>> 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*

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