perfect powers !!
amy666
its the next logical step after prime numbers.
who wants to investigate perfect powers ?
answers like
x^1/2 + x^1/3 + x^1/4 + ...
or
2log(x) + 3log(x) + ...
are not very good.
we need to do better.
regards
tommy1729
amy666
and yes , i know the goldbach-euler theorem ...
not very helpfull ...
Dave L. Renfro
> the distribution of perfect powers is not considered enough !
>
> its the next logical step after prime numbers.
>
> who wants to investigate perfect powers ?
Instead, how about investigating the following?
Any countable set of real numbers has measure zero:
Given a countable set {x_1, x_2, ...} and epsilon > 0,
there exists a collection of open intervals whose
union contains the countable set and whose lengths
have a total sum of less than epsilon -- just choose
an open interval of diameter epsilon/(2^n) that contains
x_n, for each n.
Also, there exist uncountable sets with measure zero,
such as the Cantor middle thirds set.
Suppose we strengthen the condition of having measure
zero by changing "... whose lengths have a total ..."
to "... whose (lengths)^(1/3) have a total ...". Then
every countable set will still have measure zero in
this stronger form, but the Cantor middle thirds
set no longer has measure zero in this stronger form.
However, an easy modification of the Cantor middle thirds
set gives an uncountable set that has measure zero in
this stronger form.
Both results continue to hold if we further strengthen
the notion of "measure zero" to using (lengths)^(1/17),
or for any uniformly applied root. If the intervals involved
are denoted by J_1, J_2, ... and their lengths denoted by
|J_1|, |J_2|, ..., then both results even continue to hold
if we consider the sum |J_1|^(1/1) + |J_2|^(1/2) + ...
+ |J_n|^(1/n) + ...
In fact, given any strictly increasing sequence N_1, N_2, ...
of positive integers, then both results continue to hold
if we consider the sum |J_1|^(1/N_1) + |J_2|^(1/N_2) + ...
+ |J_n|^(1/N_n) + ...
At this point, we notice that any countable set satisfies
all these notions of "measure zero" simultaneously. However,
we have to continue adjusting the Cantor set construction
method to obtain an uncountable set that satisfies each specified
version of these "measure zero" notions. Does there exist an
uncountable set that simultaneously satisfies all these notions
of "measure zero"? If not, this would provide a kind of
"measure theoretic" characterization for the collection of
countable subsets of the reals.
Dave L. Renfro
P.S. For others who know, yes I know something about this,
although my understanding isn't all that "strong".
Dave L. Renfro
> Does there exist an uncountable set that simultaneously
> satisfies all these notions of "measure zero"? If not,
> this would provide a kind of "measure theoretic"
> characterization for the collection of countable
> subsets of the reals.
It's been almost 24 hours since I asked this, so in case
someone later stumbles on my post and is interested, I thought
I'd give the answer. Sci.math's set theory enthusiasts
probably already know this, but the property I described
is a minor restatement of what it means to be a "strong
measure zero" set. Borel conjectured in 1919 (see the
second full paragraph on p. 123 of [1]) that every strong
measure zero set is countable. It's not difficult to prove
that the continuum hypothesis implies there exists an
uncountable strong measure zero set and, in fact, the
existence of such a set also follows from the weaker
assumption of Martin's axiom. On the other hand, Laver [2]
showed that it is consistent with ZFC that no uncountable
strong measure zero sets exist.
[1] Émile Borel, "Sur la classification des ensembles
de mesure nulls" [On the classification of measure
zero sets], Bulletin de la Société Mathématique
de France 47 (1919) 97-125.
http://www.numdam.org/numdam-bin/item?id=BSMF_1919__47__97_0
[2] Richard Laver, "On the consistency of Borel’s conjecture",
Acta Mathematica 137 (1976), 151-169.
Dave L. Renfro
amy666
typical.
no hard conclusions.
despite 2 proofs.
horrible !
>
> [1] Émile Borel, "Sur la classification des ensembles
> de mesure nulls" [On the classification of
> n of measure
> zero sets], Bulletin de la Société Mathématique
> de France 47 (1919) 97-125.
> http://www.numdam.org/numdam-bin/item?id=BSMF_1919__47
> __97_0
>
> [2] Richard Laver, "On the consistency of Borel’s
> conjecture",
> Acta Mathematica 137 (1976), 151-169.
>
> Dave L. Renfro
and btw still irrelevant to the thread.
drifting of towards subjects like set theory.
also
typical
sigh
tommy1729
David C. Ullrich
<renf...@cmich.edu> wrote:
>Dave L. Renfro wrote (in part):
>
>> Does there exist an uncountable set that simultaneously
>> satisfies all these notions of "measure zero"? If not,
>> this would provide a kind of "measure theoretic"
>> characterization for the collection of countable
>> subsets of the reals.
>
>It's been almost 24 hours since I asked this, so in case
>someone later stumbles on my post and is interested, I thought
>I'd give the answer. Sci.math's set theory enthusiasts
>probably already know this, but the property I described
>is a minor restatement of what it means to be a "strong
>measure zero" set. Borel conjectured in 1919 (see the
>second full paragraph on p. 123 of [1]) that every strong
>measure zero set is countable. It's not difficult to prove
>that the continuum hypothesis implies there exists an
>uncountable strong measure zero set
Heh. This is all new to me - thinking about it yesterday
I constructed a proof that there does not exist an
uncountable strong measure zero set. Then I realized
that my "proof" _implied_ CH...
(Started with the plausible but false fact that if E is
uncountable then there exist two disjoint intervals
I and J such that E intersect I and E intersect J
are both uncountable. Hence E contains a Cantor
set...)
>and, in fact, the
>existence of such a set also follows from the weaker
>assumption of Martin's axiom. On the other hand, Laver [2]
>showed that it is consistent with ZFC that no uncountable
>strong measure zero sets exist.
Thanks - I don't feel quite so dumb for not being
able to settle the question.
>[1] Émile Borel, "Sur la classification des ensembles
> de mesure nulls" [On the classification of measure
> zero sets], Bulletin de la Société Mathématique
> de France 47 (1919) 97-125.
>http://www.numdam.org/numdam-bin/item?id=BSMF_1919__47__97_0
>
>[2] Richard Laver, "On the consistency of Borel’s conjecture",
> Acta Mathematica 137 (1976), 151-169.
>
>Dave L. Renfro
David C. Ullrich
"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
David C. Ullrich
wrote:
Horrible? Things are the way they are, whether
you like it or not.
>
>>
>> [1] Émile Borel, "Sur la classification des ensembles
>> de mesure nulls" [On the classification of
>> n of measure
>> zero sets], Bulletin de la Société Mathématique
>> de France 47 (1919) 97-125.
>> http://www.numdam.org/numdam-bin/item?id=BSMF_1919__47
>> __97_0
>>
>> [2] Richard Laver, "On the consistency of Borel’s
>> conjecture",
>> Acta Mathematica 137 (1976), 151-169.
>>
>> Dave L. Renfro
>
>and btw still irrelevant to the thread.
I think a certain (non-mathematical) aspect
of Renfro's post was over your head...
>drifting of towards subjects like set theory.
>
>also
>
>typical
>
>
>sigh
>
>tommy1729
David C. Ullrich
Toni...@yahoo.com
I've heard some serious mathematicians are pondering to stop doing
actual mathematics and begin to take seriously all your stupid,
childish and nonsensical questions and inquiries, so that things will
stop to look "horrible" to you.
They'll force themselves into understanding not-properly-defined
stupidities like your last inquiries about cycles, "relations" and
other nonsenses in the complex plane, and they'll will automatically
consider interesting anything you do, like tetration or perfect powers
or, say dolls dressing...
So, please, be patient and wait a little for mathematicians to comply
with your last whinning and whims.
In the meantime, grab a chair so that you won't get too tired.
Regards
Tonio
amy666
there is nothing wrong with
tetration
perfect powers
complex plane
>
> So, please, be patient and wait a little for
> mathematicians to comply
> with your last whinning and whims.
>
> In the meantime, grab a chair so that you won't get
> too tired.
>
> Regards
> Tonio
bla bla bla
still irrelevant to the OP.
its easier to get personal than to do math isnt it ?
Toni...@yahoo.com
Nobody even came close to insinuate there is, and your very writing
that proves you're more stupid than what some here thought, or else
your reading understanding is more impaired than what I thought it
was.
The point is: people may not find interesting what you do, and you get
whinny and bratty when you don't get at once a response to some of
your questions which, by the way, many times are nonsensical (pay
attention here: your questions are nonsensical, not the subject they
deal about. For example, in your last nonsense you ask about
"relations" , "limit cylces" in the complex plane and lots of other
non-defined stuff that, PERHAPS, you know what it means, but most
other mortals don't...and this is NOT the first (nor the 2nd, the
3rd., etc.) time you do that.
This last time though, I suspect that you wanted to ask something
(what I can't yet say) about "circles" and not cycles, but you're
unable to understand this even after being said so...
And about doing maths: do you honestly, REALLY, thing that is what you
do with all these nonsenses? Common...:)
Regards
Tonio
>
> > So, please, be patient and wait a little for
> > mathematicians to comply
> > with your last whinning and whims.
>
> > In the meantime, grab a chair so that you won't get
> > too tired.
>
> > Regards
> > Tonio
>
> bla bla bla
>
> still irrelevant to the OP.
>
> its easier to get personal than to do math isnt it ?-
David Bernier
Perhaps you could comment on the following that
relates to descriptive set theory and Lebesgue
measurability.
I browsed through the book:
``The Princeton Companion to Mathematics"
by Timothy Gowers, June Barrow-Green and Imre Leader.
(Princeton University Press, 2008).
If one enters into "Google the web" :
coanalytic sets measurable cardinal
the first hit now is from that book, on page 631 from
Part IV, section 22 . It's a survey of set theory,
written by Joan Bagaria.
On pages 628 and 629, measurable cardinals, and what follows from
assuming that a measurable cardinal exists, are discussed.
I expect you know what is meant by Sigma-1-2 sets; the author
describes these as: "continuous images of coanalytic sets".
She goes on to say that Solovay showed in 1969 that if there is
a measurable cardinal, then all Sigma-1-2 sets of reals are
Lebesgue measurable.
(and if V=L , there are Sigma-1-2 sets [ in any R^n ?] that are not
Lebesgue measurable. )
I'm not very familiar with descriptive set theory, and the mini-chapter
is a brief survey.
I thought that a Lebesgue measurability result that follows from
ZFC + (exists a measurable cardinal), but not from ZFC,
was quite interesting.
David Bernier
Dave L. Renfro
> Perhaps you could comment on the following that
> relates to descriptive set theory and Lebesgue
> measurability.
>
> I browsed through the book:
> "The Princeton Companion to Mathematics"
> by Timothy Gowers, June Barrow-Green and Imre Leader.
> (Princeton University Press, 2008).
>
> If one enters into "Google the web" :
> coanalytic sets measurable cardinal
> the first hit now is from that book, on page 631 from
> Part IV, section 22 . It's a survey of set theory,
> written by Joan Bagaria.
>
> On pages 628 and 629, measurable cardinals, and what follows
> from assuming that a measurable cardinal exists, are discussed.
>
> I expect you know what is meant by Sigma-1-2 sets; the author
> decribes these as: "continuous images of coanalytic sets".
>
> She goes on to say that Solovay showed in 1969 that if there is
> a measurable cardinal, then all Sigma-1-2 sets of reals are
> Lebesgue measurable.
>
> (and if V=L , there are Sigma-1-2 sets [ in any R^n ?] that
> are not Lbesgue measurable. )
>
> I'm not very familiar with descriptive set theory, and the
> mini-chapter is a brief survey.
>
> I thought that a Lebesgue measurability result that follows
> from ZFC + (exists a measurable cardinal), but not from ZFC,
> was quite interesting.
I'm by no means an expert in this area, but I have bumped
into this subject a lot and I've spent a lot of time (and plan
to do much more in future years) researching the early history
of descriptive set theory. One of my interests has been that
the Borel/projective classification of a set is often useful
in characterizing (or at least describing some of) its
properties in certain settings (for example, the sets that
can arise as those points where a function f:interval --> R
is locally unbounded are exactly the sets that are both
countable and G_delta). Thus, for me these classifications
are like cardinality, density (or nowhere density), Lebesgue
measure, etc. -- they are very useful for describing sets of
reals and sets in more general spaces. This is also how the
subject originally developed, but very quickly (as in around
1917-1930) quite a number of seemingly fundamental questions
remained unsolved, despite the best attempts of Lusin [= Luzin],
Sierpinski, and others during this period. In fact, Lusin and
Sierpinski pretty much solved just about every natural problem
(and many that were fairly contrived) that one can think of,
and as a testiment to their mathematical power, I think it's
fair to say that for all intents and purposes the only problems
anyone came up with that they were not able to solve turned
out to be independent of ZFC.
Since you can read French, the best place to go for the
classical results is Lusin's 1930 book "Lecons sur les Ensembles
Analytiques et leurs Applications". Unfortunately, this isn't
freely available anywhere on the internet that I know of
(I haven't looked in the past year, however), but maybe
you can find it somewhere anyway. Before Lusin wrote this
book, he published a survey paper in Fundamenta Mathematica
(which I think earlier appeared in a Russian journal in
Russian), and this survey paper from 1927 _is_ freely
available on the internet:
N. Lusin, "Sur les ensembles analytiques", Fund. Math., 1927.
http://matwbn.icm.edu.pl/tresc.php?wyd=1&tom=10
http://matwbn.icm.edu.pl/ksiazki/fm/fm10/fm1011.pdf
Also, Sierpinski's general topology book (recently
re-issued by Dover) contains a lot of useful material
at a relatively pedestrian level (as compared to, say,
Kechris's "Descriptive Set Theory" text), and the second
half of Volume 1 of Kuratowski's 1966 (English version)
"Topology" is probably the most complete account anywhere
of the classical material. By the way, by "classical",
I mean the subject before the logicians got ahold of
things and transformed it to something seemingly far
removed from classical real analysis and point set theory
(but in reality not, at least I don't think so as long
as you learn their language, which I haven't).
I once posted a summary of some results related to the
continuum hypothesis and descriptive set theory, which
might be of interest to you:
sci.math -- "How bad can 2^(Aleph_0) be?" history question
(Dave L. Renfro, 5 June 2001)
http://groups.google.com/group/sci.math/msg/82a5ec796aeb1073
Dave L. Renfro
David Bernier
This is all very interesting, so thanks.
There's a two-part survey on independence results
by Harvey Friedman entitled (for part I):
``WHAT YOU CANNOT PROVE 1: before 2000"
Page 8 mentions Borel’s Conjecture (1919):
BC: Strongly measure zero is equivalent to countable.
Richard Laver is mentioned for showing that
Borel's Conjecture is independent of ZFC.
If I'm not mistaken, Borel's Conjecture was the
main topic of your earlier post in this thread.
Section 4 is about the projective hierarchy.
The analytic sets (A), co-analytic sets (CA)
and projective co-analytic sets (PCA) are
clearly explained starting from the
Borel sets in R, R^2, R^3, .... R^n , ... n in N.
The "desire" to "settle" questions on
Lebesgue-measurability and determinacy about
all sets in the projective hierarchy in a way
that is pleasant to set theorists (note:
Friedman and others don't think V = L
is a pleasant axiom to add to ZFC)
motivates considering/adding:
“the existence of Woodin cardinals”
as an axiom, to "settle" the questions.
Friedman writes:
<< Measurable cardinals will suffice for
questions about A, CA, and PCA
(sometimes CPCA), but not higher. >>
Cf.:
http://www.math.ohio-state.edu/~friedman/pdf/CCRTalk1.121905.pdf
David Bernier
amy666
you just did !!
those were the subjects i investigated and you rediculized.