Another NSA Question
Math1723
question is substantially different enough about NSA that I feel
justified starting a different thread on the topic.
Essentially one question here: Is it true that the countable sum of
infinitesimals is still infinitesimal?
In IST, infinite countable sizes are not accessible as they are
external. (Internally, Hyper-Countable sets are either finite or
uncountable.) So we are left with looking at this analytically.
For any infinitesimal e > 0, clearly lim n->inf (e*n) remains
infinitesimal. However, it seems to me possible that you could
somehow have a growing sequence of infinitesimals such that e1 + e2 +
e3 + ... is a finite number.
Unfortunately, given that *R is not Dedekind complete, we cannot
conclude that just because an increasing sequence of infinitesimals is
bounded from above, that there is even a well defined value to which
this series converges.
Perhaps the more basic question is: Can NSA even have a well-defined
definition for Countable Additivity in general?
Note: To avoid the side controversy created in my last post, let me
just say I am not upset at being told if I am wrong or have a logical
fallacy somewhere in my thinking. Feel free to be direct. I will
also endure small levels of egotism and insults (if necessary) to
become educated. But I am hoping that this can remain civil. Thanks
in advance.
David C. Ullrich
<anony...@aol.com> wrote:
>With the other thread devolving into debates on crankhood, my next
>question is substantially different enough about NSA that I feel
>justified starting a different thread on the topic.
>
>Essentially one question here: Is it true that the countable sum of
>infinitesimals is still infinitesimal?
>
>In IST, infinite countable sizes are not accessible as they are
>external. (Internally, Hyper-Countable sets are either finite or
>uncountable.) So we are left with looking at this analytically.
>
>For any infinitesimal e > 0, clearly
>(i) lim n->inf (e*n) remains infinitesimal.
_If_ you assume that for some reason the limit exsts, which
is not at all clear to me (the sentence above reads as though
you haven't read the rest of _your_ post yet) then yes,
this is clear.
> However, it seems to me possible that
>(ii) you could
>somehow have a growing sequence of infinitesimals such that e1 + e2 +
>e3 + ... is a finite number.
No, that's impossible.
I'm curious: How do you _prove_ (i)? The reason I ask is that I can't
imagine a proof of (i) that would lead you to speculate about (ii);
they seem equally clear to me. Hence the question: Is (i) really
clear to you, or are you just saying it's clear because you read
it somewhere?
How do you prove (i) anyway? If you can't say then you're
lying again when you say (i) is clear. (Just like you lied about
how those papers defined "lacunary", and about who really
wrote that amazing proof of the Beatty thing, etc. Lying
is not good.)
>Unfortunately, given that *R is not Dedekind complete, we cannot
>conclude that just because an increasing sequence of infinitesimals is
>bounded from above, that there is even a well defined value to which
>this series converges.
>
>Perhaps the more basic question is: Can NSA even have a well-defined
>definition for Countable Additivity in general?
>
>Note: To avoid the side controversy created in my last post, let me
>just say I am not upset at being told if I am wrong or have a logical
>fallacy somewhere in my thinking. Feel free to be direct. I will
>also endure small levels of egotism and insults (if necessary) to
>become educated. But I am hoping that this can remain civil.
Coming from you that's very funny.
>Thanks
>in advance.
Aatu Koskensilta
> If you can't say then you're lying again when you say (i) is
> clear. (Just like you lied about how those papers defined "lacunary",
> and about who really wrote that amazing proof of the Beatty thing,
> etc. Lying is not good.)
It seems you have some reason to think Math1732 is in fact none other
than our dear master/tommy?
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon man nicht sprechen kann, darüber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Math1723
wrote:
> On Thu, 16 Dec 2010 07:38:51 -0800 (PST), Math1723
>
>
> >(i) lim n->inf (e*n) remains infinitesimal.
>
> _If_ you assume that for some reason the limit exsts, which
> is not at all clear to me (the sentence above reads as though
> you haven't read the rest of _your_ post yet) then yes,
> this is clear.
You are right, I should have said "If the limit exists, then it is
infinitesimal". Quite possibly the limit doesn't exist.
Jesse F. Hughes
> David C. Ullrich <ull...@math.okstate.edu> writes:
>
>> If you can't say then you're lying again when you say (i) is
>> clear. (Just like you lied about how those papers defined "lacunary",
>> and about who really wrote that amazing proof of the Beatty thing,
>> etc. Lying is not good.)
>
> It seems you have some reason to think Math1732 is in fact none other
> than our dear master/tommy?
I'm sure that David mistakenly identified the two because of the similar
posting names (Math1732 and tommy1729).
--
"You are simply one person who persists in delusive thinking about
your own relative importance, as you also rationalize data you do not
wish to accept. I, unlike you, am a worldwide figure."
-- James S. Harris, on self-delusions
Math1723
>
> >(ii) you could
> >somehow have a growing sequence of infinitesimals such that e1 + e2 +
> >e3 + ... is a finite number.
>
> No, that's impossible.
>
> I'm curious: How do you _prove_ (i)? The reason I ask is that I can't
> imagine a proof of (i) that would lead you to speculate about (ii);
> they seem equally clear to me. Hence the question: Is (i) really
> clear to you, or are you just saying it's clear because you read
> it somewhere?
(i) seems very clear to me. (ii) seemed very likely to me, but I
couldn't think of how to prove it.
> How do you prove (i) anyway? If you can't say then you're
> lying again when you say (i) is clear. (Just like you lied about
> how those papers defined "lacunary", and about who really
> wrote that amazing proof of the Beatty thing, etc. Lying
> is not good.)
I am not lying. (What motive would I have for lying???) Nor am I
familiar with whatever "lacunary" is. I've never seen that word prior
to this post. Nor do I know what "the Beatty thing" refers to.
Proving (i) would go along these lines:
For every positive infinitesimal e in *R, there exists a least n0 in
*N such that e*n0 > 1. Thus for all n < n0, e*n < 1. Note that this
n0 is necessarily unlimited (that is, greater than any standard
natural number n). If it were not, then e would not be infinitesimal.
More generally, for any positive real number r, there is a least n0 in
*N such that e*n0 > r. So a proof would essentially be one of
contradiction: assume that (i) yielded some appreciable (finite but
not infinitesimal) number. Then I'd show that the sum required would
be an uncountable one (for i = 1 to n0, where n0 is unlimited).
But with (ii) I am not sure how I could modify such a proof.
First, let me acknowledge that I used poor language in my post. "lim
n->inf" refers to a Countable sum only in Standard Analysis. The way
I should have phrased it as follows:
Let { a_i } be a sequence of infinitesimals. If A = Sum (n in N)
{ a_i } converges, must it always be that A is infinitesimal?
>> Coming from you that's very funny.
Nice.
It's been at least a couple of years since you and I have been in the
same sci.math thread, so I don't know from whence your animosity
comes. Presumably, I pissed you off, or you are confusing me with
someone who had. I will assume the latter, and move on. If you
choose to continue being rude to me, I only ask that you at least
include me in on what the reasons are. Thanks.
quasi
I think you've inadvertently been mistaken for another sci.math
participant who goes by various names such as tommy1729, amy666, and
master1729, but I think we can infer from the contents of the above
reply that you and tommy1729 are not the same person. Once people
realize that, the tone of replies to you will probably get less
hostile.
quasi
David R Tribble
>> It's been at least a couple of years since you and I have been in the
>> same sci.math thread, so I don't know from whence your animosity
>
quasi wrote:
> I think you've inadvertently been mistaken for another sci.math
> participant who goes by various names such as tommy1729, amy666, and
> master1729, but I think we can infer from the contents of the above
> reply that you and tommy1729 are not the same person. Once people
> realize that, the tone of replies to you will probably get less hostile.
Indeed. I assumed initially it was Tommy, too, in the previous
thread, at least until I read the poster's entire question.
However, the tone and, more significantly, the depth of mathematical
understanding that was obvious from the poster's question led me to
conclude that I had mis-judged. (Either that, or Tommy was using
yet another user-ID but was acting like he had taken more meds
this time.)
Not the OP's fault, but he can surely understand that sci.math
has more than an adequate number of visiting cranks, and that
many of them deride NSA/IST and set theory in general.
Math1723
>
> I think you've inadvertently been mistaken for another sci.math
> participant who goes by various names such as tommy1729, amy666, and
> master1729, but I think we can infer from the contents of the above
> reply that you and tommy1729 are not the same person. Once people
> realize that, the tone of replies to you will probably get less
> hostile.
Thanks for that update, quasi. Quite understandable. I appreciate
your taking the time to clear that up for me.
Math1723
>
> Not the OP's fault, but he can surely understand that sci.math
> has more than an adequate number of visiting cranks, and that
> many of them deride NSA/IST and set theory in general.
I certainly can understand. I have been off of sci.math for a few
years (as life has its own demands), but sadly I am seeing that JSH is
still posting away (has he finally proven FLT?), Archimedes Plutonium
is still at, and other cranks seems to have continued unabated. I
apparently had the misfortune of posting an NSA question on the heals
of some crank work, hence this confusion. I am just saddened that I
must wade through crankdom to get questions posted. But I shall take
the good with the bad.
My apologies to David Ullrich if I got a little pissy. Had I kept my
head about me, I would have seen that this was an obvious case of
mistaken identity.
Tim Little
> For any infinitesimal e > 0, clearly lim n->inf (e*n) remains
> infinitesimal.
What would you mean by "lim n->inf (e*n)"? Ordinarily, it means some
L in R such that for all eps in R+, there exists k in N such that
|e*n - L| < eps for all n in N where n > k.
By that definition, L=0 suffices. But is that appropriate for NSA,
since all of R, R+, and N are external sets? If you replace "n in N"
with "n in *N", then there is no limit: e*n exceeds any bound.
> Perhaps the more basic question is: Can NSA even have a well-defined
> definition for Countable Additivity in general?
How about starting even more basic: what is the definition of
"sequence" in NSA? Then we can move on to more complex questions like
defining the sum of such a thing.
--
Tim
Transfer Principle
> Essentially one question here: Is it true that the countable sum of
> infinitesimals is still infinitesimal?
> In IST, infinite countable sizes are not accessible as they are
> external. (Internally, Hyper-Countable sets are either finite or
> uncountable.) So we are left with looking at this analytically.
> For any infinitesimal e > 0, clearly lim n->inf (e*n) remains
> infinitesimal. However, it seems to me possible that you could
> somehow have a growing sequence of infinitesimals such that e1 + e2 +
> e3 + ... is a finite number.
The poster Fred Jeffries tried to give an infinitesimal such that
one really can have infinitely many infinitesimals add up to one.
To do so, Jeffries began by finding some hypernatural number H
that is divisible by all standard natural numbers. It's easy to
find such a hypernatural in NSA. Assuming that one is defining
hyperreals as equivalence classes of sequences of reals, one
such hypernatural is the class containing the sequence:
{0!, 1!, 2!, 3!, 4!, 5!, 6!, ..., n!, ...}
Then Jeffries applies the Transfer Principle to:
1/n + 1/n + 1/n + ... + 1/n (n times) = 1
to obtain
1/H + 1/H + 1/H + ... + 1/H (H times) = 1
Not only that, but since every pofnat divides H, we can obtain
every standard positive rational as a sum of 1/H's. For example,
1/H + 1/H + 1/H + ... + 1/H (H/2 times) = 1/2
Hopefully, the poster Fred Jeffries himself will come post in
this thread, to clarify what he is doing with 1/H here.
Math1723
> On 2010-12-16, Math1723 <anonym1...@aol.com> wrote:
>
> > For any infinitesimal e > 0, clearly lim n->inf (e*n) remains
> > infinitesimal.
>
> What would you mean by "lim n->inf (e*n)"? Ordinarily, it means some
> L in R such that for all eps in R+, there exists k in N such that
> |e*n - L| < eps for all n in N where n > k.
>
> By that definition, L=0 suffices. But is that appropriate for NSA,
> since all of R, R+, and N are external sets? If you replace "n in N"
> with "n in *N", then there is no limit: e*n exceeds any bound.
You are absolutely right, that was exceedingly sloppy of me. I caught
that only after I posted (and attempted to fix that in some follow on
posts). In my head, I was using "lim n->inf" so as to cover countably
infinite n. This works in Standard Analysis, not in Non-Standard
Analysis.
> > Perhaps the more basic question is: Can NSA even have a well-defined
> > definition for Countable Additivity in general?
>
> How about starting even more basic: what is the definition of
> "sequence" in NSA? Then we can move on to more complex questions like
> defining the sum of such a thing.
Okay, here is where I am going with this. Please feel free to jump in
and point out anywhere you think I am in error. I will try to answer
your questions in my flow:
Ordinarily, a sequence or reals can be viewed as simply a function
from N to R. In NSA (by Transfer) it would be a function from *N to
*R. Similarly, the sum of the series would be the limit (if it
exists) in both cases.
So, let's say we have a sequence { a_n } for n in *N of positive
infinitesimals. Now, let M be some subset of *N. I am considering
the partial sum of A = Sum (n in M) { a_n }.
Clearly when M is a finite set, A will converge and be an
infinitesimal. This process is also well defined whenever M is an
internal set (although the partial sum may not converge). I was
considering the case where M = N (the set of standard natural
numbers). Now granted, I am aware that N is an external set, and
given that *R is Dedekind incomplete, we cannot be guaranteed that A
will converge (even when bounded from above).
It seems to me that if A exists, it must still be infinitesimal. I
can't see how to prove it, but the reason I think so can be very
sloppily described in this way: It has to do with hyper-naturals
being uncountable. Any infinitesimal is bounded above and below by 1/
n and 1/n-1 for some unlimited hyper-natural *N. Thus, I "loosely"
think of infinitesimals as the inverse of uncountable infinities, and
you can't add only a countable number of these to get back to an
appreciable (finite but not infinitesimal) size. I realize that my
words here are non-rigorous, but it is essentially my intuition about
it.
But what if the sequence of infinitesimals is monotonically increasing
in a dramatic way? Could it be that a countable infinite sum of these
become appreciable? I don't think so, but I'd like to see the proof.
I first considered something like a_n = e*2^n for e some constant
infinitesimal. Clearly the series diverges to infinity when summed
over all n in *N, and the partial series remains infinitesimal when
summed over only a finite number of n in N. I think I can prove that
when summed over all n in N, the result must be either infinitesimal
or that the series does not converge to any hyper-real number. But
could there be other sequences which can converge to some finite (non-
infinitesimal) value? I don't believe so, but I desire proof not
belief here.
Thanks for your time reading this.
Tim Little
> So, let's say we have a sequence { a_n } for n in *N of positive
> infinitesimals. Now, let M be some subset of *N. I am considering
> the partial sum of A = Sum (n in M) { a_n }.
>
> Clearly when M is a finite set, A will converge and be an
> infinitesimal. This process is also well defined whenever M is an
> internal set (although the partial sum may not converge).
I'm still not sure how the sum is defined in that case. Certainly the
principle of hypernatural induction would apply to the partial sums
Sum_(n < k) a_n for each k in *N, but I'm not sure how it generalizes
to arbitrary sets.
> I was considering the case where M = N (the set of standard natural
> numbers). Now granted, I am aware that N is an external set, and
> given that *R is Dedekind incomplete, we cannot be guaranteed that A
> will converge (even when bounded from above).
I'm not even sure that we can define the sum. Certainly we cannot use
the principle of hypernatural induction, as the upper bound on indices
is not a hypernatural.
Perhaps we can use the standard analysis concept of infinite limit,
the most straightforward translation being: the limit L in *R such
that for all eps > 0 in *R, there exists m in N such that
|L - sum_(i <= n) a_i| < eps for all n > m.
However, I think this always fails if there are infinitely many
nonzero terms. If we take k to be any unlimited hypernatural, we can
let eps be the product of all the terms up to k. Regardless of L and
m, the variation in the sequence of subsequent partial sums will be
too great.
The next most straightforward translation (restricting eps to R) also
fails, as this time all infinitesimal L (including negative ones)
suffice equally. A sequence of positive values should not have a
negative limit!
After that, I'm not sure where to go.
> I first considered something like a_n = e*2^n for e some constant
> infinitesimal.
Even more dramatic: a_n = e^(1/n). The ratio of successive terms is
infinite!
- Tim
FredJeffries
>
> Ordinarily, a sequence or reals can be viewed as simply a function
> from N to R. In NSA (by Transfer) it would be a function from *N to
> *R. Similarly, the sum of the series would be the limit (if it
> exists) in both cases.
>
> So, let's say we have a sequence { a_n } for n in *N of positive
> infinitesimals. Now, let M be some subset of *N. I am considering
> the partial sum of A = Sum (n in M) { a_n }.
>
> Clearly when M is a finite set, A will converge and be an
> infinitesimal. This process is also well defined whenever M is an
> internal set (although the partial sum may not converge). I was
> considering the case where M = N (the set of standard natural
> numbers). Now granted, I am aware that N is an external set, and
> given that *R is Dedekind incomplete, we cannot be guaranteed that A
> will converge (even when bounded from above).
>
> It seems to me that if A exists, it must still be infinitesimal.
Supposing, for the sake of argument, that A is not infinitesimal. You
agree, I take it, that what we mean by the sum of the infinite series
is the limit of the sequence of (finite) partial sums and that each of
the finite partial sums is infinitesimal.
Try something like: If A > 0 is the limit, then there must be some
(finite) partial sum between A/2 and A
FredJeffries
> from N to R. In NSA (by Transfer) it would be a function from *N to
> *R. Similarly, the sum of the series would be the limit (if it
> exists) in both cases.
>
> So, let's say we have a sequence { a_n } for n in *N of positive
> infinitesimals. Now, let M be some subset of *N. I am considering
> the partial sum of A = Sum (n in M) { a_n }.
>
> Clearly when M is a finite set, A will converge and be an
> infinitesimal. This process is also well defined whenever M is an
> internal set (although the partial sum may not converge). I was
> considering the case where M = N (the set of standard natural
> numbers). Now granted, I am aware that N is an external set, and
> given that *R is Dedekind incomplete, we cannot be guaranteed that A
> will converge (even when bounded from above).
Don't know if this will work (or if it even makes sense):
Since for each unlimited H, N is a subset of
{0, 1, 2, 3, ..., H-1, H}
perhaps you could "define" the sum over N as the inf of the sums from
0 to H for H unlimited (if such a thing exists).
Math1723
>
> Don't know if this will work (or if it even makes sense):
>
> Since for each unlimited H, N is a subset of
> {0, 1, 2, 3, ..., H-1, H}
>
> perhaps you could "define" the sum over N as the inf of the sums from
> 0 to H for H unlimited (if such a thing exists).
Given the Dedekind incompleteness of *R, I cannot assume that supremum
of the partial sums for i=0 to n (with n limited) would be the same as
the infimum of all partial sums for i=0 to n (with n unlimited). (It
would be nice if it were true, but I haven't seen that proof.)
There are times (perhaps all the time?) when the two nicely line up,
so that any partial sum that is between falls in the "holes" of *R's
Dedekind incompleteness.
For example, consider the simple series of a_n = 2^-n. The series
summed over all *N obviously gives the sum as 2. What is interesting
is looking at partial sums:
1. When summing n=0 to n with n limited gives you a result which is
less than 2. Moreover, this sum is always an appreciable (non-
infinitesimal) deitance from 2.
2. When summing n=0 to n with n unlimited gives you a result which is
less than 2. Moreover, this sum is in the shadow of 2 (an
infinitesimal distance away from 2).
3. When summing over all limited n, the sum does converge to any hyper-
real value. It can be shown that if the sum exists, it must be
greater than any hyper-real whose real part is less than 2. It can
also be shown that if this sum exists, it must be les than any hyper-
real in the shadow of 2. Since all limited hyper-reals can be
uniquely written as r + i (where r is a standard real, and i and
infinitesimal), these result prove that this partial sum does not
converge. (It essentially sums to a value which is a "hole" in *R.)
Thanks.
Han de Bruijn
> Aatu Koskensilta <aatu.koskensi...@uta.fi> writes:
> > David C. Ullrich <ullr...@math.okstate.edu> writes:
>
> >> If you can't say then you're lying again when you say (i) is
> >> clear. (Just like you lied about how those papers defined "lacunary",
> >> and about who really wrote that amazing proof of the Beatty thing,
> >> etc. Lying is not good.)
>
> > It seems you have some reason to think Math1732 is in fact none other
> > than our dear master/tommy?
>
> I'm sure that David mistakenly identified the two because of the similar
> posting names (Math1732 and tommy1729).
Sort of incredible. Even a Dutchman can see that e.g. their mastering
of the English language are (uhm ..) quite different.
Han de Bruijn
Jesse F. Hughes
Yes, that's true. I guess that David didn't read the post carefully at
all.
--
Jesse F. Hughes
"Depression hits more people than thought."
--headline in Lexington, KY newspaper, as reported on
NPR's Morning Edition
master1729
and that is really sad.
it shows that judgement is not based on content , but on attidude such as doubt on conformistic ideas and WHO is talking , rather than the content he is saying.
they cant even identify me !
but since the OP dared to question conformisitic ideas , he had to be me and say stupid things.
now that he isnt me , his content and question is - despite in my spirit - suddenly much more respected ...
how low can you go ...
abolish fractions now ( remember that ? )
still the same.
non non rien n'a changé.
Math1723
Hey, everybody screws up now and again. I take no offense. I'm sure
if/when he checks back and sees this, he'd be the first to apologize.
Thanks for having my back, you all. :-)
Han de Bruijn
But admittedly it's confusing, when reading news items in a glimpse:
Math1723
master1729
Han de Bruijn
David C. Ullrich
Not the first, I've been "away" for a day or so.
But sorry.
The problem had to do with the funny names and also the fact
that timmy's insistence that infinitesmals solve the supposed
problem of non-measurable sets. The first fact you mentioned,
that if e is infinitesmal then n*e cannot converge to a
non-infinitesmal, is precisely something he was unable or
unwilling to grasp. Now, it happens a lot that he denies
something, and then when faced with evidence he can't
deny he revises his claim to something equally absurd,
showing he didn't really understand the reason his
original claim was absurd in the first place. Your question
looked exactly like that... sorry.
sto
, is precisely something he was unable or
> unwilling to grasp. Now, it happens a lot that he denies
> something, and then when faced with evidence he can't
> deny he revises his claim to something equally absurd,
> showing he didn't really understand the reason his
> original claim was absurd in the first place.
This is odd. I've had this exact same conversation with a couple of
people that have never posted anything on the Internet. It basically
runs along the lines of first making a clearly false assertion then,
when presented with evidence that the first assertion is false, making a
second false assertion to refute the proof that the first assertion is
false and so on. When the sequence of false assertions grows long
enough, the person forgets what the original assertion was. I would
conjecture that perhaps this is the natural way to think for someone
having no training in mathematics or science, but I am not sure.
Perhaps it is some kind of combination of intrinsic personality type and
of never having had the experience of standing at the blackboard staring
out at the PhD qualifying exam committee.
Math1723
>
> Not the first, I've been "away" for a day or so.
>
> But sorry.
No problem. Sorry if I was such a jerk about it. Obviously an honest
mistake.
> The problem had to do with the funny names and also the fact
> that timmy's insistence that infinitesmals solve the supposed
> problem of non-measurable sets.
Ah, I see. I was unaware that this was even a thought. This will
absolutely not work, as assigning a non-measurable set some
infinitesimal value will still cause Countable Additivity to fail.
This is actually trivial to prove:
Construct a Vitali sets in the usual manner, so that { V_i } is a
countable collection in which U V_i = [0,1). Since Lebesque Measure
is translation invariant, if the m(V_i0) exists for some i0 and is
equal to e, then m(V_i)=e for all i. Furthermore, by Countable
Additivity, we have e + e + e + ... = 1.
Obviously no real number has that property. Suppose then we extend
the definition of Lebesque measure so that it includes Hyper-Reals and
observe the above example again. We can still yield a contradiction
because now EVERY countably infinite collection of { V_i } has measure
1. To see this, note that each V_i contains a unique rational number,
call it q(V_i). Let V1 = { V_i | q(V_i) < 0.5 } and V2 = { V_i |
q(V_i) >= 0.5 }. Since V1 and V2 are each countable collections V_i
(each of size e), we have:
m(V1) = e + e + e + ... = 1
m(V2) = e + e + e + ... = 1
And since V1 and V2 are disjoint, we are left with the contradiction 1
= m[0,1) = m(V1 U V2) = m(V1)+m(v2) = 1+1 = 2.
Placing this question within the confines are Measure Theory makes it
quite obvious now to me that no countable collection of infinitesimals
could ever yield an appreciable value.
David: you were quite right in saying that it should have been obvious
to me. I can only say I am a bit rusty (having graduated with my Math
degree over 20 years ago, I don't get to use it as much as I'd like as
a software developer).
Transfer Principle
> > I think you've inadvertently been mistaken for
> > another sci.math
> > participant who goes by various names such as
> > tommy1729, amy666, and
> > master1729, but I think we can infer from the
> > contents of the above
> > reply that you and tommy1729 are not the same person.
> > Once people
> > realize that, the tone of replies to you will
> > probably get less hostile.
> > quasi
> and that is really sad.
> it shows that judgement is not based on content , but on attidude such as doubt on conformistic ideas and WHO is talking , rather than the content he is saying.
Unfortunately, that's the reality of the situation. Both of us have
been labeled as "unwilling to learn," and because of this, we will
regularly be given more negative responses than those who have not
received such a label.
There is another math website quickly gaining prominence, called
Math Overflow. One distinguishing feature of Math Overflow is that
posters receive something called "reputation points." Similar to
Google stars, reputation points are awarded when one writes good
posts, and are deducted when one writes bad points. Only those who
already have points can award or deduct points. (Presumably at the
time the site was created, the creators of the website started out
with a large positive point total -- otherwise no one would ever
have any points at all.)
There are several ways to log in to Math Overflow -- including
using a Google account, like the account that I'm currently using
to post to Usenet. But I know better than to attempt to use my
Google account to log in to Math Overflow, for the minute that
someone recognizes me and knows what type of posts I write here
at sci.math, I'd have a negative point total. Indeed, this is
exactly what happened when another well-known sci.math "crank"
tried to log in to Math Overflow using the same account that he
uses to post to sci.math.
But, as it turns out, sci.math has the same "reputation point"
system that Math Overflow has, except it's figurative. For Math
Overflow only rigorously quantifies what sci.math users already
know about posters' reputations. We (i.e., you tommy1729 and I)
don't need a big negative number next to our names -- the others
already know us to have low reputations without a literal
"reputation point" system.
Is there anything the two of us can do to raise our reputation? It
won't be easy. For me to simply change my posting habits will be
insufficient, because others will be skeptical that I've really
changed my personality (and understandably so, since I've been
posting here for over three years). Also, those who have already
killfiled me would never see that I even changed.
The two of us could try the following steps, which are more likely
to raise our reputations:
1. Change our usernames. Of course, you (tommy1729) already tried
this, but you're easily caught unless the other steps are followed.
2. Post from a different email and IP address.
3. Post from a different Usenet interface. Since you usually use
Mathforum and I use Google, we could switch to each other's Usenet
interface, but posting from a traditional NNTP newsserver with a
traditional newsreader would be more effective.
4. Eliminate our most blatantly bad habits. For you, you would need
to use more capitals. Once again, you already tried to include more
caps when you posted as Napier2, but there were still enough caps
errors to be caught. For me, I'd need to avoid mentioning other
poster's names as often as possible. Most of the time, when I
mention another poster's name, I'm trying to guess their reason for
posting, which is likely to be wrong, and then I am accused of
misrepresenting or outright "lying" about what is being said. So
to avoid mentioning names eliminates the temptation for me to
misrepresent other posters. Sticking to first and second person
pronouns is desirable.
5. Avoid phrases like "I'm right" and "you're wrong," and use
phrases like "you're right" and "I'm wrong." Even though I prefer
to use "I disagree" to "you're wrong," even "I disagree" is to be
avoided, if my sole purpose is to gain reputation points.
Even though I'd like to raise my reputation, I don't plan on
following steps 1-5 in the immediate future, but if I ever become
desperate to have a civil math discussion, I'll have no choice
but to follow these five steps.
Han de Bruijn
How about _content_, instead of form ?
Han de Bruijn
Jesse F. Hughes
> On Dec 21, 2:35 am, Transfer Principle <lwal...@lausd.net> wrote:
>>
>> Unfortunately, that's the reality of the situation. Both of us have
>> been labeled as "unwilling to learn," and because of this, we will
>> regularly be given more negative responses than those who have not
>> received such a label.
[...]
>> Is there anything the two of us can do to raise our reputation? It
>> won't be easy. For me to simply change my posting habits will be
>> insufficient, because others will be skeptical that I've really
>> changed my personality (and understandably so, since I've been
>> posting here for over three years). Also, those who have already
>> killfiled me would never see that I even changed.
>>
>> The two of us could try the following steps, which are more likely
>> to raise our reputations:
[...]
>> Even though I'd like to raise my reputation, I don't plan on
>> following steps 1-5 in the immediate future, but if I ever become
>> desperate to have a civil math discussion, I'll have no choice
>> but to follow these five steps.
>
> How about _content_, instead of form ?
It's like you've never read Walker's posts before or something.
--
Jesse F. Hughes
"Most people don't even know what a rootkit is, so why should they
care about it."
-- Thomas Hesse, sony executive defends DRM-by-rootkit.
Han de Bruijn
> Han de Bruijn <umum...@gmail.com> writes:
>
> > On Dec 21, 2:35 am, Transfer Principle <lwal...@lausd.net> wrote:
>
> >> Unfortunately, that's the reality of the situation. Both of us have
> >> been labeled as "unwilling to learn," and because of this, we will
> >> regularly be given more negative responses than those who have not
> >> received such a label.
>
> [...]
>
> >> Is there anything the two of us can do to raise our reputation? It
> >> won't be easy. For me to simply change my posting habits will be
> >> insufficient, because others will be skeptical that I've really
> >> changed my personality (and understandably so, since I've been
> >> posting here for over three years). Also, those who have already
> >> killfiled me would never see that I even changed.
>
> >> The two of us could try the following steps, which are more likely
> >> to raise our reputations:
>
> [...]
>
> >> Even though I'd like to raise my reputation, I don't plan on
> >> following steps 1-5 in the immediate future, but if I ever become
> >> desperate to have a civil math discussion, I'll have no choice
> >> but to follow these five steps.
>
> > How about _content_, instead of form ?
>
> It's like you've never read Walker's posts before or something.
Oh, well, I'm convinced that he can do better than this.
Han de Bruijn
Transfer Principle
The form vs. content idea has come up in earlier threads. So I
accept your suggestion that I pay more attention to the
distinction between the two.
Also, I notice that I've been criticized for the way that I
judge and analyze _others'_ form/content as much as for my _own_
form/content. So, if I want to raise my reputation, then I must
focus on all four areas:
1. My judgment of others' form
2. My judgment of others' content
3. My own form
4. My own content
Let me focus on each area in more detail. Please feel free to
jump in and correct me if I fail to show understanding of the
form/content distinction.
1. My judgment of others' form
Posters regularly criticize each other's form all the time -- for
example, those who write in all lowercase (as I already mentioned
in this thread). Word choice also counts as form, so when I talk
about people who use certain five-letter words, this counts as my
judgment of others' form.
How can I improve in this area? For starters, I should avoid
worrying about the specific five-letter words as much, and focus
more on the actual criticisms. Instead of saying,
"I don't think that he should be called a 'crank' because he
prefers this to that..."
I should say something like:
'I don't think that he shows willful ignorance on the subject at
hand, just because he prefers this to that..."
2. My judgment of others' content
I interpret "content" as referring to mathematical content. So
when a poster criticizes another for, say, not understanding set
theory, classical analysis, or abstract algebra, then one is
criticizing another poster's content.
How can I improve in this area? I should try to figure out what
theory a poster is working in, since an accurate judgment of what
one is saying is highly dependent on the theory -- especially
with regards to whether one is considering a standard theory or
one that he is trying to create. I also need to find a way to ask
the poster relevant questions, if necessary, in order to make a
more accurate judgment about what a poster is saying, without
interrogating the poster. The question mark key is to the left of
the right-shift key, and this is a key combination that I need to
learn how to use in the proper situations.
3. My own form
I need to avoid any form that is likely to get me killfiled, and
employ forms that will get me out of the killfiles that I am
already in. This entails whatever combination of username, email
address, and Usenet access that will allow me to do so. I could
have lots of great content in a post, but plonkers will never see
any of it unless I improve my form here.
4. My own content
What sort of content should my posts have to make them more
relevant and interesting? For one thing, after I find out what
theory is being discussed, I should focus on that theory, and not
bring up other theories. If a post is about standard theory, then
I should have content relevant only to standard theory. If a post
mentions alternate theories, I still should avoid mentioning yet
other alternate theories, no matter how uncanny I believe the
similarities between the theory at hand and the theory I bring up
to be.
So instead of 1-5 in the earlier post, the current 1-4 is what I
would need to focus on to raise my reputation. Right now, the
situation isn't completely desperate, so I'm not ready to follow
_all_ of 1-4 (especially #3), but there's no harm in thinking
about 1-4 each time I post.
> Oh, well, I'm convinced that he can do better than this.
Thanks, HdB. And once again, if I'm mistaken about the exact
nature of the form vs. content distinction, then please correct
me, and I'll adjust 1-4 accordingly.
Jesse F. Hughes
> The form vs. content idea has come up in earlier threads. So I
> accept your suggestion that I pay more attention to the
> distinction between the two.
>
> Also, I notice that I've been criticized for the way that I
> judge and analyze _others'_ form/content as much as for my _own_
> form/content. So, if I want to raise my reputation, then I must
> focus on all four areas:
>
> 1. My judgment of others' form
> 2. My judgment of others' content
> 3. My own form
> 4. My own content
You know what else is good? Extended posts of online introspection.
That's sure to entertain others.
--
Jesse F. Hughes
"How come there's still apes running around loose and there are
humans? Why did some of them decide to evolve and some did not? Did
they choose to stay as a monkey or what?" -Kans. Board of Ed member
Han de Bruijn
How about: less verbose?
Han de Bruijn
Math1723
>
> The poster Fred Jeffries tried to give an infinitesimal such that
> one really can have infinitely many infinitesimals add up to one.
The issue is to be able to do that with only countably infinitely many
infinitesimals. Based upon the discussions I have seen here, I don't
believe you can add countably many infinitesimals to arrive at an
appreciable amount.
> To do so, Jeffries began by finding some hypernatural number H
> that is divisible by all standard natural numbers.
That's pretty easy to do. The simplest way is to choose some
unlimited n in *N, and consider n! (that is, n factorial). This
number is divisible by all values less than n (and thus all finite
naturals).
> It's easy to find such a hypernatural in NSA. Assuming that one
> is defining hyperreals as equivalence classes of sequences of
> reals, one such hypernatural is the class containing the sequence:
>
> {0!, 1!, 2!, 3!, 4!, 5!, 6!, ..., n!, ...}
Yes, this would do nicely.
> Then Jeffries applies the Transfer Principle to:
>
> 1/n + 1/n + 1/n + ... + 1/n (n times) = 1
>
> to obtain
>
> 1/H + 1/H + 1/H + ... + 1/H (H times) = 1
Note that since H is an unlimited Hyper-Natural, H times is
uncountably many. In Standard Analysis, Countable Addivity is
defined, but Uncountable Addivity is not (It cannot be, as uncountably
many zero lengthed points would require all intervals to be of measure
zero.)
By Transfer, Hyper-Countable Addivity is defined, which means that
some uncountable series can be added up (namely, those internal sets
which are Hyper-Countable), whilst others cannot (namely, those
internal sets which are Hyper-Uncountable, and any external sets).
> Not only that, but since every pofnat divides H, we can obtain
> every standard positive rational as a sum of 1/H's. For example,
>
> 1/H + 1/H + 1/H + ... + 1/H (H/2 times) = 1/2
What is a "pofnat"?
In any case, it's not clear to me that this uncountable sum is one of
the allowable ones under Hyper-Countable Addivity.
> Hopefully, the poster Fred Jeffries himself will come post in
> this thread, to clarify what he is doing with 1/H here.
I'd be interested in seeing that.
Thanks.
Math1723
The only problem with this approach is that any unlimited H would be
of uncountably infinite size, and Uncountable Additivity is not (in
general) defined. The only time is defined is in the narrow case of
Hyper-Countable Addivity (thanks to Transfer), but this obviously does
not apply to external sets.
Transfer Principle
> > 1/H + 1/H + 1/H + ... + 1/H (H times) = 1
> Note that since H is an unlimited Hyper-Natural, H times is
> uncountably many. In Standard Analysis, Countable Addivity is
> defined, but Uncountable Addivity is not (It cannot be, as uncountably
> many zero lengthed points would require all intervals to be of measure
> zero.)
OK, I see what you mean here.
> > Not only that, but since every pofnat divides H, we can obtain
> > every standard positive rational as a sum of 1/H's. For example,
> > 1/H + 1/H + 1/H + ... + 1/H (H/2 times) = 1/2
> What is a "pofnat"?
Plain old finite natural number. Some posters use that
term "pofnat" to denote _not_ a hypernatural, or a
hyperfinite natural, or a nonstandard natural, but a
plain old finite _standard_ natural number.
> > Hopefully, the poster Fred Jeffries himself will come post in
> > this thread, to clarify what he is doing with 1/H here.
> I'd be interested in seeing that.
Me too.
Transfer Principle
OK, here is a less verbose way of discussing form/content issue:
Apparently, some posts contain _no_ mathematical content, and
some of you want me to judge their opponents' posts to contain
_no_ mathematical content whatsoever.
But I'm _still_ not as willing to judge other posts to have no
mathematical content as most of you are.
Tim Little
> Based upon the discussions I have seen here, I don't believe you can
> add countably many infinitesimals to arrive at an appreciable
> amount.
It's all a question of what properties of addition you wish to give
up. It is obviously trivial to define a "Sum" operator acting on
sequences of some superset X of R, such that for any a:N->R we have
the usual definition of Sum(a), and that there exists x in X such that
a(n) = x for all n and Sum(a) = 1.
You won't find it in Robinson's hyperreals, though.
>> Not only that, but since every pofnat divides H, we can obtain
>> every standard positive rational as a sum of 1/H's. For example,
>>
>> 1/H + 1/H + 1/H + ... + 1/H (H/2 times) = 1/2
>
> What is a "pofnat"?
A sci.math-ism for "plain old finite natural" number.
- Tim
Han de Bruijn
> On Dec 22 2010, 3:22 am, Han de Bruijn <umum...@gmail.com> wrote:
>
> > On Dec 21, 10:07 pm, Transfer Principle <lwal...@lausd.net> wrote:
> > > Thanks, HdB. And once again, if I'm mistaken about the exact
> > > nature of the form vs. content distinction, then please correct
> > > me, and I'll adjust 1-4 accordingly.
> > How about: less verbose?
>
> OK, here is a less verbose way of discussing form/content issue:
The "less verbose" requirement is quite essential for an international
forum like "sci.math". Many of us are not native English speakers; I'm
getting tired rather quickly while reading those elaborate, courtroom
style argumentations. (My preferred author therefore is Robert Israel:
always to the point and as concise as possible. Even more important,
formulated in the universal language of mathematics itself !)
> Apparently, some posts contain _no_ mathematical content, and
> some of you want me to judge their opponents' posts to contain
> _no_ mathematical content whatsoever.
>
> But I'm _still_ not as willing to judge other posts to have no
> mathematical content as most of you are.
See? I simply do not understand what you mean. Obscure formulations
have no effect with us.
Han de Bruijn
Math1723
>
> The "less verbose" requirement is quite essential for an international
> forum like "sci.math". Many of us are not native English speakers; I'm
> getting tired rather quickly while reading those elaborate, courtroom
> style argumentations.
I certainly understand that you may not be completely fluent enough in
English to grasp all of the subtleties in the Mathematics posted
here. Sadly though, many times there is no way around being rigorous
in the language (and thus verbose).
Is there not a Mathematics newsgroup in your native language in which
you can fully understand the answers to your questions?
Han de Bruijn
This is what you snipped, so I guess you missed half of my argument:
> (My preferred author therefore is Robert Israel:
> always to the point and as concise as possible. Even more important,
> formulated in the universal language of mathematics itself !)
Han de Bruijn
Math1723
>
> This is what you snipped, so I guess you missed half of my argument:
>
> > (My preferred author therefore is Robert Israel:
> > always to the point and as concise as possible. Even more important,
> > formulated in the universal language of mathematics itself !)
I snipped it because it wasn't related to my question. I was asking
if there was not a newsgroup native in your own language. I wasn't
trying to be argumentative, or even suggesting you stop reading here.
Not at all. I was merely asking in the hopes that you had some other
alternatives. My apologies if you took my question the wrong way.
Don Stockbauer
********************************
"Do you guys read what's typed into your search field?"
Han de Bruijn
Yes, there are newsgroups in my own language, but I'm a globalist :-)
Han de Bruijn
Math1723
What "search field"?
FredJeffries
That was two months ago in response to your "I still believe that it's
possible in some exotic theory to make infinitesimals add up to 1"
http://groups.google.com/group/sci.math/msg/23c94da8062243de?hl=en
http://groups.google.com/group/sci.math/msg/f4b4c2e102328b6c?hl=en
The hyperfinite grid seems to be a standard non-standard analysis
tool. See for instance Nogel Cutland's "Loeb Measures in Practice:
Recent Advances"
http://books.google.com/books?id=nWffKRIgDpwC
http://www.springerlink.com/content/y7jxpeeuwj22/
For an application to Brownian Motion see Vieri Benci's
http://www.dm.unipi.it/~ultramath/slides/benci-slides.pdf
Transfer Principle
> On Jan 3, 11:35 pm, Transfer Principle <lwal...@lausd.net> wrote:
> > OK, here is a less verbose way of discussing form/content issue:
> The "less verbose" requirement is quite essential for an international
> forum like "sci.math". Many of us are not native English speakers
Good point. So I will try to be more sensitive to those whose
first language is not English.
> > But I'm _still_ not as willing to judge other posts to have no
> > mathematical content as most of you are.
> See? I simply do not understand what you mean. Obscure formulations
> have no effect with us.
Earlier, you mentioned form vs. content:
> > How about _content_, instead of form ?
Apparently, I don't fully understand the distinction between form
and content. My "obscure formulations" were my guesses as to what
that distinction is -- _wrong_ guesses, apparently.
Transfer Principle
> On Jan 3, 2:30 pm, Transfer Principle <lwal...@lausd.net> wrote:
> > Me too.
> That was two months ago in response to your "I still believe that it's
> possible in some exotic theory to make infinitesimals add up to 1"
> http://groups.google.com/group/sci.math/msg/23c94da8062243de?hl=en
> http://groups.google.com/group/sci.math/msg/f4b4c2e102328b6c?hl=en
Thanks. The issue raised here is that in that thread, you had:
1/H + 1/H + ... + 1/H + 1/H (H times) = 1
But H is usually considered to be an _uncountable_ infinity
(since the hyperreals are uncountable). So it doesn't give us
1 as the sum of _countably_ many infinitesimals.
Han de Bruijn
You wrote, for example:
> But I'm _still_ not as willing to judge other posts to have no
> mathematical content as most of you are.
I just don't get it. Can't you formulate that in a simpler way?
Han de Bruijn
FredJeffries
I'm sorry. I do not understand what the issue IS. Do you have a more
specific question?
Yes, I know that the set {1, 2, ..., H-1, H} for H unlimited is
uncountable (I believe I showed YOU the proof of that).
I don't claim to have a system or an addition wherein one can make
countably many infinitesimals add up to 1.
Math1723
>
> I'm sorry. I do not understand what the issue IS. Do you have a more
> specific question?
>
> Yes, I know that the set {1, 2, ..., H-1, H} for H unlimited is
> uncountable (I believe I showed YOU the proof of that).
>
> I don't claim to have a system or an addition wherein one can make
> countably many infinitesimals add up to 1.
Yes, that was the question (countable additivity of infinitesimals),
and thanks for answering it!
FredJeffries
On the other hand, there ARE countable nonstandard models of the
natural numbers, so if we consider only hyper-rationals we might be
able to get it...
Transfer Principle
How about this -- rather than try to explain, let me give a
specific example.
Earlier this week in another thread, the poster Tim Golden
wrote the following:
Golden, last Tuesday, approx. 2PM Greenwich:
"Wow; a zero content post from Bill Dubuque. This is a non sequitur.
Afraid to speak clearly on notational abuse, Bill?
This was followed shortly by Marshall Spight:
Spight, last Tuesday, approx. 2:30PM Greenwich:
"When you just post the same wrong things over and over and over,
it doesn't qualify as 'content.'"
To summarize:
According to Golden, Dubuque's posts contain zero content.
According to Spight, Golden's posts contain zero content.
But according to me, _both_ Dubuque's _and_ Golden's posts
contain mathematical content. Different content, of course,
but nonetheless content.
I hope that this example helps explain what I mean.
Transfer Principle
> The hyperfinite grid seems to be a standard non-standard analysis
> tool. See for instance Nogel Cutland's "Loeb Measures in Practice:
> Recent Advances"
> http://books.google.com/books?id=nWffKRIgDpwC
> http://www.springerlink.com/content/y7jxpeeuwj22/
OK, I finally had time to look at the links you gave.
First of all, now I see that it's obvious that H needs to be
an uncountable set. For Section 4 of the link reads:
"For convenience we restrict to [0,1]. Fix infinite H in *N and
let delta_t=H^-1. Let T be the hyperfinite set
T = {0, delta_t, 2delta_t, ..., Hdelta_t}. Then T gives a
representation of [0,1] via the standard part map
st = o : T -> [0,1]."
So st maps T to [0,1] -- and this map must be surjective,
lest what the author does later on with st^-1(A) makes no
sense at all. Thus T (and hence H) is uncountable.
And the author goes on to show that all of this is going to
be equivalent to Lebesgue measure. There's no attempt to
make Vitali sets measurable at all.
Bummer. So I'm forced to admit that this route to making
Vitali sets measurable is a dead end.
> On the other hand, there ARE countable nonstandard models of the
> natural numbers, so if we consider only hyper-rationals we might be
> able to get it...
For some reason, to me, all this talk about nonstandard
models of N sounds like cheating. Perhaps it's because
I've seen so many posters attempt to disprove Cantor's
Diagonal Argument that R is uncountable by mentioning
Lowenheim-Skolem, and I already know that such arguments
are always rejected.
Han de Bruijn
It does. Thank you!
Han de Bruijn
Marshall
> On Jan 4, 11:58 pm, Han de Bruijn <umum...@gmail.com> wrote:
>
> > On Jan 5, 5:19 am, Transfer Principle <lwal...@lausd.net> wrote:
> > > But I'm _still_ not as willing to judge other posts to have no
> > > mathematical content as most of you are.
> > I just don't get it. Can't you formulate that in a simpler way?
>
> How about this -- rather than try to explain, let me give a
> specific example.
>
> Earlier this week in another thread, the poster Tim Golden
> wrote the following:
>
> Golden, last Tuesday, approx. 2PM Greenwich:
> "Wow; a zero content post from Bill Dubuque. This is a non sequitur.
> Afraid to speak clearly on notational abuse, Bill?
>
> This was followed shortly by Marshall Spight:
>
> Spight, last Tuesday, approx. 2:30PM Greenwich:
> "When you just post the same wrong things over and over and over,
> it doesn't qualify as 'content.'"
So, do you have little notecards where you write down everything
I post that catches your eye? You are one creepy motherfucker.
Marshall
Jesse F. Hughes
You've heard of Google Groups?
The search function there is mighty broken, but it sometimes gives
results.
--
"Britney thought the idea of a pre-nup was vile, because she is
loved-up with Kevin and cannot envisage breaking up. However, [...] no
one in Hollywood these days get married without brokering a
deal. [...] She had a long chat with Kevin and he was cool about it."
Marshall
> Marshall <marshall.spi...@gmail.com> writes:
> > On Jan 6, 8:41 pm, Transfer Principle <lwal...@lausd.net> wrote:
> >> On Jan 4, 11:58 pm, Han de Bruijn <umum...@gmail.com> wrote:
> >> > On Jan 5, 5:19 am, Transfer Principle <lwal...@lausd.net> wrote:
> >> > > But I'm _still_ not as willing to judge other posts to have no
> >> > > mathematical content as most of you are.
> >> > I just don't get it. Can't you formulate that in a simpler way?
>
> >> How about this -- rather than try to explain, let me give a
> >> specific example.
>
> >> Earlier this week in another thread, the poster Tim Golden
> >> wrote the following:
>
> >> Golden, last Tuesday, approx. 2PM Greenwich:
> >> "Wow; a zero content post from Bill Dubuque. This is a non sequitur.
> >> Afraid to speak clearly on notational abuse, Bill?
>
> >> This was followed shortly by Marshall Spight:
>
> >> Spight, last Tuesday, approx. 2:30PM Greenwich:
> >> "When you just post the same wrong things over and over and over,
> >> it doesn't qualify as 'content.'"
>
> > So, do you have little notecards where you write down everything
> > I post that catches your eye? You are one creepy motherfucker.
>
> You've heard of Google Groups?
>
> The search function there is mighty broken, but it sometimes gives
> results.
Sure sure. Regardless of whether Transfer has a literal shrine
to me in his basement with candles and printouts of my posts
or whether he dredges them up from the Google, he's still creepy.
Whenever he quotes me I feel all dirty, and not in a good way.
Marshall
Math1723
>
> > > I don't claim to have a system or an addition wherein one can make
> > > countably many infinitesimals add up to 1.
>
> > Yes, that was the question (countable additivity of infinitesimals),
> > and thanks for answering it!
>
> On the other hand, there ARE countable nonstandard models of the
> natural numbers, so if we consider only hyper-rationals we might be
> able to get it...
Why are hyper-rationals relevant to this procedure? It seems to me
countable sums of infinitesimals will have this problem, irrespective
of whether they are hyper-rationals or hyper-irrationals.
FredJeffries
Not relevant to your interest in countable additivity, but Leonard
wants to find some system which includes infinitesimals and which some
kind of infinite sum can be defined wherein "it's possible in some
exotic theory to make infinitesimals add up to 1".
http://groups.google.com/group/sci.math/msg/7913299a4c46cbbc?hl=en
is where this all started, I believe.
FredJeffries
>
> > On the other hand, there ARE countable nonstandard models of the
> > natural numbers, so if we consider only hyper-rationals we might be
> > able to get it...
>
> For some reason, to me, all this talk about nonstandard
> models of N sounds like cheating. Perhaps it's because
> I've seen so many posters attempt to disprove Cantor's
> Diagonal Argument that R is uncountable by mentioning
> Lowenheim-Skolem, and I already know that such arguments
> are always rejected.
But it does illustrate the fundamental difference between an infinite
series where the sum is defined as the limit of the sequence of finite
partial sums and a hyper-finite sum defined using the transfer
principle:
Let H be an unlimited hyper-natural-number
In the former, the infinite series sum from 1 to infinity 1/H cannot
sum to any standard amount A since then it would require some finite
partial sum to be greater than A/2.
However since the hyper-finite sum from 1 to H of 1/H is H*(1/H) = 1.
Furthermore, the hyper-finite sum from 1 to 2*H of 1/H is (2*H)*(1/H)
= 2.
Now if we take our countable nonstandard model of the natural numbers
and construct the semi-field of quotients then the set of hyper-
natural-numbers less than or equal to H is countable and the hyper-
finite grid from 0 to H (or 2*H) with mesh 1/H is countable.
This may not be useful in your quest to measure the Vitali set, but,
to me, it is of interest that the ability of a hyper-finite sum of
infinitesimals to be appreciable is NOT due to its having uncountably
many terms.
master1729
> (My preferred author therefore
> is Robert Israel:
> always to the point and as concise as possible. Even
> more important,
> formulated in the universal language of mathematics
> itself !)
Robert Israel RULES !
if only he was more creative.
like making conjectures and similar ' creative bravery '
>
> Han de Bruijn
join the fanclub :)
tommy1729
mjc
If you look at his web site at http://www.math.ubc.ca/~israel/,
you will see that he has done quite a lot.
I guess that he prefers to publish his work at places other than here.
Can't say that I blame him - "publishing" here won't help him make
full professor.
David R Tribble
> But I'm _still_ not as willing to judge other posts to have no
> mathematical content as most of you are.
My conjecture is that people who post using their real names
are more likely to be coherent and correct in what they post
and are more likely to understand mathematics in general
than people who post using an alias.
There are a few notable exceptions to this little conjecture
of mine, obviously.
Math1723
>
> My conjecture is that people who post using their real names
> are more likely to be coherent and correct in what they post
> and are more likely to understand mathematics in general
> than people who post using an alias.
>
> There are a few notable exceptions to this little conjecture
> of mine, obviously.
I'd like to think I'm an exception. And even if you don't grant that,
surely you grant James S. Harris as an exception. :-)
Jesse F. Hughes
Surely, Archimedes Plutonium is another obvious exception.
--
Jesse F. Hughes
"Karl Popper writes of a solution to the logical dielmma of first order
theory. I have been read by his students and no understanding was
findable in my writings." -- Douglas Eagleson
Math1723
>
> Surely, Archimedes Plutonium is another obvious exception.
Yes, but artificially so. Apparently, he was born Ludwig Poehlmann,
but legally changed his name to "Archimedes Plutonium" (after two or
three other legal name changes). Or at least, that is what is
reported. Who can really know with someone this strange?
David R Tribble
>> My conjecture is that people who post using their real names
>> are more likely to be coherent and correct in what they post
>> and are more likely to understand mathematics in general
>> than people who post using an alias.
>> There are a few notable exceptions to this little conjecture
>> of mine, obviously.
>
Math1723 writes:
>> I'd like to think I'm an exception. And even if you don't grant that,
>> surely you grant James S. Harris as an exception. :-)
>
Jesse F. Hughes wrote:
> Surely, Archimedes Plutonium is another obvious exception.
As someone else said, it depends on whether you consider his
4th name change a name or an alias.
Other exceptions come to mind: Orlow, Muckenheim, de Bruijn,
Harris, et al. Exceptions on the flip side, too: Math1723, MoeBlee
(that's not your real name, is it?), pubkeybreaker, et al.
I was shooting for generality, of course, not completeness.
And don't call me Shirley.