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Misunderstanding Bateson
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Daryl McCullough
science forum Guru


Joined: 24 Mar 2005
Posts: 1167

PostPosted: Thu Mar 24, 2005 5:47 pm    Post subject: Re: Misunderstanding Bateson Reply with quote

Albert Wagner says...

Quote:
In the case of "infinite", if you already know what "bijection" means,
and you know what "proper subset" means, then a perfectly adequate
definition of "infinite" is this:

A set is defined to be infinite if there exists a bijection between
that set and a proper subset of that set.

False. That is *not* a definition of 'infinite'. That is only a
definition of an 'infinite set', assuming 'set' and 'infinite'
are previously defined.

Tell me where the word "infinite" appears in the expression

there exists a bijection between
that set and a proper subset of that set.

I think that perhaps you are confused because what is being defined
is not the *word* "infinite" but the phrase "infinite set". The word
"infinite" has several meanings, as you can discover by looking it
up in the dictionary.

--
Daryl McCullough
Ithaca, NY
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Rob Dekker
science forum beginner


Joined: 24 Mar 2005
Posts: 12

PostPosted: Thu Mar 24, 2005 5:47 pm    Post subject: Re: Inflationary Theory ; I'm confused Reply with quote

Thanks for your explanation Joseph.
A few comments and questions below.

"Joseph Lazio" <jlazio@adams.patriot.net> wrote in message news:lllla8gzeh.fsf@adams.patriot.net...
[...]
Quote:
RD> This is at the core of my 'confusion' : I don't understand why
RD> inflationary theory would be needed. I don't see that the horizon
RD> problem (...) is so hard to overcome with existing theories of BB
RD> and with proper application of GR.

Take two points on opposite sides of the sky. To very high precision,
the intensity of the cosmic microwave background that we measure from
both points is very nearly the same. At the time when the CMB formed,
about 300,000 years after the Big Bang, they were outside each other's
horizon.

Mmm. I'm not sure if that is really true.
The Universe was only 300,000 years old, and if it was/is round
(a 4-dim sphere) and expanding at light speed, then the two points
might have been very close to each other at that time at the far-side
of the sphere.... Just draw the horizon on polar paper back to the center.
How can we even know how far apart they were, and if they
did or did not share an event history ?

At that time, if the Universe was only 300,000 LYs in size, any
point on that sphere has a light cone back into time, and back into
an even smaller sphere.
So can it not be that all light cones / horizons always just touch
each other at the beginning of time/space, rather than to conclude
up-front that there must be something 'unseen' ?

Quote:
In other words, the size of the horizon or the size of the
observable Universe for an observer at each of these two points did
*not* include the observer at the other point. Therefore they should
know nothing about each other, and there is no reason to expect the
CMB from one direction to have the same intensity as the CMB from
another direction. Yet it does.


As far as the isotropy is concerned, a Universe of only 300,000
years old must have been tens of thousands of degrees hot and
very dense. It should not be so surprising that the temperature
and matter distribution was still very consistent.
It still (pretty consistent) on our side of the Universe 15 billion years later...

Quote:
--
Lt. Lazio, HTML police | e-mail: jlazio@patriot.net
No means no, stop rape. | http://patriot.net/%7Ejlazio/
sci.astro FAQ at http://sciastro.astronomy.net/sci.astro.html
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Shmuel (Seymour J.) Metz1
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Joined: 03 May 2005
Posts: 604

PostPosted: Thu Mar 24, 2005 5:47 pm    Post subject: Re: (sketch of a) Proof that the set of Real Numbers doesn't exist Reply with quote

In <ctigjh$di$1@gander.coarse.univie.ac.at>, on 02/01/2005
at 03:10 AM, piotr5@unet.univie.ac.at (Piotr Sawuk) said:

Quote:
iff a + (b-a)/3 and b - (b-a)/3 are actually both inside [a,b] (which
I can believe) and your open interval actually is an open set of the
topology in question (which needs to be proven).

The first is trivial to prove. The second is true by definition if you
are using the standard topology; if you aren't using the standard
topology then it is incumbent on you to explain what you mean by
"open".

Quote:
if the topology of real numbers where defined by "each open set contains rational and
irrational numbers"

That's not a definition. The standard definition is the set of all
unions of open intervals. The fact that each non-empty open set
contains both rational and irrational numbers is a simple theorem, not
part of the definition.

Quote:
if you would start out with equivalence-classes of cauchy-filters in
Q as the definition for R, and you would define "<" as "eventually
some U(x) and U(y) will be seperate with all elements of one set
(which are rational numbers) being smaller than all elements of the
other set", then you would first need to prove that equality is
present when neither "<" nor ">" is the case. otherwise above
paragraph could become reality!

That's true for any model involving equivalence classes; you must
prove trichotomy.

Quote:
starting from dedekind-cuts as the definition of R I would agree
with you. but as Herman Rubin said, nowdays maths does go away from
the idea of "definition" and instead uses "characterization",

It's a trivial consequence of any of the standard models.

Quote:
and cauchy-filters are a good way for creating a characterization of
real numbers,

No; they're far too complicated. They're not even a good way to
construct a model, since Cauchy sequences and Dedekind cuts are
simpler.

Quote:
I3=intersection(I3_n),I3_n=(x,q1(n)), q1(n)>x and {q1(n)} is {q>x in
Q}.

ITYM "q1(n) in {q>x in Q}."

Quote:
I4=intersection(I4_n),I4_n=(x,r(n)), r(n)>q1(n)>x and r(n) in R\Q

That still doesn't make sense. Do you intend q1 and r to be functions
of x and n or only of n? You need to indicate all of the functional
dependencies. Please try to fix collect all of your definitions, fill
in the holes, carefully proofread them and only then post them.

--
Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel>

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Shmuel (Seymour J.) Metz1
science forum Guru


Joined: 03 May 2005
Posts: 604

PostPosted: Thu Mar 24, 2005 5:47 pm    Post subject: Re: (sketch of a) Proof that the set of Real Numbers doesn't exist Reply with quote

In <ctiuls$dt$1@gander.coarse.univie.ac.at>, on 02/01/2005
at 03:11 AM, piotr5@unet.univie.ac.at (Piotr Sawuk) said:

Quote:
unfortunately after this definition did become "standard" the world
did change

No. There is a lot of Mathematics relating to algorithms, and automata
but that is in no way a replacement for the rest of Mathematics.

Quote:
what does now make sense is to define real numbers in terms of
algorithms producing them.

You're welcome to believe that, but it will cause you a good deal of
difficulty in understanding most of Analysis.

Quote:
however, if you would see open sets of rational numbers as closed
sets for which the maximum and minimum element can't get described
by a finite amount of symbols,

That would flout the finitistic program that you seem to be
advocating.

Quote:
then real numbers would seem like those rational numbers which can
get described by an infinite amount of symbols,

The above has no meaning.

Quote:
and you would get some kind of bijection

Feel free to construct it.

Quote:
not in ZFC.

Please give the axioms for the set theory in which it is true.

Quote:
no, the other way around: the C in ZFC did lift an old taboo,

No.

Quote:
depends on what you chose,

I don't "choose". I'm asking what you meant by "applying the
division-algorithm on infinite sequences of digits (0-9)", as you're
the one that claims it means something.

Quote:
they actually all are the same

What are all the same?

Quote:
find the first number,

What does that mean?

Quote:
calculate the remainder,

What does that mean?

Quote:
I can: if the theorem did not make use of the properties I did throw
out.

Throw out? If you construct a system that doesn't have the properties
of R then it is disingenuous to refer to it as R.

Quote:
for example most of Analysis does not make use of the uncountability
of real numbers, only convergence is required, and by limiting the
convergence (for example by claiming per axiom that sqrt(2) doesn't
exist)

You get a contradiction, because I can construct a series of rational
numbers that converges to sqrt(2). So if you want consistency then
you'll have to change the convergence properties as well.

Quote:
you actually get something where those theorems still hold,

What theorems still hold? Some of the obvious ones clearly don't.

Quote:
and you only need to prove a small set of theorems and assumptions.

You need to prove just about all of the theorems that you claim are
valid. As for assumptions, you don't prove those; that's why they're
called assumptions.

Quote:
the inverse operation of division!

You haven't defeind division yet.

Quote:
and I did assume that all sequences do have a maximum
element even when they are infinite,

So a "sequence" isn't a sequence?

Quote:
a mapping to our usual set of real numbers in ZFC,

What do you mean by that? Unless you have some a theory in which you
have defined models of your set theory and of ZFC, that appears
meaningless.

Quote:
no, the proper definition of open sets is as sets where for each
point in that set there exists a closed set inside of the open set,
such that the point is not on the border of this closed set.

Where on Earth did you get that notion? It's nonsense. I urge you to
learn the vocabulary ASAP.

Quote:
of course maths is much more than formulas and algorithms, maths is
mainly about translating theorems from one field to the next.

No. Mathematics is about a lot of things, and that's only a small part
of it.


Quote:
it's not about aribatary extensions,

Clearly it is; you have some sort of vague angst and believe that an
unspecified extension of Mathematics would cure it. From the
perspective of anybody outside of your skull, that's arbitrary, and if
you want somebody to actually take you seriously then you will have to
produce convincing reasons why:

1. What you want makes sense; so far it doesn't

2. What you want is interesting; so far it isn't.

Quote:
I'm looking for structure in the existing extensions,

What existing extensions? Sequences that aren't sequences? Set
theories that you haven't defined but have great expectations from?

Quote:
I really am not very good at thinking, therefore
I need to re-use other people's results...

If you want to reuse other people's results then you need to learn
their vocabulary.

--
Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel>

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Shmuel (Seymour J.) Metz1
science forum Guru


Joined: 03 May 2005
Posts: 604

PostPosted: Thu Mar 24, 2005 5:47 pm    Post subject: Re: (sketch of a) Proof that the set of Real Numbers doesn't exist Reply with quote

In <ctj4pk$ee$1@gander.coarse.univie.ac.at>, on 02/01/2005
at 03:11 AM, piotr5@unet.univie.ac.at (Piotr Sawuk) said:

Quote:
therefore I assumed the same would also be possible in maths.

In Mathematics terms have well defined technical meanings; if you want
to introduce a new term then you are obligated to define it. It's not
enough to assume that something is possible, you must prove it, and
you must either prove uniqueness or define which of the possible
somethings is the one that you meant.

Quote:
I would define "taboo" as something where an increase of the degree

What degree? And what results can you derive about tabbooness that are
important enough to justify coining a new term?

Quote:
in maths altering an axiom does have much greater impact than merely
changing the definition, it does change the properties so that even
a house could be used to ride to work in my house. the trick is just
to alter them in a way such that certain old properties still hold
and new properties get added. the field in maths which does handle
this kind of alterations in a general fashion such that other fields
in maths could re-use the results, such a field seems to be missing
in maths. or am I wrong?

What gives you the idea that altering axioms is either necessary or
desirable in order for one branch of Mathematics to use results in
another branch? Set Theory makes it very easy to define Mathematical
systems in terms of definitions rather than new axioms, and Category
Theory makes it easy to unify and generalize.

Quote:
many thanks for your corrections, I always wondered why people keep
changing variable-manes from one quantifyer to the next, and now I
realized: they are too lazy to make clear where one quantifier-scope
does end or if the quantifiers are nested...

It's the other way around; they are diligent enough to ensure that
they use unique names in order to avoid confusion. Reusing variable
names is a sign of laziness.

Quote:
for all I uncountable subset of R which has no element in common
with Q: (for all r1 and r2 in I there exists q in Q such that
r1<q<r2) => (for all r1 in I there exists q in Q such that for all
r2 in I: r2>r1 => r1<q<r2)
please prove that above statement is wrong.

You're still being lazy; fix your variable names to remove the
ambiguity. Your statements are still using multiple variables with the
same names and not making it clear which you mean where.

--
Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel>

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|-|erc
science forum Guru Wannabe


Joined: 25 Jul 2005
Posts: 269

PostPosted: Thu Mar 24, 2005 5:47 pm    Post subject: Re: sketch of PROOF OF COUNTABLE REALS Reply with quote

"|-|erc" <h@r.c> wrote in >

Quote:
sentencenumber = int(random(sentencesofsize(sentencesize))

We can use a buffer of sentences for each sentence size to ensure that even logically if also probabilistically
every sentence is still examined. BUT by switching to a RANDOM PROCESS instead of a list,
the defn_of_countable_is_still_met, the real is always found, but you_cannot_diagonalise_the_list.


Correction. Using a random array element selection just gives an equivalent function to
counting through a list, so introducing a random element won't help

But we can keep the 'semantic real' list private to some function and use the agent defn of countable.

Take a less restrictive defn of countable. (%) = A process is said to be real-countable IFF
given any real number, that process precisely describes that real after some finite number of attempts


Now to disprove the list is complete, you must specify a real completely. Most of these
will be covered in the sentences UTMp( 1234, digit ) which will evaluate to different
computable reals for different integers. Even antidiag(UTMp) is listed!

The only way to diagonalise the list is to specify the diag.

"Given the alphabet {+, -, sin, cos, 0, 1, 2, a, b, c, d...}
take the 1st digit of the 1st number produced by the 1st sentence,
then the 2nd digit of the 2nd ....
for all sentences in the formal model"

But this is just a formal statement, as such it IS on the list of reals! (but its not printable).

Herc
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Virgil
science forum Guru


Joined: 24 Mar 2005
Posts: 5536

PostPosted: Thu Mar 24, 2005 5:47 pm    Post subject: Re: how to find out the closed formula for this sequence? Reply with quote

In article <1107320053.143066.23070@l41g2000cwc.googlegroups.com>,
hfyan0@hotmail.com wrote:

Quote:
Hello there

I would like to get a closed form for the sequence

1/(1+r) + 2/(1+r)^2 + 3/(1+r)^3 + ... + n/(1+r)^n
Could anyone out there help me out?

thank you very much!!

[r+1 - (r*(n+1)+1)/(r+1)]/r^2
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Shmuel (Seymour J.) Metz1
science forum Guru


Joined: 03 May 2005
Posts: 604

PostPosted: Thu Mar 24, 2005 5:47 pm    Post subject: Re: On Well-Ordering(s) and Sets Dense in the Reals, Infinity Reply with quote

In <ctjff9$fp$1@gander.coarse.univie.ac.at>, on 02/01/2005
at 03:11 AM, piotr5@unet.univie.ac.at (Piotr Sawuk) said:

Quote:
I learned that being dense means that there exists an injective
function from one set to the other

Where? That's certainly not the standard definition in Analysis or
To-ology.

Quote:
no, reals are not continuous.

What do you mean by that statement?

Quote:
but in order for R to be continuous it would need to be a really big
set, beyond ordinal numbers probably,

Please prove that.

Quote:
otherwise one could produce a
number which is not element of the real numbers

What does that mean? What is a "number" in this context and what
relevance does it have to R?

Quote:
The whole problem did arise with mathematical attempts of measuring
sets, you could imagine real numbers as being continuous for the sake
of applying some "size" to them,

You're confusing reals with sets of reals.

Quote:
I mean, just look out of the window, the world
is a finite set of atoms and subatomic particles,

That's just supposition on your part. Certainly QFT seems to involve
an infinite set of virtual particles, and there's the question of
whether the Universe is open or closed.

--
Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel>

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Paul Sperry
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Joined: 08 May 2005
Posts: 371

PostPosted: Thu Mar 24, 2005 5:47 pm    Post subject: Re: Modules, Idempotent element help Reply with quote

In article <ctpk2o$2p4s$1@netnews.upenn.edu>, Tony
<Ttiger222@hotmail.com> wrote:

Quote:
I would highly appreciate any help to the following problem. I have done
most of it except for a small part.

If M is a left R-module, and R is a ring with 1. An element e in R is a
central idempotent if e^2 = e and er = re for all r in R.

I am asked to prove that M = eM (+) (1-e)M

Where (+) denotes direct sum.

I am almost all the way done. I have proven that eM and (1-e)M are
submodules of M by using the fact that e^2 = e and er = re for all r in R.
Now, all I have to show is that M = eM + (1-e)M where eM + (1-e)M = { m1 +
m2 : m1 is in eM and m2 is in (1-e)M}, and I have to show that every element
in
eM + (1-e)M can be written uniquely in the form a_1 + a_2 where a_1 is in eM
and a_2 is in (1-e)M. Then I will have that M = eM (+) (1-e)M.

Ok, it is pretty clear to me that M = eM + (1-e)M since M is clearly a
subset of the latter (since if m is in M, then m = em + (1-e)m ). The
latter is also a subset of M clearly.

But I can't get that every element in eM + (1-e)M can be written uniquely in
the form a_1 + a_2 where a_1 is in eM and a_2 is in (1-e)M.

Show eM /\ (1 - e)M = {0}.

Quote:
[...]

--
Paul Sperry
Columbia, SC (USA)
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Guest






PostPosted: Thu Mar 24, 2005 5:47 pm    Post subject: Re: how to find out the closed formula for this sequence? Reply with quote

thank you so much!
Could you shed some light on how you derived it? I am very curious to
know
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Pouya D. Tafti
science forum beginner


Joined: 11 Jul 2005
Posts: 8

PostPosted: Thu Mar 24, 2005 5:47 pm    Post subject: Re: Messages with TeX content Reply with quote

alancox@gmail.com wrote:
Quote:
Thunderbird is smart enough to put the entire "14" in the exponent.
Alan

It indeed is. I was in the Linux console and could not try it at that

time, sorry. Yet I remember seeing something rendered incorrectly,
perhaps it was x^0.5.

-- Pouya
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Tonico
science forum beginner


Joined: 06 May 2005
Posts: 30

PostPosted: Thu Mar 24, 2005 5:47 pm    Post subject: Re: Modules, Idempotent element help Reply with quote

Hi:
Either you prove that evey element of M can be written in a unique way
as you said, or you show that the intersection of eM and (1-e)M is
zero.
Assume x is in both eM and (1-e)M ==> there exist m,n in M s.t. x = em
= (1-e)n ==>
e(m+n) = n ==> n = ek is in eM (since k = m+n is in M), and thus:
x = em = (1-e)n = (1-e)(ek) = 0 and we're done.
I hope this helps
Regards
Tonio
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Virgil
science forum Guru


Joined: 24 Mar 2005
Posts: 5536

PostPosted: Thu Mar 24, 2005 5:47 pm    Post subject: Re: how to find out the closed formula for this sequence? Reply with quote

In article <1107323155.342804.207970@l41g2000cwc.googlegroups.com>,
hfyan0@hotmail.com wrote:

Quote:
thank you so much!
Could you shed some light on how you derived it? I am very curious to
know

I did it on my HP49+, which can find closed forms for a large number of
summation formulas.
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Jesse F. Hughes
science forum Guru


Joined: 24 Mar 2005
Posts: 801

PostPosted: Thu Mar 24, 2005 5:47 pm    Post subject: Re: ******** CAN ANYONE HERE DEFINE CHAITIN'S OMEGA ? *********** Reply with quote

examachine@gmail.com writes:

Quote:
In fact, one could regard a non self-delimiting program a rather bad PL
design. Any thoughts?

I have a thought. My thought is: one would consider self-delimitation
a positive feature for a language only if one was smitten with
Chaitin's writing.

I don't see any reason that a language ought to restrict programs to
being self-delimiting. Why would it be desirable?

It's a handy technical restriction for Chaitin's work but it doesn't
possess any practical merits for programming languages that I can see.

--
"If you have a really big idea, you can get a measure of how big it is
by how much people resist the obvious. From what I've seen, I have a
REALLY, REALLY, *REALLY*, BIG DISCOVERY!!!"
--James Harris, on being ignored
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|-|erc
science forum Guru Wannabe


Joined: 25 Jul 2005
Posts: 269

PostPosted: Thu Mar 24, 2005 5:47 pm    Post subject: Re: ******** CAN ANYONE HERE DEFINE CHAITIN'S OMEGA ? *********** Reply with quote

haha I was right.


"Jesse F. Hughes" <jesse@phiwumbda.org> wrote in
Quote:
examachine@gmail.com writes:

In fact, one could regard a non self-delimiting program a rather bad PL
design. Any thoughts?

I have a thought. My thought is: one would consider self-delimitation
a positive feature for a language only if one was smitten with
Chaitin's writing.

I don't see any reason that a language ought to restrict programs to
being self-delimiting. Why would it be desirable?

It's a handy technical restriction for Chaitin's work but it doesn't
possess any practical merits for programming languages that I can see.

--
"If you have a really big idea, you can get a measure of how big it is
by how much people resist the obvious. From what I've seen, I have a
REALLY, REALLY, *REALLY*, BIG DISCOVERY!!!"
--James Harris, on being ignored
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