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Mueen Nawaz wrote:
> Yes. Haven't studied it, but I remember that the answer was either
> "No", or "Can't be decided." Your message makes me think it's the latter.
You are correct. The truth of that conjecture can be proven to be
independent of the ZFC axioms of set theory.
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Darren New wrote:
> I bet even Gail didn't know that Oracle has solved *that* problem.
>
> http://laurentschneider.com/wordpress/2007/10/what-is-bigger-than-infinity.html
>
>
> (Check out the comments, too.)
Comments? Seems to be one comment, repeated multiple times. :-P
--
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*
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Darren New wrote:
> Warp wrote:
>> *After* you have thrown the dart, the probability to hit the number
>> the dart did hit grows to 1 (or 100%).
>
> But is the probability that it hit any rational number (vs any
> irrational number) non-zero at that point? If you do it countably many
> times, will you *ever* hit *any* rational number?
>
>> (I really wonder if this has any relation whatsoever to the so-called
>> axiom of choice.)
>
> That's pretty much exactly what the axiom of choice is. Given an
> infinite set of sets, can you create a set by picking one element from
> each subset?
>
I don't see how it directly applies completely directly, but the notion
of choosing elements from sets without need a rule by which to make the
choice is certainly the main feature of both.
As a more mind bending thing than the probability of 0 that you'll hit a
rational number, if you assume that the axiom of choice is true then you
can show that there exists subsets of the real numbers in the unit
interval such that it it's *impossible* to define *any* probability that
the dart will hit an element of it (zero or otherwise).
Essentially the real numbers are numerous enough that with the help of
the axion of choice it's possible to define subsets of the reals that
are so bizarrely structured that the notion of length, area, volume etc.
don't even apply to them. (more specifically, they're known as
non-measurable sets since they don't have a well defined measure).
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Mueen Nawaz wrote:
> In mathematics, you can prove
> the existence of certain sets, and you can also prove that you can never
> explicitly exhibit those sets (i.e. you can *only* show they exist -
> it is mathematically impossible to demonstrate them, though). So an
> argument that shows one cannot exhibit/construct a certain set does not
> imply that they don't exist.
>
> Some day I have to formally learn how that is.
It may seem a bit like cheating, but there's an easy way to define such
a set. Take whatever system you wish to use to construct sets of
natural numbers, so long as you only have a countable number of axioms
and rules. Now, each construction of a set will require a finite set of
derivation steps to define it. Since we have only a countable number of
choices for each derivation step and a finite number of steps in total,
there are the same number of such derivations as there are natural
numbers. Since there are uncountably many subsets of the natural
numbers, there must be some subsets for which there exists no derivation
to construct them. And there you go, a proof of the existence of a set
which by definition it's impossible to construct!
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Mueen Nawaz <m.n### [at] ieee org> wrote:
> In math, you can't divide by infinity.
I don't think that's correct. Maybe in *some* branches of math you can't,
but that's certainly not true for *all* branches of math.
--
- Warp
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Mueen Nawaz wrote:
>
>
> Orchid XP v8 wrote:
>> What is Ackermann's function, why does it grow so fast, and why does
>> anybody care anyway?
>
> Not sure if this will answer your question, but it's a fun read:
>
> http://www.scottaaronson.com/writings/bignumbers.html
>
>
And if you like really big numbers, The Clarkkkkson number is even bigger.
http://qntm.org/?clarkkkkson
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Mueen Nawaz wrote:
> Not sure if this will answer your question, but it's a fun read:
>
> http://www.scottaaronson.com/writings/bignumbers.html
Yeah, that IS pretty fun actually...
--
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*
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On Fri, 11 Jul 2008 21:12:29 +0100, Orchid XP v8 wrote:
>>> The question, surely, is what functions are *not* computable by a
>>> computer? ;-)
>>
>> I would think any problem that was mathematically intractable. :-)
>
> Well, if you don't know "how" to solve a problem then you can't write a
> program that does it. That's true enough. I'm not sure it makes it not a
> "computable function" though...
There's a difference, though, between knowing how to solve a problem and
being able to show - mathematically - that the problem grows at an
exponential rate as the size of the problem is increased in a linear
fashion.
>> Or put another way, some computational problems where the growth
>> function is exponential. Some of those problems are solved on a small
>> scale computationally (like the traveling salesman problem), but as the
>> scale of the problem grows, the complexity grows as well.
>>
>> At least that's what I remember from my discrete maths class in the
>> early 90's. :-)
>
> Nope. That's not "impossible" to compute, just really damned slow. ;-)
I don't think it's about a specific size for the problem, but rather the
growth pattern as the size of the problem is increased.
Doing something like calculating aircraft separation is an example of
such a problem - it's the calculation of the intersection of oblique
spheroids (because the separation is IIRC 5,000 feet horizontally and
3,000 feet vertically). With 2 plans that's easy. With 20 planes it
takes more time to calculate the intersections to determine if an
airspace violation has occurred.
With 200 planes, it takes even more time. With 2,000 even moreso.
And at some point, of course, you have to start over because planes
aren't stationary in space - they move.
> If we're talking about what a computer can *theoretically* calculate.
> It's just a matter of leaving the machine running for long enough.
>
> But even given infinite time and memory, a computer *still* can't solve
> the Halting Problem though. Even theoretically. Don't ask me why...
But this is an aspect I hadn't considered.
Jim
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On Fri, 11 Jul 2008 14:23:30 -0700, Kevin Wampler wrote:
> Jim Henderson wrote:
>>
>> *ding* - light bulb just went on here. I was trying to figure this one
>> out, and it just intuitively appeared to me.
>>
>>
> It's that sort of moment of epiphany that I love about learning this
> sort of stuff. One moment it seems totally false and the next moment it
> somehow makes sense, a bit it seems obvious!
Yeah. That is a nice feeling, isn't it? :-)
> > The weird thing is that I can't even explain it, but I understand it.
>
> As for the proof, this site explains it reasonably well i think:
> http://www.math.hmc.edu/funfacts/ffiles/30001.3-4.shtml
Cool, thanks for that.
Jim
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>> Map each real number to a positive integer.
>
> Um... like, how? There's more of them...
http://en.wikipedia.org/wiki/Reductio_ad_absurdum
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