On 7/6/2010 9:05 AM, Invisible wrote:
> JimT wrote:
>
>> though I suppose you need to be as old as me to talk about c rather
>> than Aleph_1
>
> The cardinallity of the continuum is equal to Beth-one, which is equal
> to Aleph-one if and only if the continuum hypothesis holds. And the
> continuum hypothesis is independent of the axioms of ZFC, so...

What is Aleph-null? Is it the set of all integers? or is it something a 
little different. I know it's basically a different sort of infinity...

[A Quick wiki detour later] Oh, Aleph-Null is basically any infinite 
set, Aleph-One would be a set of all ordinals (positive integers and 0) 
.. Interesting

-- 
~Mike

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Mike Raiford wrote:

> What is Aleph-null? Is it the set of all integers? or is it something a 
> little different. I know it's basically a different sort of infinity...
> 
> [A Quick wiki detour later] Oh, Aleph-Null is basically any infinite 
> set, Aleph-One would be a set of all ordinals (positive integers and 0) 
> .. Interesting

Aleph-null is the *size* of a set (specifically, the set of natural 
numbers). The technical term is "cardinality".

The set of all positive numbers (including or excluding zero) is 
Aleph-null. In fact,

   Aleph0 + x = Aleph0
   Aleph0 * x = Aleph0
   Aleph0 ^ x = Aleph0

assuming that x < Aleph0 (i.e., x is finite). For this reason, the set 
of all integers (positive and negative) has size 2 * Aleph0 = Aleph0. In 
other words, the set of all integers is THE SAME SIZE as the set of 
positive integers. (So it really isn't especially important exactly 
which set you use as your definition.)

Additionally, the set of all 2D coordinates has cardinality Aleph0 * 
Aleph0 = Aleph0, so that's the same size too. The set of all rational 
numbers also has the same size, as does the set of all algebraic numbers 
(i.e., roots of polynomials - so that includes irrational square roots 
and the like).

However, the set of all *real* numbers includes also transcendental 
numbers - numbers which are not the root of any polynomial. And *this* 
set has cardinallity Beth-one. And Beth-one > Aleph-null.

   Aleph-one = 2 ^ Aleph-null

(Note that Aleph-null ^ 2 = Aleph-null, which isn't the same thing at all!)

If the continuum hypothesis is true then Beth-one = Aleph-one.

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

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Darren New wrote:

> I think you have to worry more about it being smaller than the Plank 
> length.

What happens to objects smaller than the Plank length? Do they fall 
between the cracks and drop out of the bottom of the universe or something?

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

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Orchid XP v8 wrote:
> Darren New wrote:
> 
>> I think you have to worry more about it being smaller than the Plank 
>> length.
> 
> What happens to objects smaller than the Plank length? 

Mu.


The question is meaningless, because there is no such thing as "smaller than 
the Plank length".  It's like saying "what happens when it gets bigger than 
the universe?" or "how long did it take before time started?"




(http://en.wikipedia.org/wiki/Mu_%28negative%29)

-- 
Darren New, San Diego CA, USA (PST)
    C# - a language whose greatest drawback
    is that its best implementation comes
    from a company that doesn't hate Microsoft.

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>> What happens to objects smaller than the Plank length? 
> 
> Mu.
> 
> 
> The question is meaningless.

...as is my humour, apparently...

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

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On Tue, 06 Jul 2010 19:00:39 +0100, Orchid XP v8 wrote:

>>> What happens to objects smaller than the Plank length?
>> 
>> Mu.
>> 
>> 
>> The question is meaningless.
> 
> ...as is my humour, apparently...

I found the entire thing to be funny - and have shared it. :-)

Jim

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Orchid XP v8 wrote:
> Mike Raiford wrote:
> 

I found Invisible's summary to be a bit confusing, and also seemed to 
have a semi-important error, so here's another quick summary:

The Aleph numbers are "cardinals" which means they represent the size of 
sets.  Aleph_0 is the first infinite cardinal, which is equal to the 
size of the set of natural numbers.  Aleph_1 is the next size that a set 
can have which is larger than Aleph_0.  Beth_1 is the size of the set of 
all real numbers, and it can be proven that the size of this set is a 
"larger infinity" than the size of the set of all natural numbers (so 
Beth_1 > Aleph_0).  It cannot be proven by standard set theory whether 
or not Beth_1 = Aleph_1.  Put another way, it is impossible to prove 
whether or not there are sets which have a size between that of the 
natural numbers and that of the real numbers.

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On 7/6/2010 10:01 PM, Kevin Wampler wrote:
> Orchid XP v8 wrote:
>> Mike Raiford wrote:
>>
>
> I found Invisible's summary to be a bit confusing, and also seemed to
> have a semi-important error, so here's another quick summary:
>
> The Aleph numbers are "cardinals" which means they represent the size of
> sets. Aleph_0 is the first infinite cardinal, which is equal to the size
> of the set of natural numbers. Aleph_1 is the next size that a set can
> have which is larger than Aleph_0. Beth_1 is the size of the set of all
> real numbers, and it can be proven that the size of this set is a
> "larger infinity" than the size of the set of all natural numbers (so
> Beth_1 > Aleph_0). It cannot be proven by standard set theory whether or
> not Beth_1 = Aleph_1. Put another way, it is impossible to prove whether
> or not there are sets which have a size between that of the natural
> numbers and that of the real numbers.

Ok, that makes sense to me now. Yeah, I think my original post was a bit 
mistaken ... I realized pretty quickly after Andrew's posting what it 
meant, but I don't quite think I had the right words to express what I 
was thinking.


-- 
~Mike

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Am 06.07.2010 14:30, schrieb Invisible:

> My personal favourit:
>
> 99 bottles of TNT standing on the wall,
> 99 bottles of TNT standing on the wall,
> And if one of those bottles should accidentally fall,
> There's be no more bottles of TNT, and no more ****ing wall.

 From all I know, TNT doesn't come bottled, nor does it explode from 
just accidentally falling.

Unfortunately, "99 bottles of Nitroglycerine standing on the wall" 
doesn't quite fit the rhythm...

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On Thu, 08 Jul 2010 11:32:36 +0200, clipka wrote:

> Am 06.07.2010 14:30, schrieb Invisible:
> 
>> My personal favourit:
>>
>> 99 bottles of TNT standing on the wall, 99 bottles of TNT standing on
>> the wall, And if one of those bottles should accidentally fall, There's
>> be no more bottles of TNT, and no more ****ing wall.
> 
>  From all I know, TNT doesn't come bottled, nor does it explode from
> just accidentally falling.
> 
> Unfortunately, "99 bottles of Nitroglycerine standing on the wall"
> doesn't quite fit the rhythm...

I consider it 'artistic license'.  Just go with it. ;-)

Jim

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