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|-|erc
science forum Guru Wannabe

Joined: 25 Jul 2005
Posts: 269

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

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
Munsey
science forum beginner

Joined: 24 Mar 2005
Posts: 5

 Posted: Thu Mar 24, 2005 5:47 pm    Post subject: Re: 0 * X = null? 0 * x = 0 can be proved using limits. lim(x->0) f(x)=x*c (c is any real constant) is clearly 0. Null or undefined is used for functions like lim(x->0) f(x)=1/x. This is clearly not definable because, in this case not only is there a vertical asymptote at x=0, but the limits are different (-infinity, +infinity) depending on whether you are approaching from the right or left.
Torkel Franzen
science forum Guru

Joined: 30 Apr 2005
Posts: 639

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

"Jesse F. Hughes" <jesse@phiwumbda.org> writes:

 Quote: 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.

It's misleading anyway to refer to programming languages in this
context. It's not as though there is some restriction of bit strings
to a certain class of strings which constitute programs. In the
context of prefix-free complexity, given strings p and q which our
universal machine U accepts as input, it is essential that another
machine will accept pq as input.
science forum beginner

Joined: 28 Apr 2005
Posts: 7

Posted: Thu Mar 24, 2005 5:47 pm    Post subject: Re: 0 * X = null?

"Munsey" <munsey@gmail.com> wrote in message
 Quote: 0 * x = 0 can be proved using limits. lim(x->0) f(x)=x*c (c is any real constant) is clearly 0. Null or undefined is used for functions like lim(x->0) f(x)=1/x. This is clearly not definable because, in this case not only is there a vertical asymptote at x=0, but the limits are different (-infinity, +infinity) depending on whether you are approaching from the right or left.

I'd be hesitant to use limits to prove this statement. lim(x->0) f(x)=x/x
is 1 but 0/0 is not equal to 1
Rupert
science forum Guru

Joined: 18 May 2005
Posts: 372

Posted: Thu Mar 24, 2005 5:47 pm    Post subject: Re: sketch of PROOF OF COUNTABLE REALS

|-|erc wrote:
 Quote: rupertmccallum@yahoo.com> wrote in message To construct the anti-diagonal for the list of all computable numbers, you need to pick out the Turing machines that halt for every digit. This cannot be done by a Turing machine. Hence the anti-diagonal is not computable. This is proof that nobody here reads what they are replying to. "If the real is non-computable, then " 1/ But it is representable

I thought you were putting forth the list of computable reals as your
countable list of all real numbers. What list are you putting forward?
What does "representable" mean?

 Quote: 2/ That is not the problem of the people putting forward a countable list. If the list has incomplete digits or not it still maps to every real. It is the onus of the people asserting diagnoalisation disproves that assertion to find some rigging in which diagonalisation works. You can, just take the matrix at 'some' scale of completion, substitute an 11th digit for NULL and the diagonalisation argument is viable at any stage. The claim is reals U sequences_with_blanks = computable list --- UTM(n ranges over N, digit) mod 10 We don't need a real list, we only need to verify the membership relation is possible here : real e {UTM(index, digit)} = [T | F] As long as that is decidable we have the functionality of a real list. Herc
Jason

Joined: 24 Mar 2005
Posts: 72

Posted: Thu Mar 24, 2005 5:47 pm    Post subject: Re: Epistemology 201: The Science of Science

 Quote: You are perhaps referring to First Order Predicate Calculus (FOPC). And indeed, mathematicians do use FOPC. However, mathematics is not FOPC, and FOPC is not sufficiently expressible to allow it to be used exclusively. Given a particular system of axioms, say PA (the Peano Axioms), mathematicians could in principle use FOPC applied to those axioms. But mathematics is not confined to working within a particular axiom system. Moreover, the discussion axiom system itself is part of mathematics. Maths is an extension of FOPC, like PA. Not really. Mathematics is much older than FOPC, so it doesn't make sense to say it is an extension of FOPC.

Okay, this is really strange to me because this is so not what I've come to
understand mathematics as. These days, in mathematical reasoning, logical
arguments are used to deduce consequences (theorems) of the assumptions of maths
(axioms). Most of maths is built from sets, so the basic assumptions of maths
are the axioms of set theory, in particular ZFC set theory. [Chapter Zero -
Fundamental Notions of Abstract Mathematics, Carol Schumacher]

You are suggesting that maths is not this formal system, so I am lead to assume
that you have some sort of prior understanding of what is mathematically legal
and illegal, like most people. But is this type of reasoning informal or have
we our own set of assumptions, much like axioms, that enable us to perform
mathematical inference. When there is a disagreement, where do we turn? From
my understanding it is this formalised system of mathematics, which took root
with Whitehead and Russell in the principia mathematica. Hence its FOPC roots.

I can accept that the axioms are not often invoked in the heat of proofs, but
then neither is the road-code when we are driving. Axioms as such don't need to
be the way to go either. The more intuitive way to go are to use rules of
inference, which are equivalent and perhaps closer to the story about how we
'do' maths.

Out of interest, if maths is not this formal system then how can abstract
mathematics take place? For example, how can the continuum hypothesis be
(dis)proven, or proved not to be provable?

 Quote: and assumed, as far as I am aware. Again, not really. Mathematicians often try to make do with minimal axioms.

Which ones? The choice is critical to what is provable and what isn't.

 Quote: If another system is used in maths then people need to know about it. The ZF system without the axiom of Choice for example, can lead to the creation of two spheres out of one in topology. I'm not sure of your point there. If you happen to be making a vague reference to the Banach-Tarski paradox, then you have it wrong. Banach-Tarski does depend on the axiom of choice.

I went to a seminar on this last year and I thought the dude said the problem
went away with invoking the axiom of choice. But now having read some more I
realise I misunderstood. Okay, bad example.

How about another then. It has been proven that in ZFC set theory, the formal
system of mathematics (I honestly can't see why you flatly refuse that there is
such a system), the continuum hypotheses is can neither be proven or disproven.
So it could be asserted true or false with a new axiom and there would be two
overlapping but distinct mathematical universes to choose from.

If ZFC is assumed as the foundations of maths, it has been shown by Chaitin that
there are infinite arithmetic truths that cannot be proven in ZFC. Where does
maths as not-a-formal-system fit into this?

 Quote: The study of axioms don't take place in maths. It is meta-logic or meta-maths that deals with this. Godels theorem for example is a meta-mathematical proof. While Goedel's theorem is meta-mathematics, nevertheless a lot of mathematics is effectively a study of axioms and their consequences.

This cannot be if maths is a formal system. But I understand that you don't tak
e it to be one and this statement is contingent on this.

 Quote: Since mathematics has evolved along-side science and plays a large part in describing and predicting how the world works, then as a formal system goes, it seems to be on the money as far as capturing something about the world. That's your opinion. As a mathematician, I have a different opinion. I consider it important that mathematics is not about the world. Roughly speaking, mathematics is about what would happen if reality did not intrude. We discover a lot about reality by seeing how it differs from the mathematical ideal. Fair enough. The formal system of maths is ripe for exploration. People study it divorsed from the world. But why spend so much time on maths and not some other formal system? I think because of the close link maths has with the world. There you go again. You talk about "the formal system of maths", but there is no such formal system. Then you suggest that we should instead study some other formal system. It is gibberish.

Is ZFC set theory a small and inconsequential part of mathematics? I suppose
you don't really get into it unless you study number theory, mathematical logic
and stuff, but it was my understanding that this system what the foundation of
modern maths.
oðin
science forum Guru

Joined: 05 May 2005
Posts: 408

Posted: Thu Mar 24, 2005 5:47 pm    Post subject: Re: Surrogate factoring, out of the box

 Quote: "Surrogate Factoring For Dummies"

Just "Dummy Factoring"
OsherD
science forum Guru Wannabe

Joined: 04 May 2005
Posts: 141

Posted: Thu Mar 24, 2005 5:47 pm    Post subject: Re: Epistemology 201: The Science of Science

with all facets of science and knowledge. By and large I ope­rate out
of comp.ai.philosophy but crosspost to other groups as appro­priate.

It's been one of the more successful threads I've produced b­ut I
expect Epistemology 201 will pretty much run its course over­ the next
week. It's generated well over 200 replies but participants ­seem to
have been reduced to name calling as a substitute for argume­nts.

[Osher:] Thank you, Lester Zick. Of course I should look at the
individual contributions, though I don't quite see any reason for
ending the thread as long as even 2 people (or even 1!) type something
useful. Mathematicians are supposed to be fond of numbers, but I'm
fonder of exceptions. In fact, this may be an important clue to
Knowledge, which at least among human beings seems to either diminish
with increasing numbers of people who believe something or to increase
more slowly as more people believe something (under the appropriate
context) - at least in the first generation during which some Knowledge
is discovered.

I must plead overwork in failing to thus far read most of the other
posts in this thread. In the meanwhile, try to keep the thread alive
for the sake of at least minorities of one or two.

Osher Doctorow
Munsey
science forum beginner

Joined: 24 Mar 2005
Posts: 5

 Posted: Thu Mar 24, 2005 5:47 pm    Post subject: Re: Tree Algorithm If I understand your question correctly, you are asking about an algorithm to balance a tree? If so, AVL is probably the easiest to understand, although it might not be the most efficient. http://faculty.washington.edu/stepp/courses/2004autumn/tcss342/lectures/notes/10-balanced_bst.ppt
Jason

Joined: 24 Mar 2005
Posts: 72

Posted: Thu Mar 24, 2005 5:47 pm    Post subject: Re: Epistemology 201: The Science of Science

 Quote: Philosophy does that much, Bob. What is different about mathematics? Mathematics is rigourous. Philosophy is sloppy.

Phil is only sloppy to those poetic psudo-philosopher types ("fuzzies" or
"conties") that think philosophy is some kind of dreamy intellectually artistic
nirvana. The other extreme are the logical psudo-philosophers ("techies" or
"analytics") that think philosophy has to follow some kind of well-trodden
formal method or schema.

If maths is rigorous then it must follow rules of some kind. If these rules are
agreed upon and written down then, I argue, maths has become a formalised
system.
Álvaro Begué
science forum beginner

Joined: 13 May 2005
Posts: 23

 Posted: Thu Mar 24, 2005 5:47 pm    Post subject: Re: how to find out the closed formula for this sequence? S = 1/(1+r) + 2/(1+r)^2 + 3/(1+r)^3 + ... + n/(1+r)^n S/(1+r) = 1/(1+r)^2 + 2/(1+r)^3 + 3/(1+r)^4 + ... + n/(1+r)^(n+1) Substracting, we get S*(1-1/(1+r)) = 1/(1+r) + 1/(1+r)^2 + 1/(1+r)^3 + ... + 1/(1+r)^n - n/(1+r)^(n+1) Now just add the geometric sequence and you quickly get your closed expression.
Jim Heckman
science forum Guru Wannabe

Joined: 28 Apr 2005
Posts: 121

Posted: Thu Mar 24, 2005 5:47 pm    Post subject: Re: Groups of order 36, 40, 56

On 1-Feb-2005, "Van" <calccurve-spam123@yahoo.com>

[...]

 Quote: As for groups of order 36, the 2 coming from C_2 x C_2 acting on C_3 x C_3 being normal,

Um, there are *4* such groups, as per my previous post.

 Quote: must be S_3 x S_3 = (C_3 x| C_2) x (C_3 x| C_2) with 4 generators a,b,x,y with a^2 = b^2 = 1 = x^3 = y^3 and ab = ba, xy = yx,

Don't forget the relations axax, [a,y], [b,x] and byby, if
you're making everything explicit. (I usually leave trivial
commutators out of my own presentations, so I would typically
list only axax and byby for this group.)

 Quote: and Z_6 X S_3.

You left out the abelian group C_3 x C_3 x C_2 x C_2, and the
group ((C_3 x C_3) x| C_2) x C_2 = <x,y,a,b | a^2, b^2, x^3,
y^3, axax, ayay>. Now *why* are these the only 4 groups with
C_2 x C_2 acting on C_3 x C_3?

 Quote: There is also the other abelian C_2 x C_2 x C_9, etc., actual there must be 4 abelian groups, 2 with C_9 and 2 with C_3 x C_3, right?

Yes. So why did you leave out one of the ones with C_3 x C_3
just above? :-/

--
Jim Heckman
Torkel Franzen
science forum Guru

Joined: 30 Apr 2005
Posts: 639

Posted: Thu Mar 24, 2005 5:47 pm    Post subject: Re: Epistemology 201: The Science of Science

"Jason" <jasonstevensNOSPAM@free.net.nz> writes:

 Quote: Most of maths is built from sets, so the basic assumptions of maths are the axioms of set theory, in particular ZFC set theory. [Chapter Zero - Fundamental Notions of Abstract Mathematics, Carol Schumacher]

However, after Chapter Zero, nobody cares about the axioms of ZFC,
let alone about the rules of predicate logic.

This is not to say that ZFC does not play an important role in
contemporary mathematics.
Robert B. Israel
science forum Guru

Joined: 24 Mar 2005
Posts: 2151

Posted: Thu Mar 24, 2005 5:47 pm    Post subject: Re: Partitions of Reals

ululuca@tiscali.it wrote:
 Quote: Anyone knows anything about this problem, please There exists a non-trivial partition of the positive reals (so without 0) into sets A and B (A, B disjoint, A U B = R+) such that A and B are both closed under addition and multiplication.

It may be worth mentioning that, even if you remove the "closed under
multiplication" requirement, any such A and B would have to be Lebesgue
nonmeasurable. In fact, neither A nor B could contain a measurable set

of nonzero measure. This is because for any such set C, C+C contains
an interval (while on the other hand it's easy to show that A and B
must be dense).

A consequence is that any example will have to involve some form of
the Axiom of Choice: it's consistent with Zermelo-Frankel set theory
(without Choice) that every set is Lebesgue measurable.

Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada V6T 1Z2
Torkel Franzen
science forum Guru

Joined: 30 Apr 2005
Posts: 639

Posted: Thu Mar 24, 2005 5:47 pm    Post subject: Re: Partitions of Reals

"Robert Israel" <israel@math.ubc.ca> writes:

 Quote: A consequence is that any example will have to involve some form of the Axiom of Choice: it's consistent with Zermelo-Frankel set theory (without Choice) that every set is Lebesgue measurable.

More significantly, this is consistent (Solovay) with ZF+the countable
axiom of choice.

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