Roman Reiner wrote:
>> Yep. There's really only two infinities you get when you talk about
>> "numbers" - Aleph0 and Aleph1.
> 
> Wasn't the question whether there is a set that is larger than a countable
> infinite set and smaller than an uncountable infinite set proven to be
> unprovable?

My memory is that it was thought to be unprovable but later proven to not be 
so. However, wikipedia (Cardinal_numbers) agrees with you. Not that it's 
unprovable, but that its provability depends on the axioms of set theory you 
accept. I suspect it depends in part on things like the axiom of choice, etc.

-- 
Darren New, San Diego CA, USA (PST)
  "How did he die?"   "He got shot in the hand."
     "That was fatal?"
          "He was holding a live grenade at the time."

Post a reply to this message

On 08/03/2011 06:07 PM, Darren New wrote:
>> Wasn't the question whether there is a set that is larger than a
>> countable
>> infinite set and smaller than an uncountable infinite set proven to be
>> unprovable?
>
> My memory is that it was thought to be unprovable but later proven to
> not be so. However, wikipedia (Cardinal_numbers) agrees with you. Not
> that it's unprovable, but that its provability depends on the axioms of
> set theory you accept. I suspect it depends in part on things like the
> axiom of choice, etc.

That was my recollection - that the question is provably "independent" 
of the standard axioms of set theory. (I believe the axiom of choice is 
specifically implicated - although I have no idea WTF the axiom of 
choice is.)

-- 
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*

Post a reply to this message

Am 07.03.2011 12:58, schrieb Mike Raiford:

> Take any mass and collapse it down small enough and it'll become a black
> hole. Once it reaches a critical ration of mass/volume, then you'll have
> a black hole. It may not last too long, or it may; and if it does, it
> would be rather irresponsible to create one so close to our own planet.

Don't worry - if such an experiment can be conducted by man (and some 
say that with the newest CERN particle accelerator it might be 
possible), then nature does it all the time as well, thereby having 
already proven it a harmless prank.

Essentially, any success in creating an artificial black hole in a 
controlled environment with current technology would prove that black 
holes do decay over time, as otherwise our universe would already be 
full of them.

(I personally favor the idea that leptons, quarks and the like might be 
black holes themselves; kind of like the smallest black holes you can 
possibly get.)

Post a reply to this message

>> It is undeniably infinite. But is it countable?
>
> Tell me when you're done counting from 0 to 1 in infinitesimally small
> intervals.

"Countable" doesn't mean that you can actually count them in finite 
time. It means that you can assign a unique positive integer to each one.

Now, between 0 and 1 in the rationals, there are a countable infinity of 
values. But in the reals (which inclused irrational numbers), there is 
an uncountably infinite supply of values.

So I suppose "how many points are there on the unit square" comes down 
to "are the coordinates rational?"

>> Sure. Regular Sudoku is 9x9. But you can make 'em other sizes (with
>> other numbers of unique symbols).
>
> Right, but, even if it were 25x25 it still has a specific layout.

You can make them rectangular, you know. ;-)

> Graph
> coloring problems don't necessarily have a consistent number of borders.

True...

>> But if you have a small mass, why would it be crushed to a small size?
>
> An incredible amount of force acted upon that mass?

Oh. You mean something *outside* the mass itself causes it to become a 
black hole?

But then, wouldn't that just mean that as soon as you remove the force, 
it wouldn't be a black hole any more?

>> But the speed of sound is /not/ constant. It changes depending on the
>> motion of the source and the receiver. Light, on the other hand, has the
>> seemingly impossible property that the relative motion of source and
>> receiver somehow makes no difference to the apparent velocity of the
>> light... but somehow *does* affect its wavelength? WTF?
>
> For the purposes of Doppler shift, it has a constant rate in the medium
> it is in. The actual speed the waves move through the medium doesn't
> change. The /relative/ speed does change, however. And is the same for
> light. Sure, the relative speed of light can even be superluminal.

No, it cannot.

This is exactly what relativity states. The speed of light, no matter 
which way you measure it, is always constant. (Except as noted below.)

> And, light does travel at different rates in different media.

This part is correct. And I would expect light entering or leaving a 
different medium to exhibit Doppler shift as a result (although it 
doesn't seem to do this...)

> Umm, to keep track of what's in your account? I generally do reconcile
> my bank statement to my register (I keep mine electronically, so it's
> fairly simple to do)
>
> I think this is basic home finance 101.

Wait - you actually use a sophisticated electronic accounting package 
just for your home finance?

>> A "root" is generally a part of a plant. Whereas a "route", as in, "this
>> is how you get to this square number", would logically make more sense.
>
> Isn't the root considered the base of something? A Plants roots are at
> its base, for example. for example the root word of unbelievable is
> believe. You wouldn't say route in that context, would you?

Hmm, I suppose...

-- 
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*

Post a reply to this message

Orchid XP v8 wrote:
> although I have no idea WTF the axiom of 
> choice is.)

The axiom of choice essentially says that if you have an infinite set each 
of whose elements is a set, you can construct a new set by picking one 
element from each of the elements of the infinite set.

-- 
Darren New, San Diego CA, USA (PST)
  "How did he die?"   "He got shot in the hand."
     "That was fatal?"
          "He was holding a live grenade at the time."

Post a reply to this message

Orchid XP v8 wrote:
> So I suppose "how many points are there on the unit square" comes down 
> to "are the coordinates rational?"

Draw a dot at sqrt(2),sqrt(2).

> Oh. You mean something *outside* the mass itself causes it to become a 
> black hole?

Yes.

> But then, wouldn't that just mean that as soon as you remove the force, 
> it wouldn't be a black hole any more?

No.  Science!

(Altho there's arguments over whether quantum effects would eventually make 
a black hole too small to form on its own evaporate, but I didn't take that 
to be what you meant.)

>> Sure, the relative speed of light can even be superluminal.
> 
> No, it cannot.

It can travel faster than the speed of light in the medium in which it's 
traveling.

> This is exactly what relativity states. The speed of light, no matter 
> which way you measure it, is always constant. (Except as noted below.)

The speed of light in a vacuum is always constant. That doesn't mean it 
can't travel faster than the speed of light in the medium in which it is 
traveling.

> This part is correct. And I would expect light entering or leaving a 
> different medium to exhibit Doppler shift as a result (although it 
> doesn't seem to do this...)

That's not how light slows down. Light slows because it is being absorbed 
and re-emitted, not because the speed per se is different in different 
media.  The train always travels the same speed - but sometimes it stops at 
more stations.

> Wait - you actually use a sophisticated electronic accounting package 
> just for your home finance?

Yes. It's not all that sophisticated.

-- 
Darren New, San Diego CA, USA (PST)
  "How did he die?"   "He got shot in the hand."
     "That was fatal?"
          "He was holding a live grenade at the time."

Post a reply to this message

Am 07.03.2011 14:26, schrieb Invisible:

>> Take any mass and collapse it down small enough and it'll become a black
>> hole. Once it reaches a critical ration of mass/volume, then you'll have
>> a black hole. It may not last too long, or it may; and if it does, it
>> would be rather irresponsible to create one so close to our own planet.
>
> But if you have a small mass, why would it be crushed to a small size?

Non-gravitational external force. Smash two particles into each other 
hard enough, and you /might/ get them close enough together that they 
can't escape each other's gravitational well anymore.

>>> If the speed of light is constant, how the hell does light undergo
>>> Doppler shift?
>>
>> Same way sound Doppler shifts.
>
> But the speed of sound is /not/ constant. It changes depending on the
> motion of the source and the receiver. Light, on the other hand, has the
> seemingly impossible property that the relative motion of source and
> receiver somehow makes no difference to the apparent velocity of the
> light... but somehow *does* affect its wavelength? WTF?

That's one of the fun things about Einstein's theory of relativity 
(don't know if this one falls into the domain of special or general 
relativity).

You can put it like this: In your frame of reference, things buzzing 
towards you simply have a faster-ticking clock than things buzzing away 
from you (presuming that you define "now" based on when light from 
"somewhere" reaches you, rather than "when" it was emitted).

>>> Do spiders have a sense of smell?

Insects do, so I guess spiders do as well - though if they follow the 
insect way, then their "noses" are on their legs.

> A "root" is generally a part of a plant. Whereas a "route", as in, "this
> is how you get to this square number", would logically make more sense.

Even more generally, a "root" is something from which something bigger 
grows, so it does make sense. Also note that if it was "route", it would 
be "route to", rather than "root of".

> Probably means the original mathematician who worked all this out was
> French or something... :-P

Maybe, but he spoke (or at least) wrote Latin. The original word was 
"radix" (latin for - guess what - "root"). In German it's "Wurzel", 
which also means "root".

Post a reply to this message

On 08/03/2011 06:40 PM, Darren New wrote:
> Orchid XP v8 wrote:
>> So I suppose "how many points are there on the unit square" comes down
>> to "are the coordinates rational?"
>
> Draw a dot at sqrt(2),sqrt(2).

I suppose if we're not talking about real, physical space, but about 
some mathematical abstraction, it can have whatever properties we define 
it to have - including the point you describe not being part of the set.

>> Oh. You mean something *outside* the mass itself causes it to become a
>> black hole?
>
> Yes.

Right. So you have to take some matter and mash it somehow?

>> But then, wouldn't that just mean that as soon as you remove the
>> force, it wouldn't be a black hole any more?
>
> No. Science!

I don't know... Normally if you remove external force, pressure decreases.

>>> Sure, the relative speed of light can even be superluminal.
>>
>> No, it cannot.
>
> It can travel faster than the speed of light in the medium in which it's
> traveling.

I was under the impression that light does not require a transmission 
medium.

>> This is exactly what relativity states. The speed of light, no matter
>> which way you measure it, is always constant. (Except as noted below.)
>
> The speed of light in a vacuum is always constant. That doesn't mean it
> can't travel faster than the speed of light in the medium in which it is
> traveling.

I don't understand.

-- 
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*

Post a reply to this message

On 08/03/2011 01:36 AM, Jim Henderson wrote:

> Receipts are useful for expense reports (nee required in most cases),

Agreed. In a commercial context, you want to keep track of this kind of 
thing.

> and
> yes, I do check that what I was charged is what I paid.  That's how you
> (a) avoid getting ripped off, and (b) avoid finding yourself in an
> overdraft situation.

So, what, you literally sit down and check all of the thousands of 
transactions you make, one at a time, to make sure every single one 
matches the printed recipt?

Doesn't that take an insane amount of time?

>> Why is it called a "square root"? Surely "square route" would make far
>> more sense...
>
> It seems to have grown out of the old English "Rote" (there's a citation
> circa 1425).  Etymologically it derives from the French word
> 'racine' (which translates to 'root', natch).  "radix" also is used, and
> that's latin for 'root'.

I knew it would be French! :-P

-- 
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*

Post a reply to this message

Orchid XP v8 wrote:
> On 08/03/2011 06:40 PM, Darren New wrote:
>> Orchid XP v8 wrote:
>>> So I suppose "how many points are there on the unit square" comes down
>>> to "are the coordinates rational?"
>>
>> Draw a dot at sqrt(2),sqrt(2).
> 
> I suppose if we're not talking about real, physical space, but about 
> some mathematical abstraction, it can have whatever properties we define 
> it to have - including the point you describe not being part of the set.

Well, sure. But if you're going to start screwing with the defintion of 
"unit square" rather than using the standard "set of points whose X and Y 
are both from 0 to 1", then the answer comes out of your definition and you 
don't have to ask.

Plus, I'm not sure that physical space is actually continuous, so it's 
possible such a square doesn't have a sqrt(2),sqrt(2) point either.

>>> Oh. You mean something *outside* the mass itself causes it to become a
>>> black hole?
>>
>> Yes.
> 
> Right. So you have to take some matter and mash it somehow?

Basically.  You can do it by taking some matter and slapping it together so 
forcefully it overcomes the non-gravitational forces. I'm not sure that's 
physically possible in the universe as it exists today.

>>> But then, wouldn't that just mean that as soon as you remove the
>>> force, it wouldn't be a black hole any more?
>>
>> No. Science!
> 
> I don't know... Normally if you remove external force, pressure decreases.

It's not pressure. It's gravity.

>>>> Sure, the relative speed of light can even be superluminal.
>>>
>>> No, it cannot.
>>
>> It can travel faster than the speed of light in the medium in which it's
>> traveling.
> 
> I was under the impression that light does not require a transmission 
> medium.

It doesn't. But it does have different speeds in different media. Otherwise, 
you wouldn't have prisms and lenses and such.

>>> This is exactly what relativity states. The speed of light, no matter
>>> which way you measure it, is always constant. (Except as noted below.)
>>
>> The speed of light in a vacuum is always constant. That doesn't mean it
>> can't travel faster than the speed of light in the medium in which it is
>> traveling.
> 
> I don't understand.

The speed of light in a diamond is less than half the speed of light in free 
space/vacuum.  Hence, it's possible to move through a diamond at faster than 
the speed of light in a diamond but slower than the speed of light in a 
vacuum. Hence, superluminal travel, for some definition of that word.

http://en.wikipedia.org/wiki/Cherenkov_radiation

It's what gives the weird blue light coming out of nuclear reactors 
underwater - particles from the reactor hitting the water at speeds faster 
than the speed of light through water.

-- 
Darren New, San Diego CA, USA (PST)
  "How did he die?"   "He got shot in the hand."
     "That was fatal?"
          "He was holding a live grenade at the time."

Post a reply to this message