject,
>
> Be sure that I read your text and that I understand what Robinson was
> doing (not all of his work, of course, but the most famous part). Bu
> why repeat paragraphs which are not in question?
Because after, in my reply to you, I had mentioned the CRUCIAL
context, you CONTINUED to post by OMITTING that context. The way you
present Robinson's view of infinity is a distortion since you leave
out the CONCLUSION he comes to about it (which conclusion puts his
view in a RADICALLY different light).
MoeBlee
Ralf Bader
If you read §7 of Cantor's Beiträge zur Begründung der transfiniten
Mengenlehre then you will see that this notion was present to him. If you
don't read (or have read) Cantor's papers then it is completely pointless
to engage in speculations about what he knew or not. Moreover, I dimly
remember that the matter with those infinitely many transpositions has been
discussed previously (of course to no avail concerning Mückenheim) and that
Cantor had made an assumption there which Mückenheim, as usual, neglected.
However, Google's news archive interface has been reengineered in a way
which now makes it completely unusable and therefore the chances to pull
that discussion out of the haystack are dim. Anyhow, poking in the fog
doesn't lead anywhere. "I think from a letter to Dedekind", "I do not know
the context" - this kind of discussion is ridiculouus.
Ralf
MoeBlee
Why aren't you able?
More importantly, notice, you SKIPPED the MOST SUBSTANTIVE part of my
argument - about finite transactions - and answered only the the very
last little bit about your being welcome to advance an alternative to
set theoy.
> > The question for you is: Are you interested in understanding that or
> > are you vastly more interested in continuing to harrump that set
> > theory is "wrong"?
>
> Read chapter 7 of my book in oreder to judge about what I understand
> by set theory and how I understand it.
I read your misconceptions as you post them in these threads. Have you
ever read a set theory textbook and followed the proofs from theorem
to theorem? (I don't mean Cantor, which is pre-axiomatic, or a book
that discusses set theory from a general persepective such as the
Fraenkel, Bar-Hillel, and Levy book.)
> I am sure to know what you
> understand by it,
You are sure to know what I understand by set theory? I doubt that.
> but I am not sure that that is the correct way to
> think.
Whether set theory is "the correct way to think", it's clear you don't
understand it.
MoeBlee
MoeBlee
> On 21 Feb., 22:07, "MoeBlee" <jazzm...@hotmail.com> wrote:
>
> > Why don't you think about Robinson's mathematics and philosophy a bit
> > more so that you won't be so inclined to miss and snip the TOTALITY of
> > what he's saying rather than use him (as you've used other
> > mathematicians) as mere fodder for quotes (taken out of context and
> > misunderstood by you)
>
> That is what I mostly like and what is typical for set theorists. One
> can express ones opinions as precisely as possible, they will say that
> the opposite is meant.
Who says the opposite is meant?
> Skolem held set theory as wrog.
Months ago, I posted to you about Skolem's point by point critique.
You replied weakly about a point or two but mostly ignored what I
said.
> Banch and Taski tried to dismiss AC.
Source (other than Wikipedia's claim without a mentioned source)? And
Tarski spent a life working in set theory.
> Robinson stated as cannot be statet more clearly:
>
> Indeed, I think that there is a real need, in formalism and elsewhere,
> to link our understanding of mathematics with our understanding of the
> physical world.
>
> (i) Infinite totalities do not exist in any sense of the word (i.e.,
> either really or ideally). More precisely, any mention, or purported
> mention, of infinite totalities is, literally, meaningless. (ii)
> Nevertheless, we should continue the business of Mathematics 'as
> usual', i.e., we should act as if infinite totalities really
> existed."
Yes, after stating his rejection of a platonistic and/or a literal
view of infinity and even of the literal meaninglessness of the
notion, Robinson does endorse working in set theory, as he goes on to
explain in his essay.
> Hey, dreamer, wake up! "any mention, or purported mention, of infinite
> totalities is, literally, meaningless." Can you understand the meaning
> of these words, literally?
>
> I did not misunderstand them. You are sleepwalking.
I've posted in other threads a fair amount on the subject.
If there is a SPECIFIC thing I've said about infinity that you
consider to be incorrect, then you are welcome to quote me and state
your basis for disagreement. But just spewing empty rhetoric like
"dreamer" and "sleepwalking" is not a basis for a discussion on the
subject.
MoeBle
Virgil
muec...@rz.fh-augsburg.de wrote:
If it was all that secret, how did WM get hold of it?
Virgil
muec...@rz.fh-augsburg.de wrote:
As applied to WM's brainchildren, such criticism is apt.
Virgil
muec...@rz.fh-augsburg.de wrote:
As WM has still not discovered for himself the by now well known fact
that the set of all paths of the complete infinite binary tree is
uncountable, he is hardly in a position to criticize other's
discoveries.
Virgil
Alois Steindl <Alois....@tuwien.ac.at> wrote:
Agreed! And it might help to have both reviewed by some mathematicians
as well.
Virgil
muec...@rz.fh-augsburg.de wrote:
>
> The natural numbers are created step by step by the Peano axioms,
> i.e., by induction. And every application has to take this into
> account.
If WM's claim were true, one would need as many axioms as there are
naturals. In fact the inductive axiom does in one step what WM claims
must be done in infinitely many separate steps.
Doing it WM's way would prohibit it ever being done at all, which is
what he wants. But, fortunately, WM has not the power to impose his will
on everyone, only on his poor hag-ridden students..
> Nothing but wrong belief.
That's WM all over, full of nothing but wrong belief.
> >
> > All replacements can be done in parallel.
>
> Nothing but wrong belief. Absolutely unjustified.
According to the Peano postulates, absolutely justified. So it is WM who
is exhibiting all those wrong beliefs.
> >
> > You need not start anywhere, you can start with all of them at once.
>
> If you could, you would see that there are not infinitely many.
What WM claims to be able to see, or not see, does not affect the vision
of those not wearing his blinders.
Virgil
muec...@rz.fh-augsburg.de wrote:
> On 21 Feb., 21:50, "MoeBlee" <jazzm...@hotmail.com> wrote:
>
> > ZFC is not a theory of physics. I don't know much about the Tarski-
> > Banach theorem, but I have not read that it entails anything contrary
> > to physics. As I understand, the theorem does not render that a
> > PHYSICAL object may be perform in the way the ABSTRACT object in the
> > theorem performs.
>
> It is about geometry. Geometry is about physics.
Only to physicists. To psychologists geometry is about psychology, and
so on for each area of study.
> If the dissection as
> proven possible by Banach and Tarski (but, of course, not concretely
> defined --- see well ordering of the reals) could be applied in
> geometry, then it could be applied in physics.
If geometry were only a part of physics then that disection of physical
spheres would be a reality, but the fact that it is not a reality means
that geometry is not a subsidiary of physics.
> Obviously this result
> of set theory contradicts set theory
On the contrary, it merely contradicts WM's dicta.
>
> Sorry, I am neither able nor willing to answer every contribution in
> detail. Therefore I must snip much. But if you have a serious request,
> please repeat it. As far as I remember you wanted to do some proofs
> concerning the binary tree, which from my diskussion with Dik should
> be clear to anybody.
Except that Dik is right and WM, as usual, is wrong in those discussions.
>
> > And I notice that is a habit you have of brushing off from among
> > the more obviously solid points of rebuttal to you. Of course, I
> > understand that no one can be expected to answer every line of every
> > post.
>
>
> In particular if he is in a position as I am here.
>
> > But I do note that I've written quite a bit to you, with a lot
> > of tissue of explanation, yet you demur from following up on those
> > substantive matters as, what seems to me, to be a way of avoiding an
> > engagement of ideas that would lead you into considerations beyond
> > your program of denunciations of set theory.
>
> In ordinary mathematics it is sufficient to show one contradiction.
> There are many. To say that, is not a denunciation.
When we show contradictions in WM's arguments, he ignores them, and
merely repeats the false arguments, but when WM makes an argument, he
expects the world to take note and bow down, however faulty that
argument proves to be.
>
> We were discussing the binary tree. Please continue. Or whatever you
> want to discuss in detail.
The complete infinite binary tree has one separate path for each
infinite binary sequence, of which there are uncountably many.
If WM claims the set of all of them countable, let him show a count of
them.
Virgil
muec...@rz.fh-augsburg.de wrote:
Because they are in question. Whatever Robinson's aside against
infinities said , his whole work is based on exactly the opposite of
that aside, so that to ignore the whole in favor of that one remark is
intentionally misleading.
Virgil
muec...@rz.fh-augsburg.de wrote:
WM seems excessively axious to ignore (ii) in his anxiety to see (i).
According to Robinson, we should ignore (i), however "true" it may be.
According to WM, we must cleave to (i), however costly to mathematics it
may turn out to be.
While I do not agree with Robinson about (i), I am in agreement with
Robinson that (regardless of (i)'s validity) we should ignore it.
muec...@rz.fh-augsburg.de
In twenty-one (and last) the authors tries to prove the invalidity
in
Hessenberg's argument that there is no surjection from a set to
its
powerset (in this case exemplified with the set of natural
numbers).
This is actually a reformulation of an earlier argument by Cantor
(see above). The argument is that given a mapping f: S -> P(S),
we
can define:
T = {x | x in S and x not in f(x)}
and obviously T can not be in the image of f (I wonder why the
author
calls the set N_k, with subscript k, the k comes out of thin air).
WM:
k is the number which is to be mapped on the set N_k.
Das ist jenes Element N_k der Potenzmenge P(N) der natürlichen Zahlen
N, das seinen "Namen", d. h. seine Nummer k, nicht selbst enthält,
falls es sie enthält, und sie enthält, falls es sie nicht enthält.
In the example given later, k = 1 is set. So it is needed as an index.
Because the presence of that set in the map is a requirement for a
surjection, f is not a surjection.
WM:
By the example given later we see that the set does not say anything
about a bijection. It simply is impredicatively defined, i.e., it is
not well defined and cannot be in any mapping, be it a surjection or
not.
As this is so for any map
f, there are no surjections from S to P(S). The author confuses
the
issue by starting with mappings from N to P(N), and argues that
(as
in the von Neumann model every integer is a set) it is also a
question
whether k itself is in the image.
WM: No word about von Neumann here. What book did you read?
I wonder what that 'k' is, because
his set N_k depends in no way on a specific integer.
He sees it as
a paradox because a triple (N_k, k, f) does not exist. It still
leaves me wondering.
WM: Yes, obviously you did not yet understand it. Please read it again
or read the explanation gien above.
If the set is subscripted I would subscript it
by f (because it depends on f),
WM: The mapping depends on every element of the mapping. Correct. This
is an impredicative definition. But in order to distinguuish a special
set under the mapping f: N --> P(N), it is not useful to index it as
f.
so I get a triple (N_f, k, f) which
does not exist for any k in N. That is not a paradox, because N_f
can be in the image of other mappings, and so a triple (N_f, k, g)
can exist.
WM: But this triple has not the deserved impredicative definition. By
the way: By means of two mappings, f and g, there can be a common
surjection from F: N and, say g: {a,b,c} onto P(N). At least
Hessenberg's proof would fail to contradict it.
But the triple itself is non-existing and that argues
that so f is not a surjection, because in a surjection such a
triple
must exist.
WM: The set of all sets which do not contain themselves does no exist.
This is the mapping of all mappings which map onto sets which do not
contain their pre-image.
Next the author states that the existence of such a triple is not
excluded by the requirement of surjectivity.
WM: i.e., the triple is not excluded by the fact that there are more
elements in the range than in the domain of the mapping. How can one
so much misunderstand this plain text?
That is obviously
true, because surjectivity *requires* the existence of such a
triple.
He tries to clarify this with the maps from
{1, a} to {{}, {1}}
and shows that there are two possible mappings and that the triple
does not exist there either. But this is a red herring,
WM: Why? Here it is clearly demonstrated that, irrespective of
cardinality, this impredicatively defined set is not existing.
the triple
is only relevant in the case of mapping a set to its powerset, and
so the requirement for existence by surjectivity is only valid in
such a context.
Wow, twenty-one arguments and all show a basic misunderstanding by
the author.
WM: because of a basic misunderstanding of the reviewer.
Regards, WM
muec...@rz.fh-augsburg.de
>
> Because after, in my reply to you, I had mentioned the CRUCIAL
> context, you CONTINUED to post by OMITTING that context. The way you
> present Robinson's view of infinity is a distortion since you leave
> out the CONCLUSION he comes to about it (which conclusion puts his
> view in a RADICALLY different light).
That is not a conclusion but pure resignation.
Regards, WM
Franziska Neugebauer
> REPLY TO THE NONSECRET REVIEW OF CHAPTER 9 (PART 4 of the Reply)
[...]
> Twenty shows a common fallacy. The computable numbers are
> countable,
> so they can be put in a list, we can compute the diagonal, so that
> is
> a computable number not in the list, showing that there is a
> paradox.
> The problem here is that it can be shown that
>
> WM:
> that the reviewer coflates computable and constructable.
Are the author and the reviewer talking at cross purposes? I quote from
WolframMathWorld, please note the "i" in "constructible".
Computable Number
A number which can be computed to any number of digits desired by a
Turing machine. Surprisingly, most irrationals are not computable
numbers!
Constructible Number
A number which can be represented by a finite number of additions,
subtractions, multiplications, divisions, and finite square root
extractions of integers. Such numbers correspond to line segments
which can be constructed using only straightedge and compass.
All rational numbers are constructible, and all constructible numbers
are algebraic numbers (Courant and Robbins 1996, p. 133). If a cubic
equation with rational coefficients has no rational root, then none of
its roots is constructible (Courant and Robbins, p. 136).
> Leaving alone that it conflates constructable with computable, the
> answer is no. It is only yes when all numbers in the list are
> computable.
>
> Please do not conflate constructable with computable. A constructable
> number is a number which can be constructed like pi or Cantor's
> diagonal.
According to the definition in MathWorld pi is *not* a constructible
number at all. Do you have any contemporary mathematical reference
supporting your use of the word "constructable"?
F. N.
--
xyz