Encapsulation, inheritance and polymorphism

[Erik Max Francis]

On 07/20/2012 03:28 AM, BartC wrote:
> "Erik Max Francis" <max at alcyone.com> wrote in message
> news:GsKdnWOQPKOOvZTNnZ2dnUVZ5s2dnZ2d at giganews.com...
>> On 07/20/2012 01:11 AM, Steven D'Aprano wrote:
>>> On Thu, 19 Jul 2012 13:50:36 -0500, Tim Chase wrote:
>
>>> I'm reminded of Graham's Number, which is so large that there aren't
>>> enough molecules in the universe to write it out as a power tower
>>> a^b^c^d^..., or even in a tower of hyperpowers a^^b^^c^^d^^... It was
>>> the
>>> provable upper bound to a question to which experts in the field thought
>>> the most likely answer was ... six.
>>>
>>> (The bounds have since been reduced: the lower bound is now 13, and the
>>> upper bound is *much* smaller than Graham's Number but still
>>> inconceivably ginormous.)
>>
>> You don't even need to go that high. Even a run-of-the-mill googol
>> (10^100) is far larger than the total number of elementary particles in
>> the observable Universe.
>
> But you can write it down, even as a straightforward number, without any
> problem. Perhaps a googolplex (10^10^100 iirc) would be difficult to
> write it down in full, but I have just represented it as an exponent
> with little difficulty.
>
> These bigger numbers can't be written down, because there will never be
> enough material, even using multiple systems of exponents.

But that's true for precisely the same reason as what I said.  If you're 
going to write a number down in standard format (whatever the base), 
then the number of digits needed scales as the logarithm of the number 
(again, whatever the base).  log_10 10^100 is trivially 100, so a rough 
order of magnitude in that form is easy to write down.  But the log_10 
10^10^100 is 10^100 = a googol, which is already more than the number of 
elementary particles in the observable Universe.

> (A few years ago the biggest number I'd heard of was Skewes' Number
> (something like 10^10^10^34), but even that is trivial to write using
> conventional exponents as I've just shown. Graham's Number is in a
> different
> class altogether.)

Anything's trivial to "write down."  Just say "the number such that ..." 
and you've written it down.  Even "numbers" that aren't really numbers, 
such as transfinite cardinals!

-- 
Erik Max Francis && max at alcyone.com && http://www.alcyone.com/max/
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