the tommy1729 website
master1729
http://sites.google.com/site/tommy1729/home
enjoy.
master1729
warning : controversial content. not your average website.
Zop Pitting Me
"master1729" <tomm...@gmail.com> wrote in message
news:1249914151.17370.12857...@gallium.mathforum.org...
Learn how to put spaces between adjacent words, you fucking imbecile.
Zop Pitting Me
"master1729" <tomm...@gmail.com> wrote in message
news:1249914151.17370.12857...@gallium.mathforum.org...
Learn how to spell 'basically' you fucking idiot.
Zop Pitting Me
"master1729" <tomm...@gmail.com> wrote in message
news:1249914151.17370.12857...@gallium.mathforum.org...
Don't put a comma after a space, you blithering moron.
hagman
Re "debunkiung nonmeasurable sets"
IF you allow infinitesimal numbers, e.g. Robinson's hyperreals R^*
(constructed using - d'oh! - the axiom of choice) you should also
build up a measure theory of R^* instead of R.
Perform the Vitali construction starting from R^* instead of R.
What do you get?
Also, are you aware that the Vitali sets alrady carry in them the
Banach Tarsky problem?
Let B_{2k} = A_k and B_{2k+1} = (A_k translated by 1).
Then (assuming measurabilty of Vitali sets) mu(B_k) = mu(A_1) and mu(U
B_k) = 2.
In your arithmetic it follows that 1 = oo * mu(A_1) = 2
Jesse F. Hughes
Yeah, certainly not your average website.
,----
| andrica proved !
|
| yes i proved andrica.
|
| in fact a statement that is a bit stronger.
|
| i understand the primes better than anyone now.
`----
You must be very proud.
--
Jesse F. Hughes | "There's no other star but one star
| and you want it to make light,
| but it's not making light."
| -- A blues tune by Quincy P. Hughes
master1729
> On 29 Sep., 16:18, master1729 <tommy1...@gmail.com>
> wrote:
> > the tommy1729 website
> >
> > http://sites.google.com/site/tommy1729/home
> >
> > enjoy.
>
> Re "debunkiung nonmeasurable sets"
> IF you allow infinitesimal numbers, e.g. Robinson's
> hyperreals R^*
> (constructed using - d'oh! - the axiom of choice) you
> should also
> build up a measure theory of R^* instead of R.
we have infinitesimal measures.
>
> Perform the Vitali construction starting from R^*
> instead of R.
> What do you get?
>
> Also, are you aware that the Vitali sets alrady carry
> in them the
> Banach Tarsky problem?
yes im aware of that.
> Let B_{2k} = A_k and B_{2k+1} = (A_k translated by
> 1).
> Then (assuming measurabilty of Vitali sets) mu(B_k) =
> mu(A_1) and mu(U
> B_k) = 2.
> In your arithmetic it follows that 1 = oo * mu(A_1)
> = 2
not quite , we get 1 = 1/2 + 1/2 = 1 !
thanks for your reply.
tommy1729
master1729
Transfer Principle
> Re "debunkiung nonmeasurable sets"
> IF you allow infinitesimal numbers, e.g. Robinson's hyperreals R^*
> (constructed using - d'oh! - the axiom of choice) you should also
> build up a measure theory of R^* instead of R.
Ah yes, tommy1729's nonmeasurability page.
Both hagman and tommy1729 are discussing the Vitali paradox, so
there exist countably infinitely many congruent disjoint sets
whose union is the unit interval. The sets can't be null since
the measure of the countable union of null sets is null, and the
sets can't have positive measure, since the measure of the
countable union of sets with the same positive measure must
necessarily be infinite. Thus, ZFC concludes that these sets
must be non-Lebesgue measurable.
So tommy1729's solution would be to assign an infinitesimal
measure to each set. But then hagman points out that if we're
assigning infinitesimal measures, then the sets which we are
measuring ought to contain infinitesimal elements as well.
> Perform the Vitali construction starting from R^* instead of R.
> What do you get?
This is a tricky question. The standard Vitali construction
consists of choosing from elements of R/Q (i.e., the quotient
group R mod Q). Now hagman tells us to replace standard R with
R^*, but what about Q? Shouldn't we have to replace Q with Q^*,
often known as the hyperrationals?
But here's the kicker -- as it turns out, there actually exist
_uncountably_ many hyperrationals! (Indeed, there actually exist
uncountably many hyper_naturals_!) And since Lebesgue measure is
only countably additive, we can simply assign each Vitali set the
Lebesgue measure zero without any contradictions.
Of course, I have a feeling hagman meant R^*/Q -- i.e, using the
hyperreals mod the _standard_ rationals. So he's asking for the
infinitesimal that tommy1729 would assign to these sets.
Let's see. There are countably infinitely many (i.e., aleph_0)
sets, and their measures must add up to 1. So the necessary
infinitesimal would be something like 1/aleph_0. And therein lies
the problem -- 1/aleph_0 is not a hyperreal. The hyperreals are
not set up so that one can add up countably infinitely many
infinitesimals to obtain 1.
Of course, all this proves is that tommy1729 never had the
hyperreals in mind when he mentions infinitesimals. Then again,
we should've already known this, since tommy1729 rejects AC, and
hagman already pointed out how AC is necessary to construct the
hyperreals in the first place.
So how should our 1/aleph_0 work? For example, one might try to
add 1/aleph_0 to itself, which would give us the measure of the
union of two Vitali sets. One might think that it makes sense
to have 1/aleph_0 + 1/aleph_0 = 2/aleph_0, but then again, maybe
the sum should be 1/aleph_0, to parallel the known cardinal
addition fact aleph_0 + aleph_0 = aleph_0.
Furthermore, what should 1 + 1/aleph_0 be? On one hand, perhaps
it should equal 1, to parallel aleph_0 + 1 = aleph_0, but then
again, the disjoint union of an interval of unit length and a
Vitali set is still nonmeasurable.
Finally, what exactly is [0,1/aleph_0]? It appears to be an
interval of measure 1/aleph_0. But then what values, if any,
lie strictly between zero and 1/aleph_0? If there are none, then
[0,1/aleph_0] is a finite set with two elements, and so its
measure ought to be zero.
So many interesting questions. But I do find the idea of trying
to assign infinitesimal measures to Vitali sets worth thinking
about further. And so I will continue to consider answers to
these above questions.
Dan Cass
master1729
> What is "andrica"???
quite a famous problem actually !?
i assumed everyone knew.
http://www.wolframalpha.com/entities/famous_math_problems/andrica's_conjecture/lr/4d/sy/
btw , i looked at all other 'proofs' found via google and sci.math , all of them are wrong.
as far as i know im the only one who knows a proof.
tommy1729
master1729
the irony is that when i started posting about infinitesimals on the continuum between 0 and 1 - they imho connect the reals ( points ) to form the continuum - , the reply was :
there are no infinitesimals ( or hyperreals ) on my real number line.
- then what is on your real line / continuum ? -
reply : on my real line are only reals. duh !
now measure is defined on " the real line " .
but now since I STARTED using infinitesimal measure ,
!!suddenly!! there is need to assign measure to the infinitesimals on the real line / continuum.
thats weird. suddenly the real line / continuum changed , and just to try to prove me wrong ...
seems the real line is time-dependend and has a personality , the personality of my critics.
my critics seem to have special powers , like mathekinetics or such.
--- thats one thing ---
** second **
you cannot just ADD or REDEFINE the continuum or real line and expect everything else to be the same.
depending on what you do it might affect ZFC or whatever you are using / you accept.
it might effect cardinals or ordinals , or whatever you are using / you accept.
" third "
i see no paradox by adding / accepting / defining or redefining infinitesimals to be on the real line .. with my measure.
indeed i do not use hyperreals , but put all my money on newtons infinitesimals aka classic (!) calculus and analysis.
the irony ( and paradox ) is here again that many claim that " the subject parts of maths actually form 1 big whole subject , all fields are related ! " but classic calculus and modern set theory analysis do not ??!?
they contradict eachother , basicly telling people to FORGET about the other ?
thats not only paradoxal and ironic but also inconsistant and confusing for students and giving a bad name to math in general to wise logical students ...
i wonder if you expected this kind of reply.
sorry if its not exactly what you wanted to hear :)
regards
tommy1729
The Pumpster
> Dan Cass wrote :
>
> > What is "andrica"???
>
> quite a famous problem actually !?
>
> i assumed everyone knew.
>
>
> btw , i looked at all other 'proofs' found via google and sci.math , all of them are wrong.
>
> as far as i know im the only one who knows a proof.
>
> tommy1729
I have certainly not seen your "proof" and it is more
than a little unkind of me to say so, but I would wager
anything you are prepared to risk that it is incorrect...
de P
Jesse F. Hughes
Man, are *you* a sucker!
Don't you realize that Tommy understands the primes better than anyone
else? I saw it on the web.
--
"So now you see a math person coming out to talk about *his* program
which is fast as he says it can count over 89 billions primes in less
than a second. How is that objective? It's childish."
-- James S. Harris, on objective facts.
The Pumpster
But if Tommy understands the primes better than anyone else, how can
the same
be true of JSH, Inverse19 and good ole ArchieP? Most puzzling.....
de P
Transfer Principle
> the irony is that when i started posting about infinitesimals on the continuum between 0 and 1 - they imho connect the reals ( points ) to form the continuum - , the reply was :
> there are no infinitesimals ( or hyperreals ) on my real number line.
> - then what is on your real line / continuum ? -
> reply : on my real line are only reals. duh !
> now measure is defined on " the real line " .
> but now since I STARTED using infinitesimal measure ,
> !!suddenly!! there is need to assign measure to the infinitesimals on the real line / continuum.
> thats weird. suddenly the real line / continuum changed , and just to try to prove me wrong ...
> seems the real line is time-dependend and has a personality , the personality of my critics.
> my critics seem to have special powers , like mathekinetics or such.
It appears what they're doing is saying, if the reals don't include
(nonzero) infinitesimals, then one can't assign infinitesimals as
measures of sets of reals. But if one wants to include infinitesimal
measures, then one must include infinitesimals in the sets of reals
being measured.
Can there be infinitesimal measures? Do the sets being measured
contain infinitesimals. The answer must be "both" or "neither," but
tommy1729 is saying "the former but not the latter," and this is why
hagman objects.
> i see no paradox by adding / accepting / defining or redefining infinitesimals to be on the real line .. with my measure.
> indeed i do not use hyperreals , but put all my money on newtons infinitesimals aka classic (!) calculus and analysis.
Does tommy1729 mean something like "dx" and "dy" as the infinitesimals
of Newton? I agree -- it might be interesting to assign "dx" as the
measure of a Vitali set and proceed from there.
master1729
no , the choice to have infinitesimals on the real line is a matter of taste and/or independent of the measures on it.
i objected to the ' mathekinetics ' some people use immorally and inconsistantly for personal benefit or personal argument.
>
> > i see no paradox by adding / accepting / defining
> or redefining infinitesimals to be on the real line
> .. with my measure.
> > indeed i do not use hyperreals , but put all my
> money on newtons infinitesimals aka classic (!)
> calculus and analysis.
>
> Does tommy1729 mean something like "dx" and "dy" as
> the infinitesimals
> of Newton? I agree -- it might be interesting to
> assign "dx" as the
> measure of a Vitali set and proceed from there.
yes dx and dy :)
tommy1729
master1729
you got yourself a deal.
im not afraid to put money on it.
perhaps you are confused about some of my posts regarding prime twins or hardy-littlewood conjecture , but those were just arguments , never called them proofs ( unless retracted later maybe ).
the proof does not require the zeta function or complex numbers. so if you are hoping i made a mistake there , you will be dissapointed.
nevertheless the proof is subtle.
regards
tommy1729
The Pumpster
Well then post it or mail it to me and, assuming it is mathematics, I
will
indicate the (first) error for you for free. This is a one-time offer
-- act
now or be disappointed!
de P
David C. Ullrich
wrote:
I on the other hand object to poeple assuming that they can give an
example of whatever they like just by saying "infinitesimal".
Regardless of the silliness above, your amazing idea of making that
set measurable by saying its measure is an infinitesimal simply
doesn't work. Note this is independent of the fact that you have
a lot of infrastucture to create first in order to make your measures
with infinitesimals "consistant". It simply doesn't work:
We have countably many sets A_n, each of which must
have the same measure, if they have any measure at all.
And [0,1] is the union of the A_n. So they cannot be measurable.
If the measure was a real number m then m = 0 implies the measure
of [0,1] is 0, while m > 0 implies that the measure of [0,1] is
infinite. Saying that m is an infinitesimal doesn't magically fix this
problem.
Look. If m is infinitesimal then 2m, 3m, etc are all infinitesimal.
So each finite union A_1 union ... union A_n has infinitesimal
measure. In particular the measure of A_1 union ... union A_n
is less than 1/2. So taking the limit as n tends to inifinity
it follows that the measure of [0,1] is <= 1/2.
Similarly the measure of [0,1] is <= 1/3, etc.
master1729
i will probably regret replying to david ..sigh ..
>
> I on the other hand object to poeple assuming that
> they can give an
> example of whatever they like just by saying
> "infinitesimal".
i on the other hand object to people assuming that they can give an example of whatever they like just by saying " axiom of choice " or " zfc "
really that comment didnt make sense.
i didnt just say " infinitesimal " i explained it.
i justified it on my site and i clarified here that it is newtons classical infinitesimal.
>
> Regardless of the silliness above,
what silliness ? your silliness ?
your reply up till now started with 2 insults and nothing else , i dont know if you even realize that anymore.
> your amazing idea
thats better .. though you dont mean that.
> of making that
> set measurable by saying its measure is an
> infinitesimal simply
> doesn't work.
i gave a justification.
why doesnt it work ?
i guess you will stop insulting and try to clarify below.
> Note this is independent of the fact
> that you have
> a lot of infrastucture to create first in order to
> make your measures
> with infinitesimals "consistant". It simply doesn't
> work:
again you say : it doesnt work.
and still without explaining.
saying it more doesnt make your argument stronger.
well , not that you already gave an argument ... it doesnt make your opinion more convincing i meant.
>
> We have countably many sets A_n, each of which must
> have the same measure, if they have any measure at
> all.
> And [0,1] is the union of the A_n. So they cannot be
> measurable.
> If the measure was a real number m then m = 0 implies
> the measure
> of [0,1] is 0, while m > 0 implies that the measure
> of [0,1] is
> infinite. Saying that m is an infinitesimal doesn't
> magically fix this
> problem.
and again you repeat yourself !!
saying it doesnt " magically " work now instead of saying it doesnt work.
whats next , saying it doesnt " " mathematically " work ?
>
> Look. If m is infinitesimal then 2m, 3m, etc are all
> infinitesimal.
yes 2m and 3m are infinitesimal.
and in general for finite n - yes finite ! - , nm is infinitesimal.
thats just a property of infinitesimals ...
> So each finite union A_1 union ... union A_n has
> infinitesimal
> measure.
for finite n yes.
> In particular the measure of A_1 union ...
> union A_n
> is less than 1/2. So taking the limit as n tends to
> inifinity
> it follows that the measure of [0,1] is <= 1/2.
wrong. it holds only for finite n.
your limit is not justified , not even for infinitesimals.
look.
lim m -> oo of n/m is always smaller than 1/2 for finite n.
but if n becomes oo this no longer necc holds.
your use of limits , infinitesimals and infinity is inconsistant. ( with calculus )
> Similarly the measure of [0,1] is <= 1/3, etc.
only for finite n.
>
>
so what is your argument ? i fear that wrongly computed limit was your argument.
>
oh thats it ?
to further convince you that " for all integer n " , lim n -> oo and oo are not necc the same consider this :
1 is rational
1 + 1/2 is rational
1 + 1/2 + 1/6 is rational
1 + 1/2 + 1/6 + 1/4! is rational
.. is rational
e - 1 is transcendental !
think about it before you try another limit argument !
tommy1729
" in fact i admire him " galathaea
master1729
lol , im not giving away the proof.
so that you ( or one of your friends ) can post a proof of " your own " in a few years and dont mention me ?
that one-time offer is meant to make me decide quickly without thinking !?
im not stupid.
master1729
it isnt true for the others.
JSH is still strugling with 7 P(x) and 9 G(x) whatever P and G are.
inverse 19 is on med.
and archie never made such claims ...
why not add |-|erc to the list as well ?? lol
The Pumpster
Actually, there's a certain amount of evidence to contradict your
last statement. In any case, it matters little to me whether you
make your proof public or not; I have no time for such silliness.
You are welcome to maintain whatever private fantasy of mathematical
competence you wish. Just remember that your having claimed to prove
Andrica's conjecture is roughly analogous to posting a claim to have
run a 3 minute mile. Others will regard it with, shall we say, more
than a hint of skepticism.
All the best,
de P
Transfer Principle
> On Fri, 01 Oct 2010 16:04:30 EDT, master1729 <tommy1...@gmail.com>
> wrote:
> >no , the choice to have infinitesimals on the real line is a matter of taste and/or independent of the measures on it.
> >i objected to the ' mathekinetics ' some people use immorally and inconsistantly for personal benefit or personal argument.
> I on the other hand object to poeple assuming that they can give an
> example of whatever they like just by saying "infinitesimal".
This isn't the first time that Ullrich has posted something similar
to the above statement. He made a similar criticism of me more than
a year ago when I tried to use "infinitesimals." He implies that
known infinitesimals (i.e., Robinson's) don't work exactly the way
that tommy1729 and I want them to, and simply using the word
"infinitesimal" doesn't make them work the way we want them to.
The reality is that Ullrich prefers classical analysis and Lebesgue
measure to any sort of "infinitesimals." So he'll criticize any of
us who attempts to use "infinitesimals" anywhere that classical
analysis and Lebesgue measure will suffice.
> Regardless of the silliness above
In other words, Ullrich considers the use of any analysis other
than classical analysis and any measure other than Lebesgue measure
to be just plain "silliness."
There is, however, one line of Ullrich's with which I do agree:
> [Y]ou have
> a lot of infrastucture to create first in order to make your measures
> with infinitesimals "consistant" [sic].
Yes. Since Robinson's infinitesimals won't work, we do have to
create a lot of infrastructure to come up with infinitesimals that
do work the way we want them to. Indeed, I wouldn't mind devoting
the next several posts in this thread to exactly that -- creating
the infrastructure to make tommy1729's infinitesimals work.
But chances are high that by the time we've created all the
necessary infrastructure, Ullrich will have long stopped reading
the discussion at all. After all, he has no incentive to read
about all of our infrastructure. He doesn't object to the existence
of nonmeasurable sets, and so he has no problem with the standardly
used L-measure. Only posters who object to nonmeasurable sets, like
tommy1729, have an incentive to consider the infrastructre.
If Ullrich or anyone else has no problem with nonmeasurable sets,
then fine. Just allow those who do object to nonmeasurable sets to
figure out how to assign them measures without condemning the whole
idea as being "silly."
> Saying that m is an infinitesimal doesn't magically fix this
> problem.
Ullrich's use of the word "magically" brings to mind a similar
proposal -- the "magic bullets" of Fred Jeffries. Below is the
description of these magic bullets by Jeffries:
"So you are looking for the Magic Bullet? That entity so small that
any
finite number of them strung together is infinitesimal but infinitely
many together have a finite size? The solution to "since the sum from
1 to n of 1/n is 1, take the limit as n goes to infinity"? A uniform
distribution for the natural numbers? "
-- Fred Jeffries, 7th August, approximately 3PM Greenwich
So Jeffries's magic bullet is exactly what tommy1729 needs. And
so the measure of each Vitali set is exactly a magic bullet.
Interestingly enough, Jeffries proceeds:
"Maybe your theory can somehow ban non-initial infinite ordinals?"
And tommy1729's theory does exactly that. We already know that
tommy1729 is opposed to ordinals.
If Ullrich still believes that this is all "silly," I say that
it's no sillier than the idea of delta functions which take on
the value zero almost everywhere, yet integrate to one.
Transfer Principle
> David Ullrich wrote :
> > I on the other hand object to poeple assuming that
> > they can give an
> > example of whatever they like just by saying
> > "infinitesimal".
> i on the other hand object to people assuming that they can give an example of whatever they like just by saying " axiom of choice " or " zfc "
The fact that tommy1729 is opposed to both AC and nonmeasurable
sets raises another idea.
It's usually stated that the existence of a nonmeasurable set
depends on AC. So what would happen if we were to consider a
theory such as ZF+"all sets are measurable"? Presumably ~AC
would be provable in this theory.
If that doesn't work, then try Z+"all sets are measurable". In
one of these theories, we should be able to make every set
measurable without resorting to infinitesimals or magic bullets.
Of course, tommy1729 works in his own set theory TST, and not
a theory like Z+"all sets are measurable." Still, when trying to
make TST rigorous, Z+"all sets are measurable" might be a good
starting point.
Jesse F. Hughes
> On Oct 2, 10:58 am, David C. Ullrich <ullr...@math.okstate.edu> wrote:
>> On Fri, 01 Oct 2010 16:04:30 EDT, master1729 <tommy1...@gmail.com>
>> wrote:
>> >no , the choice to have infinitesimals on the real line is a matter of taste and/or independent of the measures on it.
>> >i objected to the ' mathekinetics ' some people use immorally and inconsistantly for personal benefit or personal argument.
>> I on the other hand object to poeple assuming that they can give an
>> example of whatever they like just by saying "infinitesimal".
>
> This isn't the first time that Ullrich has posted something similar
> to the above statement. He made a similar criticism of me more than
> a year ago when I tried to use "infinitesimals." He implies that
> known infinitesimals (i.e., Robinson's) don't work exactly the way
> that tommy1729 and I want them to, and simply using the word
> "infinitesimal" doesn't make them work the way we want them to.
>
> The reality is that Ullrich prefers classical analysis and Lebesgue
> measure to any sort of "infinitesimals." So he'll criticize any of
> us who attempts to use "infinitesimals" anywhere that classical
> analysis and Lebesgue measure will suffice.
It's real funny how your second paragraph is so totally unrelated to
your first.
But you're absolutely right. Tommy should have every right to pretend
that infinitesimals do just what he wants them to do, without fear of
criticism. David Ullrich is nothing better than a five letter word.
--
Jesse F. Hughes
"Well, I guess that's what a teacher from Oklahoma State University
considers proper as Ullrich has said it, and he is, in fact, a teacher
at Oklahoma State University." -- James S. Harris presents a syllogism
Jesse F. Hughes
> There is, however, one line of Ullrich's with which I do agree:
>
>> [Y]ou have
>> a lot of infrastucture to create first in order to make your measures
>> with infinitesimals "consistant" [sic].
>
> Yes. Since Robinson's infinitesimals won't work, we do have to
> create a lot of infrastructure to come up with infinitesimals that
> do work the way we want them to. Indeed, I wouldn't mind devoting
> the next several posts in this thread to exactly that -- creating
> the infrastructure to make tommy1729's infinitesimals work.
>
> But chances are high that by the time we've created all the
> necessary infrastructure, Ullrich will have long stopped reading
> the discussion at all.
So far, it seems that chances are high that by the time you give an
explicit theory satisfying any significant bit of Tommy's (or anyone
else's) theory, the universe will have long since ended.
But that's probably Ullrich's fault, too.
--
Jesse F. Hughes
"As you can see, I am unanimous in my opinion."
-- Anthony A. Aiya-Oba (Poeter/Philosopher)
OwlHoot
>
> [..]
>
> lol , im not giving away the proof.
>
> so that you ( or one of your friends ) can post a proof of " your own " in a few years and dont mention me ?
>
> that one-time offer is meant to make me decide quickly without thinking !?
>
> im not stupid.
Post it on viXra [ http://www.vixra.org ], if it actually exists
outside your addled head, and isn't as vacuous as your daft website.
Cheers
John Ramsden
OwlHoot
>
> so that you ( or one of your friends ) can post a proof of " your own " in a few years and dont mention me ?
>
> that one-time offer is meant to make me decide quickly without thinking !?
>
> im not stupid.
Post it on viXra [ http://www.vixra.org ], if it actually exists
OwlHoot
>
> so that you ( or one of your friends ) can post a proof of " your own " in a few years and dont mention me ?
>
> that one-time offer is meant to make me decide quickly without thinking !?
>
> im not stupid.
Post it on viXra [ http://www.vixra.org ], if it actually exists
David C. Ullrich
wrote:
If every member of an increasing sequence is < 1/2
then the limit, if it exists, is <= 1/2.
>look.
>
>lim m -> oo of n/m is always smaller than 1/2 for finite n.
>
>but if n becomes oo this no longer necc holds.
Huh? The limit of n/m as m tends to infinity has nothing
to do with this.
What's relevant is the limit of n*m as n tends to infinity.
That limit is 0 if m is 0, infinite if m is a positive real,
and (if it exists at all, which would depend on
details) infinitesimal if m is infinitesimal. No way it
equals 1.
>your use of limits , infinitesimals and infinity is inconsistant. ( with calculus )
>
>> Similarly the measure of [0,1] is <= 1/3, etc.
>
>only for finite n.
>
>>
>>
>
>so what is your argument ? i fear that wrongly computed limit was your argument.
>
>
>>
>
>oh thats it ?
>
>
>to further convince you that " for all integer n " , lim n -> oo and oo are not necc the same consider this :
>
>1 is rational
>
>1 + 1/2 is rational
>
>1 + 1/2 + 1/6 is rational
>
>1 + 1/2 + 1/6 + 1/4! is rational
>
>.. is rational
>
>e - 1 is transcendental !
Giggle. This has nothing to do with anything here.
>think about it before you try another limit argument !
Giggle. Can you even tell me what it _means_ to say
that L is the limit of a sequence a_n? Hint: Of course
that can happen with all the a_n rational and L
transcrendental. But it cannot happen with all the
a_n < 1/2 and L > 1/2. That's extremely easy to prove.
David C. Ullrich
Heh-heh. You're missing the point to that quote, which
is that there is no such magic bullet.
>And
>so the measure of each Vitali set is exactly a magic bullet.
>
>Interestingly enough, Jeffries proceeds:
>
>"Maybe your theory can somehow ban non-initial infinite ordinals?"
>
>And tommy1729's theory does exactly that. We already know that
>tommy1729 is opposed to ordinals.
Very curious that you snipped the last paragraph,
where I explained exactly _why_ saying "the measure
is infinitesimal" doesn't help the supposed problem.
The proof that the set cannot be measurable works
just as well with an infinitesimal measure, as I explained:
Look. If m is infinitesimal then 2m, 3m, etc are all infinitesimal.
So each finite union A_1 union ... union A_n has infinitesimal
measure. In particular the measure of A_1 union ... union A_n
is less than 1/2. So taking the limit as n tends to inifinity
it follows that the measure of [0,1] is <= 1/2.
Similarly the measure of [0,1] is <= 1/3, etc.
>If Ullrich still believes that this is all "silly," I say that
>it's no sillier than the idea of delta functions which take on
>the value zero almost everywhere, yet integrate to one.
Another giggle. People talk about such delta "functions"
in elementary courses where the students don't have the
background required to follow the explanation of what
"delta" actually is, but in actual math nobody ever speaks
of such a function - the delta "function" is not a function
at all, it's a measure. Or a distribution (in either of two
common senses of that word).
Han de Bruijn
H(w) = [ ( delta(w-w_0)-delta(w+w_0) ) / ( delta(w-w_0)-delta(w
+w_0) ) ]
* e^( j*(w/w_0)*phi )
Where w, w_0 and phi belong to R; j is the imaginary unit.
For what values of w is H(w) defined and what are its values there?
Han de Bruijn
Jesse F. Hughes
>> Another giggle. People talk about such delta "functions"
>> in elementary courses where the students don't have the
>> background required to follow the explanation of what
>> "delta" actually is, but in actual math nobody ever speaks
>> of such a function - the delta "function" is not a function
>> at all, it's a measure. Or a distribution (in either of two
>> common senses of that word.
>
> H(w) = [ ( delta(w-w_0)-delta(w+w_0) ) / ( delta(w-w_0)-delta(w
> +w_0) ) ]
> * e^( j*(w/w_0)*phi )
>
> Where w, w_0 and phi belong to R; j is the imaginary unit.
>
> For what values of w is H(w) defined and what are its values there?
What is the point of your question? Are you merely curious about the
answer and incapable of finding it yourself? Or is this a silly test
regarding Ullrich's computational capabilities?
In either case, it seems that your question is fairly irrelevant, no?
--
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
David C. Ullrich
The expression makes no sense; there's no such thing as the reciprocal
of a delta "function".
>Han de Bruijn
Dan Cass
>
> > What is "andrica"???
>
> quite a famous problem actually !?
>
> i assumed everyone knew.
>
> http://www.wolframalpha.com/entities/famous_math_probl
>
> btw , i looked at all other 'proofs' found via
> google and sci.math , all of them are wrong.
>
> as far as i know im the only one who knows a proof.
>
> tommy1729
So at the website I found it said that Andrica's
Conjecture was:
For p_n the nth prime number, sqrt(p_(n+1))-sqrt(p_n)<1.
Just to be sure, you claim to have a proof of this
statement, is that right?
Transfer Principle
> Transfer Principle <lwal...@lausd.net> writes:
> > This isn't the first time that Ullrich has posted something similar
> > to the above statement. He made a similar criticism of me more than
> > a year ago when I tried to use "infinitesimals." He implies that
> > known infinitesimals (i.e., Robinson's) don't work exactly the way
> > that tommy1729 and I want them to, and simply using the word
> > "infinitesimal" doesn't make them work the way we want them to.
> that infinitesimals do just what he wants them to do, without fear of
> criticism. David Ullrich is nothing better than a five letter word.
We know that in classical analysis, Vitali sets are nonmeasurable. And
the posts of Ullrich and hagman imply that Vitali sets must remain
nonmeasurable in NSA as well. Thus, Vitali sets are nonmeasurable in
all known forms of analysis which prove that such sets must exist in
the first place.
But tommy1729 doesn't accept this fact. My point is that tommy1729
shouldn't be bound by what classical analysis states. If one doesn't
accept a result from classical analysis, then one should be allowed to
use an analysis other than classical analysis. And if all known forms
of analysis prove the undesirable result, then one should be allowed
to attempt to come up with a new form of analysis which avoids the
undesirable result.
(And of course, I don't mean something trivial like declaring all
Vitali sets to have measure 1729 by fiat. I mean a theory which
assigns a measure to Vitali sets while maintains as many properties
of Lebesgue measure as possible without contradicting measurability.)
In another post, Hughes suggests that it might take a very long time,
perhaps until the end of the universe, before anyone can come up with
a theory satisfying tommy1729's desiderata. The implication of course
is that it's a waste of time trying to come up with such a nonstandard
theory, so one should just accept classical analysis and its results.
(Note: the argument that tommy1729, like byron and WM, is simply
discussing classical analysis and is therefore "wrong" fails since
tommy1729 explicitly mentions the name of his proposed theory, TST.)
But naturally posters like hagman, Hughes, and Ullrich will find
alternate theories satisfying tommy1729's a waste of time because
they _don't_ find nonmeasurability undesirable. TST isn't for posters
who find the idea of a nonmeasurable set harmless. It's for posters
who find the idea of a nonmeasurable set _harmful_.
Thus, so what if it takes forever to come up with a rigorous
axiomatization for tommy1729's desiderata? No amount of time is too
long for someone who finds classical results undesirable. And this
includes both the nonmeasurability of certain sets and the existence
of cardinals greater than aleph_aleph_0.
Jesse F. Hughes
> On Oct 2, 6:38 pm, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
>> Transfer Principle <lwal...@lausd.net> writes:
>> > This isn't the first time that Ullrich has posted something similar
>> > to the above statement. He made a similar criticism of me more than
>> > a year ago when I tried to use "infinitesimals." He implies that
>> > known infinitesimals (i.e., Robinson's) don't work exactly the way
>> > that tommy1729 and I want them to, and simply using the word
>> > "infinitesimal" doesn't make them work the way we want them to.
>> But you're absolutely right. Tommy should have every right to pretend
>> that infinitesimals do just what he wants them to do, without fear of
>> criticism. David Ullrich is nothing better than a five letter word.
>
> We know that in classical analysis, Vitali sets are nonmeasurable. And
> the posts of Ullrich and hagman imply that Vitali sets must remain
> nonmeasurable in NSA as well. Thus, Vitali sets are nonmeasurable in
> all known forms of analysis which prove that such sets must exist in
> the first place.
>
> But tommy1729 doesn't accept this fact. My point is that tommy1729
> shouldn't be bound by what classical analysis states. If one doesn't
> accept a result from classical analysis, then one should be allowed to
> use an analysis other than classical analysis. And if all known forms
> of analysis prove the undesirable result, then one should be allowed
> to attempt to come up with a new form of analysis which avoids the
> undesirable result.
You see no difference between
(1) coming up with a new form of analysis and proving the result you
desire holds and
(2) simply stating the result you desire holds?
To be sure, Tommy is "allowed" to do either of these. Who could stop
him? But it is surely reasonable to criticize him for the latter.
> (And of course, I don't mean something trivial like declaring all
> Vitali sets to have measure 1729 by fiat. I mean a theory which
> assigns a measure to Vitali sets while maintains as many properties
> of Lebesgue measure as possible without contradicting measurability.)
>
> In another post, Hughes suggests that it might take a very long time,
> perhaps until the end of the universe, before anyone can come up with
> a theory satisfying tommy1729's desiderata. The implication of course
> is that it's a waste of time trying to come up with such a nonstandard
> theory, so one should just accept classical analysis and its results.
Er, no. I suggested that it will take *you* a very long time to come up
with a theory, judging by the speed at which your previous results have
thus far poured in. My comment was not about Tommy's desired theory.
Rather, I'm simply entertained by the amount of time you spend claiming
that you want to work on something fascinating compared to the apparent
time you spend actually doing so.
> (Note: the argument that tommy1729, like byron and WM, is simply
> discussing classical analysis and is therefore "wrong" fails since
> tommy1729 explicitly mentions the name of his proposed theory, TST.)
Oh, well that's all right then. It's got a name. No axioms, but a
name. That's totally awesome.
> But naturally posters like hagman, Hughes, and Ullrich will find
> alternate theories satisfying tommy1729's a waste of time because
> they _don't_ find nonmeasurability undesirable. TST isn't for posters
> who find the idea of a nonmeasurable set harmless. It's for posters
> who find the idea of a nonmeasurable set _harmful_.
I have absolutely no opinion on the desirability of nonmeasurable sets.
> Thus, so what if it takes forever to come up with a rigorous
> axiomatization for tommy1729's desiderata? No amount of time is too
> long for someone who finds classical results undesirable. And this
> includes both the nonmeasurability of certain sets and the existence
> of cardinals greater than aleph_aleph_0.
Well, take your time. But surely, one should have the theory at hand
before proclaiming its results.
And at some point, talking about how good it would be to work on such a
theory and what a shame others suppress such projects must give way to
actual work.
Note also that Tommy has recently claimed to have proved Andrica's
conjecture, if I recall correctly. I hope that he understands that
Andrica's conjecture is a conjecture in classical mathematics. *Even
if* Tommy has developed an alternative theory (Ho! Ho! It is to laugh!),
a proof of (the analogue of) Andrica's conjecture in that theory is
simply not the solution to the "real" conjecture.
--
"You can do 'math' until you drop dead, but if you're wrong, you're
still wrong, and the world will keep on turning regardless. Heaven
will not open up and God shout at you to stop, you'll just live in
error until one day you die." -- A James S. Harris self-reflection
MoeBlee
> In another post, Hughes suggests that it might take a very long time,
> perhaps until the end of the universe, before anyone can come up with
> a theory satisfying tommy1729's desiderata. The implication of course
> is that it's a waste of time trying to come up with such a nonstandard
> theory, so one should just accept classical analysis and its results.
NO, Jesse did NOT imply that the futility of trying to make
tommy1729's writings into a theory entails that one should accept
classical analysis.
> Thus, so what if it takes forever to come up with a rigorous
> axiomatization for tommy1729's desiderata? No amount of time is too
> long for someone who finds classical results undesirable.
No amount of time is too long for me to make sure I have food to eat.
That doesn't entail that I'd start by going through garbage cans when
there are at least a few fairly well stocked grocery stores in my
neighborhood. If one wants an alternative to classical analysis, it's
better to turn to the literature of mathematics in which there are
many different kinds of non-classical proposals by people who actually
know what they are talking about rather than turn to someone as
ignorant, horribly confused, and (to boot) and conceptually,
notationally, and grammatically juvenile as tommy1729.
> And this
> includes both the nonmeasurability of certain sets and the existence
> of cardinals greater than aleph_aleph_0.
And you think tommy1729 is a good place to start for this?
MoeBlee
MoeBlee
> Oh, well that's all right then. It's got a name. No axioms, but a
> name. That's totally awesome.
Actually, it has "axioms". Remember we went over it with him. Of
course, tommy1729 has his own notion, whatever it might be, of what an
axiom is.
MoeBlee
MoeBlee
> It's for posters
> who find the idea of a nonmeasurable set _harmful_.
Harmful in what way? Causes soap scum in the shower? Makes your teeth
rot? Contributes to global warming? Damns souls to perdition?
MoeBlee
Jesse F. Hughes
I suppose I repressed this memory.
And these days, Google wants me to log in before it will show me posts.
To hell with them. I want a database of Usenet posts, nothing more.
--
"Even if [...] a communistic regime should come [to China], the old
tradition [...] will break Communism and change it beyond recognition,
rather than Communism [...] break the old tradition. It must be so."
-- Lin Yutang on "Socialism with Chinese characteristics" in 1935
Transfer Principle
> On Sat, 2 Oct 2010 13:48:26 -0700 (PDT), Transfer Principle
> Heh-heh. You're missing the point to that quote, which
> is that there is no such magic bullet
... in either classical analysis or NSA, that is. But since
tommy1729 wants a magic bullet, he eschews both classical
analysis and NSA in favor of an analysis that does provide
for a magic bullet. And it is likely that we'll have to leave
ZFC to find such an analysis.
And as I told Hughes earlier, I'm willing to spend however long
it takes in order to find the axioms that will prove the
existence of the magic bullet.
WARNING: Below is the first post in which I will begin searching
for the rigorous axiomatization. Those like Ullrich who aren't
interested in such a theory should stop reading this post!
So what axiom should we try? Perhaps we might try an axiom which
simply states by fiat that magic bullets exist.
But then one might ask, if infinitely many magic bullets add up
to one, how many bullets add up to 1/2, or 1/3? (This is related
somewhat to Ullrich's objections.)
Previous sci.math posters have tried looking for magic bullets
(which is why Jeffries was discussing them), but many of them
want to change the how infinity works, as well. For example, if
"infinity" magic bullets add up to 1, then the number of bullets
adding up to 1/2 must be "infinity/2", the number of bullets
adding up to 1/3 must be "infinity/3", and so on. (The most
notable advocate of such a theory is RF, and he named the
function mapping the number of bullets to their measure "the
Equivalency Function.)
But tommy1729's desiderata appear to rule out something like the
EF, since to him, the only infinite numbers which exist are the
cardinals aleph_0, aleph_1, aleph_2, ..., aleph_aleph_0. So
there appears to be no room for aleph_0/2 or anything like that.
But, we try to append a new element, which we'll call "d" (in
order to bring to mind Newton's intended infinitesimals, which
tommy1729 desires) and declare:
d(aleph_0) = (aleph_0)d = 1
But I recall that tommy1729 accepts most of the rules of
cardinal arithmetic, including:
aleph_0 + aleph_0 = aleph_0
Multiply both sides by d:
d(aleph_0) + d(aleph_0) = d(aleph_0)
d(aleph_0) + d(aleph_0) = 1
a blatant contradiction.
Now tommy1729 didn't want this to turn into another set theory
thread, but it appears that we have to consider the set theory
axioms themselves in order to avoid this problem. (Jeffries
did warn that this was necessary.)
So we have to look at what axioms we want to include in TST. I
often suggest ZF, or even Z, as a starting point (since
tommy1729 often distrusts Choice) and take it from there.
And so we start from Z. One of the axioms still needs to be
modified, though, in order to guarantee that there's a
largest possible cardinal, without leading to Russell's
paradox or anything like that. It'll take time for me to
figure out what axiom we need, and so I'll consider this in a
subsequent post.
Transfer Principle
> Transfer Principle <lwal...@lausd.net> writes:
> > In another post, Hughes suggests that it might take a very long time,
> > perhaps until the end of the universe, before anyone can come up with
> > a theory satisfying tommy1729's desiderata. The implication of course
> > is that it's a waste of time trying to come up with such a nonstandard
> > theory, so one should just accept classical analysis and its results.
> Er, no. I suggested that it will take *you* a very long time to come up
> with a theory, judging by the speed at which your previous results have
> thus far poured in.
Once again, I never claimed that it will be _easy_ to come up with
the correct axioms.
Sometimes, I want to start stringing together random logical symbols
and "e" and variables and calling it an axiom for the sole purpose of
showing Hughes that I'm devoting a great deal of time writing axioms
and not on other stuff. But if I were to do so, one of two things
will inevitably happen. Either Hughes will discover that the axiom
leads to inconsistency in mere _minutes_ (if not _seconds_), or he
will question how the axiom will even approach proving any result
claimed by tommy1729 (or another sci.math poster).
I must put thought into the axioms in order to at least show that
the axioms eventually lead to tommy1729's claims. But right now, I
don't _know_ what axioms we need. And since I don't _know_ what
axioms to post, Hughes will keep saying that I like to talk about
writing theories but not actually write them.
I just don't know what I can do to prove that I want to spend the
majority of my time on sci.math coming up with theories. If I write
axioms hastily, they're likely to be inconsistent, but if I take the
time to write them, it looks as if I like talking about theories but
not writing them.
Sometimes I wish I could just snap my fingers and have intelligent
axioms appear.
> Note also that Tommy has recently claimed to have proved Andrica's
> conjecture, if I recall correctly. I hope that he understands that
> Andrica's conjecture is a conjecture in classical mathematics.
OK, I'll concede that when tommy1729 mentions Andrica he's referring
to classical mathematics. But when he refers to TST he isn't.
Transfer Principle
> On Oct 4, 3:00 pm, Transfer Principle <lwal...@lausd.net> wrote:
> > Thus, so what if it takes forever to come up with a rigorous
> > axiomatization for tommy1729's desiderata? No amount of time is too
> > long for someone who finds classical results undesirable.
> better to turn to the literature of mathematics
...which is inaccessible to me, as MoeBlee already knows.
> > And this
> > includes both the nonmeasurability of certain sets and the existence
> > of cardinals greater than aleph_aleph_0.
> And you think tommy1729 is a good place to start for this?
I think that tommy1729 is an _accessible_ place for me to start. If I
can access tommy1729's work but not the other mathematical literature,
then I'm going to start with tommy1729's work.
(Of course there's one exception -- tommy1729's work on Andrica is
definitely _not_ accessible, to me or anyone else.)
Jesse F. Hughes
Part of the problem is that your desiderata are mostly unnatural
mishmashes of random pronouncements. Even if there *is* a largest
cardinal, why should it be aleph_aleph_0? And what does this have to do
with infinitesimal measures? It is not surprising that this odd
collection of claims is difficult to axiomatize in a natural manner.
This is quite different than, say, anti-well-founded set theory. I
don't recall exactly why the theory was first developed, but it seems
quite natural to notice that the foundation axiom is not used all that
much in ZFC. If we simply omit it, then we can prove pretty much the
same theorems, perhaps relativized to the well-founded part.
Now, one might ask, is there a sensible way to negate the axiom? Rather
than say there are no non-well-founded sets, let's say there are lots of
them. To be sure, it took more insight than I have to do this
(essentially stating that every set of equations of a certain sort has a
unique solution). I'm not claiming that what Aczel did was easy, but it
was fairly naturally motivated and it was not surprising that a nice
theory came from it.
I see no reason to think that Tommy's pronouncements will lead to a
similarly pleasing result.
Since you sometimes misinterpret me, let me be clear: if you want to
work on a theory for Tommy's claims, you are perfectly free to do so.
If you want to keep talking about how you would like to work on a
theory, rather than just talking, no one can stop you. Do whatever you
like.
But I'm mighty puzzled why you think that Tommy's notions will lead to
anything worthwhile.
--
Jesse F. Hughes
"You shouldn't hate Mother Mathematics."
-- James S. Harris
Han de Bruijn
> Han de Bruijn <umum...@gmail.com> writes:
>
> >> Another giggle. People talk about such delta "functions"
> >> in elementary courses where the students don't have the
> >> background required to follow the explanation of what
> >> "delta" actually is, but in actual math nobody ever speaks
> >> of such a function - the delta "function" is not a function
> >> at all, it's a measure. Or a distribution (in either of two
> >> common senses of that word.
>
> > H(w) = [ ( delta(w-w_0)-delta(w+w_0) ) / ( delta(w-w_0)-delta(w
> > +w_0) ) ]
> > * e^( j*(w/w_0)*phi )
>
> > Where w, w_0 and phi belong to R; j is the imaginary unit.
>
> > For what values of w is H(w) defined and what are its values there?
>
> What is the point of your question? Are you merely curious about the
> answer and incapable of finding it yourself? Or is this a silly test
> regarding Ullrich's computational capabilities?
>
> In either case, it seems that your question is fairly irrelevant, no?
No. Ullrich actually _is_ talking about delta functions here. This is
a fairly simple test to see how a "standard" mainstream mathematician
would solve a little problem of the kind. And the outcome is that he
_denies_ that it can be solved. I find it somewhat confusing, because
sort of a solution is suggested by the original poster, preceding the
following individual message, where I am constructing such one:
http://groups.google.nl/group/sci.math/msg/28ca7be180c2052f
Han de Bruijn
Jesse F. Hughes
Oh, a test.
I see. Ain't you a presumptuous li'l pain in the ass.
I'm not sure that Ullrich exactly denied that it could be solved. He
denied that the question made any sense at all.
I have no opinion about the matter myself, knowing nothing about delta,
though if I have to guess whether you or he are more likely to be
correct, it's clear where my money is.
--
Jesse F. Hughes
"Marriage.. ..is the union of two persons of different sex for
life-long reciprocal possession of their sexual faculties."
-- Immanuel Kant, who died an unmarried virgin
Han de Bruijn
> Han de Bruijn <umum...@gmail.com> writes:
>
> > On Oct 4, 1:40 pm, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
>
> >> What is the point of your question? Are you merely curious about the
> >> answer and incapable of finding it yourself? Or is this a silly test
> >> regarding Ullrich's computational capabilities?
>
> >> In either case, it seems that your question is fairly irrelevant, no?
>
> > No. Ullrich actually _is_ talking about delta functions here. This is
> > a fairly simple test to see how a "standard" mainstream mathematician
> > would solve a little problem of the kind. And the outcome is that he
> > _denies_ that it can be solved. I find it somewhat confusing, because
> > sort of a solution is suggested by the original poster, preceding the
> > following individual message, where I am constructing such one:
>
> >http://groups.google.nl/group/sci.math/msg/28ca7be180c2052f
>
> Oh, a test.
>
> I see. Ain't you a presumptuous li'l pain in the ass.
>
> I'm not sure that Ullrich exactly denied that it could be solved. He
> denied that the question made any sense at all.
It's not even _my_ idea that the question _could_ make sense.
Wild guess: the original posting is sort of homework from a textbook
on electronics. I've always been interested in the grey area between
mathematics and technology.
> I have no opinion about the matter myself, knowing nothing about delta,
Exactly.
> though if I have to guess whether you or he are more likely to be
> correct, it's clear where my money is.
It's clear that your sympathies are _always_ with the established.
Anyway, why don't you just mind your own business, li'l "philosopher"?
Han de Bruijn
Jesse F. Hughes
> It's clear that your sympathies are _always_ with the established.
> Anyway, why don't you just mind your own business, li'l "philosopher"?
Oh, what a horrible insult.
You know, mocking me for being a philosopher is a bit silly, since I
have more formal training in mathematics than you.
Not that I count myself as either a mathematician or a philosopher. I'm
simply a somewhat overeducated housewife.
--
"Now for once I might actually have an audience that realizes that
[my proof of Fermat's Last Theorem is correct], because you see,
they'll finally know what's in it for them--cold, hard cash."
--James Harris embarks on a new mathematical strategy.
Han de Bruijn
> Han de Bruijn <umum...@gmail.com> writes:
>
> > It's clear that your sympathies are _always_ with the established.
> > Anyway, why don't you just mind your own business, li'l "philosopher"?
>
> Oh, what a horrible insult.
>
> You know, mocking me for being a philosopher is a bit silly, since I
> have more formal training in mathematics than you.
FORMAListic training, yes.
> Not that I count myself as either a mathematician or a philosopher. I'm
> simply a somewhat overeducated housewife.
Green vases are evidence for black ravens. No?
Han de Bruijn
William Hughes
> Green vases are evidence for black ravens. No?
Yes, *extremely* weak evidence, but evidence.
- William Hughes
MoeBlee
> On Oct 4, 2:55 pm, MoeBlee <jazzm...@hotmail.com> wrote:
>
> > On Oct 4, 3:00 pm, Transfer Principle <lwal...@lausd.net> wrote:
> > > Thus, so what if it takes forever to come up with a rigorous
> > > axiomatization for tommy1729's desiderata? No amount of time is too
> > > long for someone who finds classical results undesirable.
> > If one wants an alternative to classical analysis, it's
> > better to turn to the literature of mathematics
>
> ...which is inaccessible to me, as MoeBlee already knows.
Right, I forgot you're held in solitary confinement in a Mogadishu
jail with an Internet connection but one that blocks all online
booksellers. If it weren't for tommy1729, alternative mathematics
would vanish in the bleak darkness of your cell where there is only
sci.logic and the occasional chirping of a lonely sparrow outside your
window to remind you that life goes on...
MoeBlee
Aatu Koskensilta
> On Oct 4, 2:55 pm, MoeBlee <jazzm...@hotmail.com> wrote:
>
>> If one wants an alternative to classical analysis, it's better to
>> turn to the literature of mathematics
>
> ...which is inaccessible to me, as MoeBlee already knows.
There's no university library nearby? You have no money to spend on a
subject you're obviously very interested in? If so, this is unfortunate,
but, alas, not something we can really do much anything about (although
I personally am perfectly willing to send you electronic copies of
papers accessible through JSTOR and other such databases).
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon man nicht sprechen kann, darüber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Aatu Koskensilta
> Now, one might ask, is there a sensible way to negate the axiom?
> Rather than say there are no non-well-founded sets, let's say there
> are lots of them. To be sure, it took more insight than I have to do
> this (essentially stating that every set of equations of a certain
> sort has a unique solution). I'm not claiming that what Aczel did was
> easy, but it was fairly naturally motivated and it was not surprising
> that a nice theory came from it.
With the considerable benefit of hindsight we can say that Aczel's
approach is the obvious one. That is, once informed of the basic idea
pretty much any logically inclined mathematician could fill in the
details and make sure everything fits together. This is true of many
important pieces of mathematics.
Aatu Koskensilta
R> On Oct 5, 10:56 am, Han de Bruijn <umum...@gmail.com> wrote:
>
>> Green vases are evidence for black ravens. No?
>
> Yes, *extremely* weak evidence, but evidence.
We wouldn't ordinarily regard a non-X as evidence that all X's are
Y. How to mold such intuitions into a systematic theory of evidence is a
perennial philosophical conundrum.
Ki Song
I'm sorry, but it looks like you didn't carefully read the post that
you've provided the link to.
In the post, it says
> As I've said, just replace delta by delta' , where:
> delta'(t) = exp(-(t/sigma)^2/2) / (sigma.sqrt(2.Pi))
delta is not a function, but this delta' is FUNCTION.
And so,
> [ ( delta'(w-w_0)-delta'(w+w_0) ) / ( delta'(w-w_0)-delta'(w+w_0) ) ]
is not same as the expression you wrote up.
James Burns
> On Oct 4, 2:55 pm, MoeBlee <jazzm...@hotmail.com> wrote:
>>On Oct 4, 3:00 pm, Transfer Principle <lwal...@lausd.net> wrote:
>>>Thus, so what if it takes forever to come up with a rigorous
>>>axiomatization for tommy1729's desiderata? No amount of time is too
>>>long for someone who finds classical results undesirable.
>>
>>If one wants an alternative to classical analysis, it's
>>better to turn to the literature of mathematics
>
> ...which is inaccessible to me, as MoeBlee already knows.
Transfer Principle, how much are you willing to spend
accessing mathematical literature? Please be specific.
I notice that the email address you use has "lausd.net"
as a host, or "Los Angeles Unified School District".
There are a number of excellent universities in
Los Angeles, with excellent library systems, I'm sure.
I note that UCLA's library offers free borrowing priveleges
to certain "external users". Those eligible include
'Certificated K-12 public school employees", which may
include you, for all I can tell. Or you might be taking
classes at a Community College, which could also qualify
you for free privelges.
http://www.library.ucla.edu/service/385.cfm
http://www.library.ucla.edu/service/2024.cfm
For $100 per year, any member of the general public can
obtain borrowing privleges.
http://www.library.ucla.edu/service/2027.cfm
There is also the Los Angeles Public Library. Although
its acquisitions are probably weighted more heavily toward the
more popular/less technical, I would expect a wide variety
in a system that size, including some titles you may well
find useful.
For example, a title that has shown up in discussions
here in sci.math, Torkel Franzen, Goedel's theorem : /
/an incomplete guide to its use and abuse / (2005)
shows up in their catalog. (I'm one for one!)
Also, a subject search on "set theory" could be useful.
Some titles I found:
- Set theory for the working mathematician
- The foundations of mathematics in the theory of sets
- In search of infinity
- Philosophical introduction to set theory
- Mass terms and model-theoretic semantics
- Naive set theory (Halmos)
- The roots of reference (Quine)
- Contributions to the founding of the theory of transfinite numbers
(Cantor)
Let me remind you that the use of these are all /free/ to you.
It may well be that there are options available to you.
The more you share about your situation, the more likely
that someone will have a workable solution for you.
That is what you want, a solution, isn't it?
Jim Burns
David C. Ullrich
He hasn't done that.
>(And of course, I don't mean something trivial like declaring all
>Vitali sets to have measure 1729 by fiat.
_That_ is close to what he's done here! He doesn't develop
any theory, he simply declares that the measure of the set
in question should be some "infinitesmal". Alas it's supposed
to be an infinitesmal m such that the sum of countably many
m's is equal to 1, and there is no such thing.
William Hughes
> William Hughes <wpihug...@hotmail.com> writes:
>
> R> On Oct 5, 10:56 am, Han de Bruijn <umum...@gmail.com> wrote:
>
>
>
> >> Green vases are evidence for black ravens. No?
>
> > Yes, *extremely* weak evidence, but evidence.
>
> We wouldn't ordinarily regard a non-X as evidence that all X's are
> Y. How to mold such intuitions into a systematic theory of evidence is a
> perennial philosophical conundrum.
Indeed (and black ravens are de riguer), and my reply was quite
intentionally
meant to be controversial.
I don't know much about the problem (when was that ever a barrier,
especially on sci.math) but my analysis goes somethng like this.
To determine the answer to the question, "are all ravens black",
I consider two sets, A the set of ravens, and B the set of non-black
things. I am interested in the size of the intersection (in
particular
whether it is empty). An observation of a black raven gives me
information
about set A, An observation of a non-black non-raven gives me
information
about set B. Since B is much bigger than A, an observation of a raven
gives
me much more evidence than the observation of a non-black thing, but
the
observation of a non-black thing is evidence, just not very strong.
I strongly suspect that this argument is known, Since the problem is
not settled I also suspect that there are counterarguments.
If I keep this up I will lose my sci.math license to post.
- William Hughes
Jesse F. Hughes
>> You know, mocking me for being a philosopher is a bit silly, since I
>> have more formal training in mathematics than you.
>
> FORMAListic training, yes.
My Master's degree in mathematics was in a fairly standard program, no
more formal than most.
My training in logic came later.
--
"A recruitment consultant I know thinks the most important quality in
a winner is to be lucky. To avoid wasting his time with unlucky
applicants, he takes half the resumes piled on his desk and throws
them straight in the bin." -- John Ramsden
hagman
> Transfer Principle <lwal...@lausd.net> writes:
> > But naturally posters like hagman, Hughes, and Ullrich will find
> > alternate theories satisfying tommy1729's a waste of time because
> > they _don't_ find nonmeasurability undesirable. TST isn't for posters
> > who find the idea of a nonmeasurable set harmless. It's for posters
> > who find the idea of a nonmeasurable set _harmful_.
>
> I have absolutely no opinion on the desirability of nonmeasurable sets.
>
Me neither, or at least no absolute opinion.
However, for me the Lebesgue measure is a partial function from the
power set P(R) to the set R_{>=0} u {\infty} that obeys certain
rules (e.g. sigma-additivity).
It seems that tommy wants to have total function isntead.
I guess there are essentially three methods to to achieve this:
1) Make P(R) smaller by forbidding non-measurable sets. This requires
at least to abondon the axiom of choice. That is something I can
tolerate
as a valid opinion (though different from mine)
2) Drop some of the rules (Ulrich gave an argument that tommy's
idea has problems with sigma-additivity)
3) Change the range of the function to something bigger than
nonnegative reals plus infinity.
Apparently tommy now makes a try at the third method.
However, it seems that method 3) alone does not work - you must
also take a grain of 2).
hagman
David R Tribble
>> indeed i do not use hyperreals , but put all my
>> money on newtons infinitesimals aka classic (!)
>> calculus and analysis.
>
L Walker wrote:
>> Does tommy1729 mean something like "dx" and "dy" as
>> the infinitesimals of Newton?
>
tommy1729 wrote:
> yes dx and dy
Are these (Tommy's) infinitesimals nilpotent?
Gerry Myerson
"Jesse F. Hughes" <je...@phiwumbda.org> wrote:
> Han de Bruijn <umu...@gmail.com> writes:
>
> > It's clear that your sympathies are _always_ with the established.
> > Anyway, why don't you just mind your own business, li'l "philosopher"?
>
> Oh, what a horrible insult.
>
> You know, mocking me for being a philosopher is a bit silly, since I
> have more formal training in mathematics than you.
>
> Not that I count myself as either a mathematician or a philosopher. I'm
> simply a somewhat overeducated housewife.
That would be the wife of a somewhat overeducated house?
--
Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email)
Han de Bruijn
No evidence at all.
Han de Bruijn
Han de Bruijn
Sure. And _after_ that I corrected myself and wrote:
I'm not satisfied with the solution. The reason is that the following
(_non_ smooth) blocks are valid test functions as well. But reasoning
should be _independent_ of whatever test function we take. So here we
go again.
: with blocks as tet functions.
> delta is not a function, but this delta' is FUNCTION.
Informally speaking, delta is a "limit" of a (class of) function(s),
therefore it IS a function. Do you deny that the limit of a sequence
of numbers is still a number? (I _know_ that the official doctrine is
different from this, so don't try to correct me. I am perfectly able
to parrot what the standard theory says)
> And so,
>
> > [ ( delta'(w-w_0)-delta'(w+w_0) ) / ( delta'(w-w_0)-delta'(w+w_0) ) ]
>
> is not same as the expression you wrote up.
Indeed, but then we take the limit for spread of this function -> 0 .
A weak point in the argument is that it is implicitly assumed that the
delta' functions are all the _same_. Upon substitution of _different_
test functions (denoted with ' ) my argument becomes clearly invalid:
( delta'(w-w_0)-delta''(w+w_0) )/( delta'''(w-w_0)-delta''''(w+w_0) )
So, in the end, Ullrich may be quite right when he says that division
of delta functions is undefined, namely if the latter interpretation
is adopted. (Note: a physics argument, apart from "pure" mathematics)
Han de Bruijn
Jesse F. Hughes
I've no time for your jokes. I've got to get breakfast on the table and
the goat's not feeling well at all. The child has a fever and the cat's
on fire.
Don't you try to distract me.
--
Jesse F. Hughes
"LOL. How arrogant you are. Now when you realize that I DID prove
Goldbach's conjecture and that I proved Fermat's Last Theorem as well,
how are you going to feel then?" -- James Harris
William Hughes
Why not? The set of all non-black things is large but finite. In
principle,
I could establish "all ravens are black" by examining all non-black
things.
- William Hughes
Han de Bruijn
Excuse moi: by examining all green vases.
Han de Bruijn
William Hughes
No by examining all non-black things, This includes green vases,
Examining all green vases is only a very small part of the task,
but it is part of the task. Examining one green vase is a very small
part of examining all green vases, hence "*extremely* weak".
- William Hughes
part of the task
>
> Han de Bruijn
Frederick Williams
>
> [...] The child has a fever and the cat's
> on fire.
>
> Don't you try to distract me.
Dip the cat in a bucket of water and use it to mop the child's fevered
brow.
--
Needle, nardle, noo.
Ki Song
No. I don't know what kind of definition of "function" you are using.
f_n(x) = Sin(nx)
What function does this sequence converge to?
> Do you deny that the limit of a sequence
> of numbers is still a number? (I _know_ that the official doctrine is
> different from this, so don't try to correct me. I am perfectly able
> to parrot what the standard theory says)
Please tell me what number the limit of the sequence given by a_n =
Sin(n) is.
>
> > And so,
>
> > > [ ( delta'(w-w_0)-delta'(w+w_0) ) / ( delta'(w-w_0)-delta'(w+w_0) ) ]
>
> > is not same as the expression you wrote up.
>
> Indeed, but then we take the limit for spread of this function -> 0 .
Sorry, I don't really follow?
>
> A weak point in the argument is that it is implicitly assumed that the
> delta' functions are all the _same_. Upon substitution of _different_
> test functions (denoted with ' ) my argument becomes clearly invalid:
>
> ( delta'(w-w_0)-delta''(w+w_0) )/( delta'''(w-w_0)-delta''''(w+w_0) )
>
> So, in the end, Ullrich may be quite right when he says that division
> of delta functions is undefined, namely if the latter interpretation
> is adopted. (Note: a physics argument, apart from "pure" mathematics)
>
> Han de Bruijn
...so in the end, all I understood from your posting is that
1) Your definition of function is non-standard. (Any "limit" of a
sequence of functions is a function.)
2) Your definition of number is non-standard. (Any "limit" of a
sequence of numbers is a function.)
You are free to adopt a well-defined notion of "division" by a "delta
function."
But you can't just say "division by delta function is well-defined
because it is a function," and expect it to make any sense to people.
master1729
> <tomm...@gmail.com>
> wrote:
>
> >David Ullrich wrote :
> >
> >> On Fri, 01 Oct 2010 16:04:30 EDT, master1729
> >> <tomm...@gmail.com>
> >> wrote:
> >>
> >> >lwalke wrote :
> >> >
> >> >> On Sep 30, 1:59 pm, master1729
> >> <tommy1...@gmail.com>
> >> >> wrote:
> >> >> > the irony is that when i started posting
> about
> >> >> infinitesimals on the continuum between 0 and 1
> -
> >> >> they imho connect the reals ( points ) to form
> the
> >> >> continuum - , the reply was :
> >> >> > there are no infinitesimals ( or hyperreals )
> on
> >> my
> >> >> real number line.
> >> >> > - then what is on your real line / continuum
> ? -
> >> >> > reply : on my real line are only reals. duh !
> >> >> > now measure is defined on " the real line " .
> >> >> > but now since I STARTED using infinitesimal
> >> measure
> >> >> ,
> >> >> > !!suddenly!! there is need to assign measure
> to
> >> the
> >> >> infinitesimals on the real line / continuum.
> >> >> > thats weird. suddenly the real line /
> continuum
> >> >> changed , and just to try to prove me wrong ...
> >> >> > seems the real line is time-dependend and has
> a
> >> >> personality , the personality of my critics.
> >> >> > my critics seem to have special powers , like
> >> >> mathekinetics or such.
> >> >>
> >> >> It appears what they're doing is saying, if the
> >> reals
> >> >> don't include
> >> >> (nonzero) infinitesimals, then one can't assign
> >> >> infinitesimals as
> >> >> measures of sets of reals. But if one wants to
> >> >> include infinitesimal
> >> >> measures, then one must include infinitesimals
> in
> >> the
> >> >> sets of reals
> >> >> being measured.
> >> >>
> >> >> Can there be infinitesimal measures? Do the
> sets
> >> >> being measured
> >> >> contain infinitesimals. The answer must be
> "both"
> >> or
> >> >> "neither," but
> >> >> tommy1729 is saying "the former but not the
> >> latter,"
> >> >> and this is why
> >> >> hagman objects.
> >> >
> >> >no , the choice to have infinitesimals on the
> real
> >> line is a matter of taste and/or independent of
> the
> >> measures on it.
> >> >
> >> >i objected to the ' mathekinetics ' some people
> use
> >> immorally and inconsistantly for personal benefit
> or
> >> personal argument.
> >
> >i will probably regret replying to david ..sigh ..
> >
> >
> >>
> >> I on the other hand object to poeple assuming that
> >> they can give an
> >> example of whatever they like just by saying
> >> "infinitesimal".
> >
> >i on the other hand object to people assuming that
> they can give an example of whatever they like just
> by saying " axiom of choice " or " zfc "
> >
> >really that comment didnt make sense.
> >
> >i didnt just say " infinitesimal " i explained it.
> >
> >i justified it on my site and i clarified here that
> it is newtons classical infinitesimal.
> >
> >
> >>
> >> Regardless of the silliness above,
> >
> >
> >what silliness ? your silliness ?
> >
> >your reply up till now started with 2 insults and
> nothing else , i dont know if you even realize that
> anymore.
> >
> >
> >> your amazing idea
> >
> >thats better .. though you dont mean that.
> >
> >
> >> of making that
> >> set measurable by saying its measure is an
> >> infinitesimal simply
> >> doesn't work.
> >
> >i gave a justification.
> >
> >why doesnt it work ?
> >
> >i guess you will stop insulting and try to clarify
> below.
> >
> >
> >
> >
> >> Note this is independent of the fact
> >> that you have
> >> a lot of infrastucture to create first in order to
> >> make your measures
> >> with infinitesimals "consistant". It simply
> doesn't
> >> work:
> >
> >again you say : it doesnt work.
> >
> >and still without explaining.
> >
> >saying it more doesnt make your argument stronger.
> >
> >well , not that you already gave an argument ... it
> doesnt make your opinion more convincing i meant.
> >
> >
> >>
> >> We have countably many sets A_n, each of which
> must
> >> have the same measure, if they have any measure at
> >> all.
> >> And [0,1] is the union of the A_n. So they cannot
> be
> >> measurable.
> >> If the measure was a real number m then m = 0
> implies
> >> the measure
> >> of [0,1] is 0, while m > 0 implies that the
> measure
> >> of [0,1] is
> >> infinite. Saying that m is an infinitesimal
> doesn't
> >> magically fix this
> >> problem.
> >
> >and again you repeat yourself !!
> >
> >saying it doesnt " magically " work now instead of
> saying it doesnt work.
> >
> >whats next , saying it doesnt " " mathematically "
> work ?
> >
> >
> >>
> >> Look. If m is infinitesimal then 2m, 3m, etc are
> all
> >> infinitesimal.
> >
> >yes 2m and 3m are infinitesimal.
> >
> >and in general for finite n - yes finite ! - , nm is
> infinitesimal.
> >
> >thats just a property of infinitesimals ...
> >
> >
> >> So each finite union A_1 union ... union A_n has
> >> infinitesimal
> >> measure.
> >
> >for finite n yes.
> >
> >
> >> In particular the measure of A_1 union ...
> >> union A_n
> >> is less than 1/2. So taking the limit as n tends
> to
> >> inifinity
> >> it follows that the measure of [0,1] is <= 1/2.
> >
> >wrong. it holds only for finite n.
> >
> >your limit is not justified , not even for
> infinitesimals.
>
> If every member of an increasing sequence is < 1/2
> then the limit, if it exists, is <= 1/2.
>
> >look.
> >
> >lim m -> oo of n/m is always smaller than 1/2 for
> finite n.
> >
> >but if n becomes oo this no longer necc holds.
>
> Huh? The limit of n/m as m tends to infinity has
> nothing
> to do with this.
>
> What's relevant is the limit of n*m as n tends to
> infinity.
> That limit is 0 if m is 0, infinite if m is a
> positive real,
> and (if it exists at all, which would depend on
> details) infinitesimal if m is infinitesimal. No way
> it
> equals 1.
>
> >your use of limits , infinitesimals and infinity is
> inconsistant. ( with calculus )
> >
> >> Similarly the measure of [0,1] is <= 1/3, etc.
> >
> >only for finite n.
> >
> >>
> >>
> >
> >so what is your argument ? i fear that wrongly
> computed limit was your argument.
> >
> >
> >>
> >
> >oh thats it ?
> >
> >
> >to further convince you that " for all integer n " ,
> lim n -> oo and oo are not necc the same consider
> this :
> >
> >1 is rational
> >
> >1 + 1/2 is rational
> >
> >1 + 1/2 + 1/6 is rational
> >
> >1 + 1/2 + 1/6 + 1/4! is rational
> >
> >.. is rational
> >
> >e - 1 is transcendental !
>
> Giggle. This has nothing to do with anything here.
>
> >think about it before you try another limit argument
> !
>
> Giggle. Can you even tell me what it _means_ to say
> that L is the limit of a sequence a_n? Hint: Of
> course
> that can happen with all the a_n rational and L
> transcrendental. But it cannot happen with all the
> a_n < 1/2 and L > 1/2. That's extremely easy to
> prove.
>
> >
> >
> >tommy1729
> >
> >" in fact i admire him " galathaea
>
i give up
you dont know what an infinitesimal is !
lwalke might explain you , but i wont do the effort.
master1729
no.
but i am worried a bit about 2^dx since 2^ is powerset but dx is suppose to be newtons infinitesimal ...
master1729
> On Oct 3, 8:51 am, David C. Ullrich
> <ullr...@math.okstate.edu> wrote:
> > On Sat, 2 Oct 2010 13:48:26 -0700 (PDT), Transfer
> Principle
> > >So Jeffries's magic bullet is exactly what
> tommy1729 needs.
> > Heh-heh. You're missing the point to that quote,
> which
> > is that there is no such magic bullet
>
> ... in either classical analysis or NSA, that is. But
> since
> tommy1729 wants a magic bullet, he eschews both
> classical
> analysis and NSA in favor of an analysis that does
> provide
> for a magic bullet. And it is likely that we'll have
> to leave
> ZFC to find such an analysis.
newton is the keyword.
>
> And as I told Hughes earlier, I'm willing to spend
> however long
> it takes in order to find the axioms that will prove
> the
> existence of the magic bullet.
>
> WARNING: Below is the first post in which I will
> begin searching
> for the rigorous axiomatization. Those like Ullrich
> who aren't
> interested in such a theory should stop reading this
> post!
>
> So what axiom should we try? Perhaps we might try an
> axiom which
> simply states by fiat that magic bullets exist.
newton satisfies this.
>
> But then one might ask, if infinitely many magic
> bullets add up
> to one, how many bullets add up to 1/2, or 1/3? (This
> is related
> somewhat to Ullrich's objections.)
>
> Previous sci.math posters have tried looking for
> magic bullets
> (which is why Jeffries was discussing them), but many
> of them
> want to change the how infinity works, as well. For
> example, if
> "infinity" magic bullets add up to 1, then the number
> of bullets
> adding up to 1/2 must be "infinity/2", the number of
> bullets
> adding up to 1/3 must be "infinity/3", and so on.
> (The most
> notable advocate of such a theory is RF, and he named
> the
> function mapping the number of bullets to their
> measure "the
> Equivalency Function.)
>
> But tommy1729's desiderata appear to rule out
> something like the
> EF, since to him, the only infinite numbers which
> exist are the
> cardinals aleph_0, aleph_1, aleph_2, ...,
> aleph_aleph_0. So
> there appears to be no room for aleph_0/2 or anything
> like that.
not so fast.
infinity/3 is still consistant with newton.
infinity/3 = 1/(3h) where h is the infinitesimal.
and yes , things like 3h DO OCCUR IN NEWTONS CALCULUS.
dont look to far : newton !
as for the aleph interpretations :
cardinalities and ordinals only occur as measures for RELATIVE SIZE !
not ABSOLUTE SIZE !
that is why i object to x = x + 1 , w - 1 , w + 1 both their existance and their non-existance.
this RELATIVE size of things is fundamental for infinity.
it is why 1/(h+1) = 1/h and the motivation of newtons infinitesimal h !
it is probably the most important thing you missed so far in my posts.
so , newtons infinitesimal + 3-valued logic + only RELATIVE sizes for infinity => forms 3 important basics of TST ( without contradicting newton and leibniz ! )
>
> But, we try to append a new element, which we'll call
> "d" (in
> order to bring to mind Newton's intended
> infinitesimals, which
> tommy1729 desires) and declare:
>
> d(aleph_0) = (aleph_0)d = 1
>
> But I recall that tommy1729 accepts most of the rules
> of
> cardinal arithmetic, including:
>
> aleph_0 + aleph_0 = aleph_0
see my comments above , addition is typically related to ABSOLUTE size , which consistant infinities are not ...
>
> Multiply both sides by d:
>
> d(aleph_0) + d(aleph_0) = d(aleph_0)
> d(aleph_0) + d(aleph_0) = 1
>
> a blatant contradiction.
if you mix correct things with wrong things , you usually end up with wrong things.
( again see above for explaination )
>
> Now tommy1729 didn't want this to turn into another
> set theory
> thread, but it appears that we have to consider the
> set theory
> axioms themselves in order to avoid this problem.
> (Jeffries
> did warn that this was necessary.)
>
> So we have to look at what axioms we want to include
> in TST. I
> often suggest ZF, or even Z, as a starting point
> (since
> tommy1729 often distrusts Choice) and take it from
> there.
>
> And so we start from Z. One of the axioms still needs
> to be
> modified, though, in order to guarantee that there's
> a
> largest possible cardinal, without leading to
> Russell's
> paradox or anything like that. It'll take time for me
> to
> figure out what axiom we need, and so I'll consider
> this in a
> subsequent post.
>
im not afraid of Jeffries nor Robinsons oo.
i start from newton , you start from Z , i dont know if those roads will cross.
thanks for your time though.
regards
tommy1729
" the cat is on fire "
Tim Little
> "LOL. How arrogant you are. Now when you realize that I DID prove
> Goldbach's conjecture and that I proved Fermat's Last Theorem as
> well, how are you going to feel then?" -- James Harris
This is an especially rich quote as James now believes that Goldbach's
Conjecture is false.
- Tim
OwlHoot
>
> [..]
>
> but i am worried a bit about 2^dx since 2^ is powerset but dx is suppose to be newtons infinitesimal ...
Well obviously you should define that as 2 - 0.99999...
Cheers
John Ramsden
P.S. Let's hope Google Groups doesn't have a fit of hiccups again when
I post this.
Transfer Principle
> lwalke 3 wrote :
> > And it is likely that we'll have
> > to leave ZFC to find such an analysis.
> newton is the keyword.
Thank you. As it turns out, there's another poster, Androcles,
who's interested in Newton's infinitesimals. Look up recent
posts by "Androcles" for more information.
In the Androcles thread, the poster James Burns provided us
with a link to Newton's Principia Mathematica:
http://rack1.ul.cs.cmu.edu/is/newton/
Hopefully, we will be able to use this information to our
advantage and use the infinitesimals of Newton himself. The
searchable part of the translation doesn't work on my computer
(404 and 500 errors), but maybe it might work on tommy1729's
own computer. Meanwhile, I'll have to read the non-searchable
portion of the site, page by page, to learn more about how
Newton intended his infinitesimals to work.
It's my hope that by reading Newton's work, we'll be able to
answer some of tommy1729's questions directly from Newton. For
example, tommy1729 asks for 2^h or 2^dx. (Come to think of it,
isn't 2^dx Leibniz? So it should be 2^h -- of course, this is
exactly what I hope to find out at that link.)
So what exactly is 2^h? Don't answer unless you're one of the
three of us (Androcles, tommy1729, myself) working on reading
Newton and using his infinitesimals. In particular, I reject
OwlHoot's answer, unless OwlHoot professes to be interested in
Newton's infinitesimals -- though I do admit that OwlHoot's
answer is at least interesting.
> it is probably the most important thing you missed so far in my posts.
Thanks to Androcles, I'm not missing it any more.
Jesse F. Hughes
> So what exactly is 2^h? Don't answer unless you're one of the
> three of us (Androcles, tommy1729, myself) working on reading
> Newton and using his infinitesimals. In particular, I reject
> OwlHoot's answer, unless OwlHoot professes to be interested in
> Newton's infinitesimals -- though I do admit that OwlHoot's
> answer is at least interesting.
Sorry, I just want to repeat the most fascinating part of your post.
I reject OwlHoot's answer, unless OwlHoot professes to be
interested in Newton's infinitesimals.
You are utterly weird.
--
"Tempted and tried we're oft made to wonder
Why it should be thus all the day long
When there are others living about us
Never molested though in the wrong." -- Bad Livers, "Farther Along"
Transfer Principle
>David Ullrich wrote :
> > Giggle. Can you even tell me what it _means_ to say
> > that can happen with all the a_n rational and L
> > transcrendental. But it cannot happen with all the
> > a_n < 1/2 and L > 1/2. That's extremely easy to
> > prove.
> you dont know what an infinitesimal is !
> lwalke might explain you , but i wont do the effort.
I mentioned a religious analogy earlier, but now I finally
know the correct term to use. (I mentioned this religious
analogy in the Androcles thread, but I now avoid it there
to the religiously charged environment in that thread,
sparked by Androcles's anti-Semitic remarks.)
That term is "Great Apostasy."
Many Protestants believe that the Catholic church doesn't
represent the faith founded by Christ. Similarly, Androcles
and tommy1729 don't believe that Bolzano-Weierstrass deltas
and epsilons represent the infinitesimal calculus founded
by Newton.
So, when Ullrich asks:
> > Giggle. Can you even tell me what it _means_ to say
> > that L is the limit of a sequence a_n?
he is most likely expecting an answer in terms of epsilons
and deltas, such as for every epsilon > 0 , there exists a
natural number m s.t. for all n > m, abs(L-a_n) < epsilon.
But that answer presumes that tommy1729 was using classical
analysis as given by Bolzano, Cauchy, and Weierstrass. But we
see that tommy1729 was really using the infinitesimal analysis
of _Newton_ all along!
Adherents of ZFC and classical analysis, including Ullrich,
assumed that tommy1729 was referring to standard analysis, but
tommy1729 rejects classical analysis due to its perceived
unfaithfulness to the infinitesimals of Newton.
Just as I use the word "adherents," I will now use the word
"apostates" to refer to posters like Ullrich who, from the
perspective of Androcles and tommy1729, are using a form of
analysis that's not faithful to the analysis of Newton.
Do I myself consider Bolzano, Cauchy, and Weierstrass to be a
"Great Apostasy" for the infinitesimals of Newton? Of this, I
wish to remain neutral. I consider classical analysis to be _a_
rigorous formulation of Newton's calculus, but I'm also willing
to help those who reject it as well. If someone believes that
Bolzano, Cauchy, and Weierstrass represent a "Great Apostasy"
from Newton, then he should be allowed to come up with another
formulation of analysis that he feels is more faithful to the
original Newtonian formulation.
This is why tommy1729 writes:
> you [Ullrich] dont know what [a Newtonian] infinitesimal is !
And when he writes:
> lwalke might explain [it to] you , but i wont do the effort.
I gave the above explanation, by introducing this religious
analogy of a Great Apostasy.
Jesse F. Hughes
> Many Protestants believe that the Catholic church doesn't
> represent the faith founded by Christ. Similarly, Androcles
> and tommy1729 don't believe that Bolzano-Weierstrass deltas
> and epsilons represent the infinitesimal calculus founded
> by Newton.
>
> So, when Ullrich asks:
>
>> > Giggle. Can you even tell me what it _means_ to say
>> > that L is the limit of a sequence a_n?
>
> he is most likely expecting an answer in terms of epsilons
> and deltas, such as for every epsilon > 0 , there exists a
> natural number m s.t. for all n > m, abs(L-a_n) < epsilon.
>
> But that answer presumes that tommy1729 was using classical
> analysis as given by Bolzano, Cauchy, and Weierstrass. But we
> see that tommy1729 was really using the infinitesimal analysis
> of _Newton_ all along!
Right.
Do you think Tommy has really *read* Newton? And that Tommy's
pronouncements are naturally more faithful to Newton's calculus than
either classical analysis or non-standard analysis?
Really?
Because from where I'm sitting, it appears that Tommy is making shit up
as he goes, and invoking Newton's name only to lend credibility to his
own addled thoughts regarding infinitesimals. I find it very hard to
believe that these addled thoughts come from a reasoned reading of
Newton.
--
Jesse F. Hughes
"Usenet allows people to live in complete fantasy."
-- James S. Harris
Transfer Principle
> Sorry, I just want to repeat the most fascinating part of your post.
> I reject OwlHoot's answer, unless OwlHoot professes to be
> interested in Newton's infinitesimals.
> You are utterly weird.
Here's why I mentioned OwlHoot's post.
I said that I wanted to answer tommy1729's question about what 2^h
is, and that I would find out by reading Newton. But it will take
me a while to read Newton, and I'm afraid that in the meantime,
someone (like Burns or Little, for example) will just post the
answer and add a comment about how I'm too _lazy_ to read Newton
(cf. those who post homework questions on sci.math), or that I
don't really care about Newton when in reality, it takes _time_ to
read Newton to find the answer. (And it's possible that Newton
doesn't mention 2^h at all anywhere in his book.)
But what about OwlHoot, the poster who already gave an answer? His
answer is interesting, I admit. But if I just accepted OwlHoot's
answer, Burns and Little might take that as an excuse for me not to
read Newton itself (since OwlHoot has already given an answer) and
then repeat that I never cared about reading Newton all along.
Furthermore, if OwlHoot's answer has nothing to Newton, then his
respond doesn't answer tommy1729's question -- which is about
Newton's 2^h.
Therefore, whether I should accept OwlHoot's answer depends on his
stance with regards to Newton. If he is using _Newton_'s h, then
tommy1729 should accept it as an answer to his question. If not,
then we should read Newton to find the answer.
(Since Hughes emphasized my response to OwlHoot, I now feel like
discussing OwlHoot's response in more detail, but I will do so in
a separate post, addressed directly to OwlHoot.)
Transfer Principle
> > but i am worried a bit about 2^dx since 2^ is powerset but dx is suppose to be newtons infinitesimal ...
> Well obviously you should define that as 2 - 0.99999...
> Cheers
> John Ramsden
(I was calling him "OwlHoot," his username, but his signature gives
the name "Ramsden," so I'll call him "Ramsden" from now on.)
So far, I haven't had time to read Newton yet (and hopefully I will
have finished reading it by this time next week), but if I had to
guess what 2^h is, I might try something like:
e^h = 1 + h + h^2/2 + h^3/6 + ...
= 1 + h
2^h = 1 + h ln 2
So 2^h would be infinitesimally greater than 1. But we note that
there are two Newtonian infinitesimalists here -- tommy1729 and
Androcles -- and they might give different answers to 2^h.
Although 1 + h ln 2 appears acceptable to tommy1729, it can't be
acceptable to Androcles because he considers h to be the smallest
possible infinitesimal -- so in particular, it can't be multiplied
by ln 2 (which is less than 1).
But now consider Ramsden's response of 2-0.999.... We know that
many posters who believe in a smallest infinitesimal also consider
0.999... to be strictly less than 1 and furthermore for 1-0.999...
to equal that smallest infinitesimal.
But Androcles doesn't refer to 0.999... in his thread. I do not
know what his opinion of 0.999..., but three possibilities are the
most likely:
1) Androcles's 0.999... = 1 (the classical result)
2) Androcles's 0.999... = 1-h (a popular infinitesimalist result)
3) Androcles's 0.999... = 1-2h
Why possibly 1-2h? Recall that to Androcles, the real numbers
alternate between rational and irrational. Since 1 is obviously
rational, this makes 1-h irrational and 1-2h rational. But he
might notice that the digits of 0.999... are periodic, which
indicates that the number is rational. So he might reject 1-h in
favor of 1-2h. Thus 0.999... would be the largest _rational_ less
than 1, with one irrational, 1-h = 0.999+h, strictly between the
two rationals 0.999... and 1.
In either case, since ln 2 is closer to 1 than to 0, Androcles
might round off 1 + h ln 2 to 1+h. Then, if we consider the
middle case above, we have:
2^h = 1+h
= 2-(1-h)
= 2-0.999...
which is exactly Ramsden's result.
So there's a possibility that Ramsden and Androcles agree here. Of
course, the ultimate arbiter is Newton.
Transfer Principle
> Transfer Principle <lwal...@lausd.net> writes:
> > But that answer presumes that tommy1729 was using classical
> > analysis as given by Bolzano, Cauchy, and Weierstrass. But we
> > see that tommy1729 was really using the infinitesimal analysis
> > of _Newton_ all along!
> Right.
> Do you think Tommy has really *read* Newton?
If he hasn't, then he can read the link given by Burns:
http://rack1.ul.cs.cmu.edu/is/newton/
to find out once and for all whether his ideas agree with Newton's.
> Because from where I'm sitting, it appears that Tommy is making [crap] up
> as he goes, and invoking Newton's name only to lend credibility to his
> own addled thoughts regarding infinitesimals.
Once again, both tommy1729 and I can read Newton's book, and we'll
find out for sure whether tommy1729 is making crap up or actually
gaining insight into Newton's original intentions.
And even if he really did make crap up, he can nonetheless be
legitimately opposed to Bolzano-Weierstrass epsilons as faithful to
Newton's original ideas.
(I hope to have finished reading Newton by this time next week!)
Transfer Principle
Oops, I forgot about the 3-valued logic -- which was my original
discussion point for making TST rigorous in the first place!
As I said earlier, at first I didn't want to consider TST with
its 3-valued logic because of the difficulties inherent in
coming up with a new logic. But then I wanted to take a trick
from Srinivasan, who also came up with a new logic (NAFL), by
using a temporary patch to replace the new logic (namely the
axiom "D=0"), and then, once the set theory is complete, go
back and use the new logic to make the new axiom redundant.
And so let me do the same with TST. (And if I'm really lucky,
tommy1729's other desiderata will immediately follow from the
patch used to replace the 3-valued logic.)
WARNING: This marks the second post in this thread in which I
attempt to make TST rigorous. Adherents of ZFC who don't care
about making TST rigorous might as well ignore the remainder of
this post.
Now earlier, the poster George Greene noted that 3-valued logic
is unnecessary, because one can replace the three truth values
with two bits in two-valued logic.
This suggests that for our patch, instead of having a theory
with only one primitive "e", we add a second primitive "f". We
now have two bits, which actually gives us four set theoretic
relations between two sets x and y:
1) xey & xfy
2) xey & ~xfy
3) ~xey & xfy
4) ~xey & ~xfy
To collapse these four possibilities into three truth values,
we can add the following axiom:
Axiom:
Axy (xfy -> xey)
This eliminates case 3). Eventually, one will invent a new
formula, xEy, with three truth values, corresponding to the
three possibilities 1, 2), and 4) above -- but for now, there
will be two primitives, "e" and "f".
What about the other axioms of Z? For some axioms, we might
try writing them with both "e" and "f":
Extensionality:
Axy (x=y <-> Az ((zex <-> zey) & (zfx <-> zfy)))
This tells us that two sets are equal when both their
e-elements and f-elements agree.
The empty set should have neither e-elements nor f-elements,
so this gives us:
Empty Set:
Ex (Ay (~yex & ~yfx))
But what about Pairing? The unordered pair {a,b} ought to have
two elements, a and b, but should these e-elements, f-elements,
or even one of each? Suppose we had:
Pairing:
Aab (Ex (aex & bfx))
Given sets a and b this gives us both the set with e-element a,
f-element b and the set with e-element b, f-element a. If a=b
then this gives us a set with a single element that is both an
e-element and an f-element. But this does not yet give us a set
with a single e-element and no f-elements.
To be continued...
David C. Ullrich
Really? What would give you that idea?
Guffaw.
Transfer Principle
> To be continued...
WARNING: This marks the third post in this thread in which I
attempt to make TST rigorous. Adherents of ZFC who don't care
about making TST rigorous might as well ignore the remainder of
this post.
After thinking about it for a while, I realize that some of the
axioms above contain errors that need fixing, so let me do so
right now,
To review, the three possibilities are:
1) ~xey (and ~xfy, but this is redundant)
2) xey & ~xfy
3) xfy (and xey, but this is redundant)
So we notice that for the middle "truth value," we must mention
both primitives "e" and "f", but for the other two, one of them
is redundant.
For one thing, this means that the Empty Set Axiom can be
written identically to its counterpart in Z:
Empty Set Axiom:
Ex (Ay (~yex))
Pairing is a bit tricky. We need something like:
Aab (Ex (Ay ((yfx <-> y=a) & (yex <-> (y=a v y=b)))))
so as to make a the "full element" (i.e., an f-element) of x
and b a "partial element" (i.e., e-element but not f-element)
of x.
What's interesting about this version of Pairing is that we
should be able to define an _ordered_ pair without resorting
to the Kuratowski pair.
So let's define (a,b) to be the set whose existence is
guaranteed by the Pairing Axiom. Then we should be able to
prove that (a,b)=(c,d) iff a=c and b=d (with Extensionality
as given in my last post featuring prominently in the proof).
What about the other axioms? For example, what exactly is a
Union in 3-valued logic? In standard 2-valued logic, the
elements of U(a) are exactly the elements of the elements of
the set a. So maybe in 3-valued logic, the e-elements of U(a)
are the e-elements of the e-elements of a, and likewise for
the f-elements:
Union:
Aa (Ex (Ay ((yex <-> Ez (yez & zex))
& (yfx <-> Ez (yfz & zfx)))))
To be continued...
FredJeffries
Whom do you cast in the role of the Spanish Inquisition?
FredJeffries
> On Oct 3, 8:51 am, David C. Ullrich <ullr...@math.okstate.edu> wrote:
>
> > On Sat, 2 Oct 2010 13:48:26 -0700 (PDT), Transfer Principle
> > >So Jeffries's magic bullet is exactly what tommy1729 needs.
> > Heh-heh. You're missing the point to that quote, which
> > is that there is no such magic bullet
>
> ... in either classical analysis or NSA, that is. But since
> tommy1729 wants a magic bullet, he eschews both classical
> analysis and NSA in favor of an analysis that does provide
> ZFC to find such an analysis.
>
> it takes in order to find the axioms that will prove the
> existence of the magic bullet.
>
> WARNING: Below is the first post in which I will begin searching
> for the rigorous axiomatization. Those like Ullrich who aren't
> interested in such a theory should stop reading this post!
>
> So what axiom should we try? Perhaps we might try an axiom which
> simply states by fiat that magic bullets exist.
>
> to one, how many bullets add up to 1/2, or 1/3? (This is related
> somewhat to Ullrich's objections.)
>
> Previous sci.math posters have tried looking for magic bullets
> (which is why Jeffries was discussing them), but many of them
> want to change the how infinity works, as well. For example, if
> "infinity" magic bullets add up to 1, then the number of bullets
> adding up to 1/2 must be "infinity/2", the number of bullets
> adding up to 1/3 must be "infinity/3", and so on. (The most
> notable advocate of such a theory is RF, and he named the
> function mapping the number of bullets to their measure "the
> Equivalency Function.)
You're overlooking the obvious solution: you don't change the "number"
of magic bullets, you change their sizes.
They're like quarks that never appear individually or fractals. You
always have the same "infinite number" of them occuring togther.
You need to think holistically: When you cut the unit interval in half
you are simultaneously cutting each of the magic bullets in half.
FredJeffries
> On Oct 6, 2:41 pm, master1729 <tommy1...@gmail.com> wrote:
>
> > so , newtons infinitesimal + 3-valued logic + only RELATIVE sizes for infinity => forms 3 important basics of TST ( without contradicting newton and leibniz ! )
>
> Oops, I forgot about the 3-valued logic -- which was my original
> discussion point for making TST rigorous in the first place!
>
> As I said earlier, at first I didn't want to consider TST with
> its 3-valued logic because of the difficulties inherent in
> coming up with a new logic. But then I wanted to take a trick
> from Srinivasan, who also came up with a new logic (NAFL), by
> using a temporary patch to replace the new logic (namely the
> axiom "D=0"), and then, once the set theory is complete, go
> back and use the new logic to make the new axiom redundant.
>
You need to look at Professor John L Bell's papers wherein he examines
and compares all of the different infinitesimal conconcepts throughout
history:
http://publish.uwo.ca/~jbell/
Regarding the use of non-standard logic when dealing with the
continuum see
http://publish.uwo.ca/~jbell/New%20lecture%20on%20infinitesimals.pdf
page 6
"a faithful account of the truly continuous would involve abandoning
the unrestricted applicability of the law of excluded middle".
For Newton, see chapter 2 of "The Continuous and the Discrete in the
History of Thought"
http://publish.uwo.ca/~jbell/Chapter%202.pdf
starting at page 16
David C. Ullrich
<fredje...@gmail.com> wrote:
>On Oct 10, 12:30 am, Transfer Principle <lwal...@lausd.net> wrote:
>> On Oct 6, 2:41 pm, master1729 <tommy1...@gmail.com> wrote:
>>
>> > so , newtons infinitesimal + 3-valued logic + only RELATIVE sizes for infinity => forms 3 important basics of TST ( without contradicting newton and leibniz ! )
>>
>> Oops, I forgot about the 3-valued logic -- which was my original
>> discussion point for making TST rigorous in the first place!
>>
>> As I said earlier, at first I didn't want to consider TST with
>> its 3-valued logic because of the difficulties inherent in
>> coming up with a new logic. But then I wanted to take a trick
>> from Srinivasan, who also came up with a new logic (NAFL), by
>> using a temporary patch to replace the new logic (namely the
>> axiom "D=0"), and then, once the set theory is complete, go
>> back and use the new logic to make the new axiom redundant.
>>
>
>You need to look at Professor John L Bell's papers wherein he examines
>and compares all of the different infinitesimal conconcepts throughout
>history:
>http://publish.uwo.ca/~jbell/
And in particular he needs to find a version where there exists
a sequence of infinitesimals which converges to 1.
Semms to me like this will not be easy, but who knows?
After all, I don't know what an infinitesimal is.
Giggle.
Ross A. Finlayson
> On Oct 10, 12:30 am, Transfer Principle <lwal...@lausd.net> wrote:
>
> > On Oct 6, 2:41 pm, master1729 <tommy1...@gmail.com> wrote:
>
> > > so , newtons infinitesimal + 3-valued logic + only RELATIVE sizes for infinity => forms 3 important basics of TST ( without contradicting newton and leibniz ! )
>
> > Oops, I forgot about the 3-valued logic -- which was my original
> > discussion point for making TST rigorous in the first place!
>
> > As I said earlier, at first I didn't want to consider TST with
> > its 3-valued logic because of the difficulties inherent in
> > coming up with a new logic. But then I wanted to take a trick
> > from Srinivasan, who also came up with a new logic (NAFL), by
> > using a temporary patch to replace the new logic (namely the
> > axiom "D=0"), and then, once the set theory is complete, go
> > back and use the new logic to make the new axiom redundant.
>
> You need to look at Professor John L Bell's papers wherein he examines
> and compares all of the different infinitesimal conconcepts throughout
> history:http://publish.uwo.ca/~jbell/
>
> Regarding the use of non-standard logic when dealing with the
> page 6
> "a faithful account of the truly continuous would involve abandoning
> the unrestricted applicability of the law of excluded middle".
>
> For Newton, see chapter 2 of "The Continuous and the Discrete in the
> History of Thought"http://publish.uwo.ca/~jbell/Chapter%202.pdf
> starting at page 16
Modern standard real analysis is great and works perfectly.
(Countable additivity in methods of exhaustion, is classical with the
modern demurral, where Dedekind/Cauchy/Weierstrass formalized modern
classical analysis before using set theory, ie without it.) There are
considerations of the non-measurable sets that follow from Vitali from
Cantor and corresponding results from topology that given the standard
yield what for some are surprising results (Banach-Tarski, that "area"
or measure in the 1-D case isn't "conserved").
The impetus to go beyond the standard is to discover some new features
of these mathematical objects that are so ubiquitous in every modern
formulation of application. The idea for those who seek to learn new
features of these fundamental mathematical (and fundamentally maybe
not just mathematical) objects is that besides aesthetically in
knowledge for its own sake that in the real and concrete: novel
applications of novel features of the objects as they most closely
represent everything around us offer means to better understand and
cast into formalism the natural philosophy, or physics.
Newton had infinitesimals, fluxions to the unit fluents, and the
fluxion is the differential, but he also sees that fluxion simply the
fluent to comparatively again infintesimal fluxions, preserving some
fixed infinite scale between them. It is instead Leibniz'
infinitesimals in terms of attributed founders of the integral
calculus (or infinitesimal analysis, as it was and is known), with the
differential, that seems to be the infinitesimals that conglomerate
(in a functional framework) from zero to one in the straight line, the
unit line segment. Leibniz' notation is still used, the integral bar
S for summation of those (countable additivity). Int_0^1 x dx = 1.
Any proposed extension to the definitions of the real numbers that
doesn't preserve all reasonably expected results from analysis is
worthless. (Thus additivity is countable.) Then as to whether there
could ever be discovered or formulated suitable extensions to
definition or redefinitions of the real numbers to actually provide
applicable results unavailable to the standard, particularly as to
where those can be used in physical experiment to falsifiably verify
concise and general models of physical systems, whether or not it may
be, that is the goal (for those who would discover those things).
Yes, it's nice that Bell's SIA is another modern precedent to talk
about the infinite and infinitesimals, where basically projection is
complete, that also points to the standard definition of "continuous"
and how in delta-epsilonics it applies to, for example, the rationals.
Physicists have a lot in place in the frameworks in terms of
"continuum analysis", they might well believe that whatever they could
learn and show would be about "the" real continuum. Surely they'd be
very interested in mathematical developments about the continuum, and
surely they've looked and tried to make use of the trans-finites.
Also many of them see delta and step as functions, and their inventors/
discoverers did, where of course they readily defer to the standard in
their formalisms.
Nobody expects the Spanish Inquisition. Then in terms of the bridge
of fools or pons asinorum to Cantor's paradise, where he would have
none of those bacilli as they were termed, where there are real
features of the real numbers beyond standard theory, which Goedel
guarantees there are, here there are a variety of soi disant Black
Knights for whom: none shall pass.
Regards,
Ross Finlayson
MoeBlee
wrote:
> Dedekind/Cauchy/Weierstrass formalized modern
> classical analysis before using set theory
What do you mean by 'formalized'? Ordinarily, I would think
'formalize' means to put in a formal system with a formal language,
formal inference rules, formal axioms, and formal definitions. I'm not
aware that there was a formal system for analysis provided by those
men.
MoeBlee
Transfer Principle
> > As I said earlier, at first I didn't want to consider TST with
> > its 3-valued logic because of the difficulties inherent in
> > coming up with a new logic. But then I wanted to take a trick
> > from Srinivasan, who also came up with a new logic (NAFL), by
> > using a temporary patch to replace the new logic (namely the
> > axiom "D=0"), and then, once the set theory is complete, go
> > back and use the new logic to make the new axiom redundant.
> You need to look at Professor John L Bell's papers wherein he examines
> and compares all of the different infinitesimal conconcepts throughout
> history:
> http://publish.uwo.ca/~jbell/
Thanks for the links. Unfortunately, right now I'm in the middle of
reading the Newton's Principia link that has been recommended to me
by another poster. So far, I've finished Book I, Section 1, and am
in the middle of Section 2.
Hopefully, I'll be able to read those links when I have time.
Transfer Principle
wrote:
> On Sat, 09 Oct 2010 20:44:41 -0400, "Jesse F. Hughes" wrote:
> > [I]t appears that Tommy is
> > [...] invoking Newton's name only to lend credibility to his
> > own addled thoughts regarding infinitesimals. I find it very hard to
> > believe that these addled thoughts come from a reasoned reading of
> > Newton.
Well, let's try to make a "reasoned reading of Newton" now.
OK, to reach the relevant page, we type in "94" into the box given at
Burns's link and click "Go." (For some reason, there's some sort of
fencepost error, and when "94" is typed in, page 95 appears.)
Notice that Lemma 1 on this page was quoted directly by Androcles in
the OP of his thread. So we know at the very least that _Androcles_
(not sure about tommy1729) has read Newton.
Androcles writes:
"Irrational Bonehead insists h can be divided by 2 (despite the
definition
that it cannot) and therefore he would conclude pi can be exactly
divided
by 2 and is rational.
Bonehead challenges the mathematics of Newton, who stated {Lemma I]."
So according to Androcles, Lemma I is contradicted by classical
analysis
and its assertion that 2 divides pi in standard R. Thus, Androcles
rejects
classical analysis and desires an alternate analysis that he feels is
more
faithful to Newton and his Lemma I.
Here is Newton's Lemma I as quoted by Androcles:
"LEMMA I.
Quantities, and the ratios of quantities, which in any finite time
converge
continually to equality, and before the end of that time approach
nearer the
one to the other than by any given difference, become ultimately
equal.
If you deny it, suppose them to be ultimately unequal, and let D be
their
ultimate difference. Therefore they cannot approach nearer to equality
than
by that given difference D; which is against the supposition. -- Sir
Isaac
Newton, Principia Mathematica."
Now Androcles mentions an infinitesimal, which he calls h. Is he
trying to
equate his "h" with the "D" mentioned in Lemma I? On one hand, the "D"
in
Newton Lemma appears in a proof by contradiction, and so that "D"
doesn't
actually exist. But perhaps Androcles claims that the sequence:
1, 1/2, 1/3, 1/4, 1/5, ..., 1/n, ...
does not have a limit of zero, since there is a value of D such that
all
values of that sequence are greater than D, and that D equals h.
Hmmm. Newton's Lemma II, on the same page, appears to describe the
method
of approximating an integral via Riemann sums. Newton doesn't mention
the
rectangles (or "parallelograms") as having width D or h, but simply
states
that the "ultimate ratios" of the Riemann sum and the area under the
curve
"are ratios of equality."
Similarly, on page "102" (actually 103), at the end of Book I Section
1,
Newton describes these "evanescent quantities" and "ultimate ratios":
"For those ultimate ratios with which quantities vanish are not truly
the
ratios of ultimate quantities, but limits towards which the ratios of
quantities decreasing without limit do always converge; and to which
they
approach nearer than any given difference, but never go beyond, nor in
effect attain to, till the quantities are diminished _in_infinitum_."
(emphasis Newton's)
At this point, Newton's use of the word "limit" sounds as if it can
refer
to the classical notion of limit, but also there's no reason that
there
can't be a smallest difference called "D" or "h."
(Note: recall that I have no problem with saying that Newton's ideas
are
formalized by classical analysis -- but right now I'm trying to help
two
posters who believe that it doesn't.)
I believe that I need to read a few more pages of Newton, to see
whether
any concept like "D" or "h" appears again.
Transfer Principle
Do not blame RF for mentioning Cauchy and Weierstrass. He was only
repeating what I was saying about the formalization of analysis.
So MoeBlee is telling us that Dedekind, Cauchy, and Weierstrass
preceded the formalization of analysis. I already knew that Newton
and Leibniz are pre-formal, but I'm surprised that Cauchy, and
especially Dedekind, are still considered pre-formal. After all, why
do we call them "Cauchy sequences" and "Dedekind cuts" if Cauchy and
Dedekind are not responsible for the formalization of analysis?
And if Cauchy and Dedekind weren't responsible for the formalization
of analysis, then who was? Don't tell me who _wasn't_ responsble for
formalization, tell me who _was_!
BTW, before anyone criticizes me for not checking Google before
asking the question, a Google search for "who formalized analysis"
(including the quotes) retrieved a single result:
http://www.mth.kcl.ac.uk/staff/eb_davies/primes14.pdf
which contained the statement:
"The mathematicians who formalized analysis in the late nineteenth
century laid the grounds for proving the existence of continuous
but nowhere differentiable functions of a real variable."
without mentioning who those mathematicians are.
At this point, my guess is that Hilbert -- whom I know to have
formalized Euclidean geometry -- is mainly responsible for the
formalization of analysis. But since I was wrong to attribute
formalization to Cauchy and Weierstrass, I'm likely still wrong to
attribute it to Hilbert as well.
Then again, going back to my original point, it doesn't really
matter who it was who finally formalized analysis. We already agree
that _Newton_ was pre-formal, and that _somebody_ formalized his
vision of analysis. And tommy1729 and Androcles reject this
classical formalization as being faithful to Newton's vision.
Jesse F. Hughes
> So according to Androcles, Lemma I is contradicted by classical
> analysis and its assertion that 2 divides pi in standard R. Thus,
> Androcles rejects classical analysis and desires an alternate analysis
> that he feels is more faithful to Newton and his Lemma I.
>
> Here is Newton's Lemma I as quoted by Androcles:
>
> "LEMMA I.
> Quantities, and the ratios of quantities, which in any finite time
> converge continually to equality, and before the end of that time
> approach nearer the one to the other than by any given difference,
> become ultimately equal. If you deny it, suppose them to be
> ultimately unequal, and let D be their ultimate difference. Therefore
> they cannot approach nearer to equality than by that given difference
> D; which is against the supposition. -- Sir Isaac Newton, Principia
> Mathematica."
So, is Androcles right, in your opinion? Does this lemma entail that pi
is not divisible by 2 in R?
--
"Eventually the truth will come out, and you know what I'll do then?
Probably go to the beach. I'll also hang out in some bars. Yup, I'll
definitely hang out in some bars, preferably near a beach."
-- JSH on the rewards of winning a mathematical revolution
Ross A. Finlayson
Blame?
Oh, I wasn't repeating what you said, per se, about that the standard
reals as they are used were made "rigorous" enough to use, formally.
That's in the formal system where there are definitions of the
standard real numbers as they are basically today what with having the
defined terms and the axioms about them, with those generally today
surely fitting to modern mathematics, because measure theory was
fitted to it. There's plenty of formal in the classical.
Interesting read today: 0.999... not necessarily equal to 1 in
primary/secondary education?
http://news.slashdot.org/story/10/10/14/135219/Proving-0999-Is-Equal-To-1
Regards,
Ross F.