 |
|
|
|
|
|
| |
| |
|
|
|
|
| |
| |
|
|
Am 19.07.2013 20:54, schrieb Nekar Xenos:
> Maybe I should start with something simpler.
>
> Infinity + Infinity = ?
>
> Is there any other answer than 2(Infinity)?
Yes: Infinity + Infinity = Infinity.
> If so please explain.
I can't. I guess it's just the way infinity is defined.
Note that there's no such thing as infinity in real life, so there's no
"natural" way of dealing with infinity. It's an artificial construct,
and as such is defined in a way that applying the "natural" rules of
arithmetic to it does't yield results too useless for too many
applications too much of the time.
Post a reply to this message
|
|
| |
| |
|
|
|
|
| |
| |
|
|
On Fri, 19 Jul 2013 21:13:39 +0200, clipka <ano### [at] anonymous org> wrote:
> Am 19.07.2013 20:54, schrieb Nekar Xenos:
>
>> Maybe I should start with something simpler.
>>
>> Infinity + Infinity = ?
>>
>> Is there any other answer than 2(Infinity)?
>
> Yes: Infinity + Infinity = Infinity.
This is what I thought, but didn't have any evidence for it.
--
-Nekar Xenos-
Post a reply to this message
|
|
| |
| |
|
|
|
|
| |
| |
|
|
Le 19/07/2013 20:54, Nekar Xenos nous fit lire :
> I don't know.
>
> Maybe I should start with something simpler.
>
> Infinity + Infinity = ?
>
> Is there any other answer than 2(Infinity)?
Well, if anything else, that's never the answer.
Infinity + Infinity = Infinity.
>
> If so please explain.
You have to be aware that some Infinity are bigger than some other
Infinity. (rahh... not now!)
To ease the understanding, the smallest infinity is called Aleph-0.
(well, that still a controversial subject, linked to the Axiom of Choice)
The natural integer (0, 1, 2, .. and so on) are in the N set, whose
cardinality is indeed Aleph-0.
Imagine one hotel of infinite room, each room numbered uniquely from N
set. And now you build another one next to it, of same size.
You then renumber the room of the first hotel x--> 2x, and of the second
hotel y--> 2y+1. You now have a single hotel-company of the same size as
one of the original.
You can also split it in multiple (finite number) hotels, all of
infinite size... they were already there!
Post a reply to this message
|
|
| |
| |
|
|
|
|
| |
| |
|
|
On 19/07/2013 07:54 PM, Nekar Xenos wrote:
> I don't know.
>
> Maybe I should start with something simpler.
>
> Infinity + Infinity = ?
>
> Is there any other answer than 2(Infinity)?
>
> If so please explain.
>
> -Unlike Warp I don't know the answer and really want to know :)
"Infinity" is rather vague.
Let's talk specifics. Suppose we have the set of all natural numbers
(i.e., all positive whole numbers). There are infinity of these.
This set is commonly denoted by the latter N. I'm going to refer to the
size of this set as being A0.
Now, suppose we compare N, the set of all *positive* integers, to Z, the
set of *all* integers, negative and positive. Clearly there are twice as
many of these. For every item in N, there are two corresponding items in
Z, one negative and one positive. So the size of Z is A0 + A0.
But wait...
I can map every odd number in N to a negative number in Z, and every
even number in N to a positive number in Z. In this way, every element
of N is mapped to *one* element of Z, and every element of Z is mapped
to *one* element of N.
The existence of this mapping _proves_ that the sets N and Z are
ACTUALLY THE SAME SIZE! O_O
In other words, A0 + A0 = A0, exactly unchanged.
This seems utterly bizarre. But it's a consequence of infinitely large
quantities. To understand, try doing the same trick to prove that the
set {1 .. 5} is the same size as the set {-5 .. +5}. If you try this,
you'll discover the following:
{1 .. 5} {-5 .. +5}
1 <-> -1
2 <-> +1
3 <-> -2
4 <-> +2
5 <-> -3
+3
-4
Oh bugger! +4
-5
+5
It doesn't work. You can't match them up. Because - AS IS FRIGGING
OBVIOUS - the sets aren't the same size. Basically you run out of
elements of the left set before you've matched up all the elements of
the right set.
The ONLY reason this works with infinite sets is that, because they're
infinite, you will never "run out" of stuff to match up. Because of
this, even "doubling" the size of an infinite set actually leaves its
size unchanged - bizarre as that is!
If adding a set to itself once does nothing, doing this twice also does
nothing. That is,
A0 + A0 + A0 = A0
In general,
x * A0 = A0
This is clearly true for any finite x. But what if x ISN'T FINITE? What
happens if we do this:
A0 * A0 = ???
Remember that A0 is the size of N, the set of all integers. So if we
construct the set of all PAIRS of integers, like this...
(1, 1)
(1, 2)
(1, 3)
.
.
.
infinity
(2, 1)
(2, 2)
(2, 3)
.
.
.
infinity
(3, 1)
(3, 2)
(3, 3)
.
.
.
infinity
Surely, the size of this set is CLEARLY equal to A0 * A0.
But wait...
If I take both numbers, add zeros to pad them out to the same length,
and then interleave their digits, I get this:
... ...
... ...
(123, 456) <-> 142536
(123, 457) <-> 142537
(123, 458) <-> 142538
... ...
... ...
Every number on the right matches exactly one pair on the left. Every
pair matches on number on the right. THE SETS HAVE THE SAME SIZE! We
just proved that A0 * A0 = A0.
Just when it's starting to look like A0 is A0 no matter what you do to
it, something interesting happens.
The size of N is A0. If we construct all possible subsets of N, then for
each element of N, a given subset of N either DOES or DOES NOT contain
that element. That's 2 possibilities, independently for each element of
N. So that's a total of 2 ^ A0 possible subsets.
It turns out that (2 ^ A0) > A0.
Yes sir, there are actually DIFFERENT SIZES of infinity. Let us write
A1 = 2 ^ A0
A1 > A0
But, by the same algorithm, 2 ^ A1 > A1, so we actually have
A(n+1) = 2 ^ A(n)
A(n+1) > A(n)
In summary, infinity comes in different sizes. (Now you see why the
original question "what is infinity plus infinity?" is vague; there's
one than one infinity you may ask about.)
In general,
A(x) + A(y) = A(y) iff y >= x
That is, if you add two infinities, the result is the size of whichever
is the largest infinity. (Thus, when you add a particular infinity to
itself, nothing happens.)
Post a reply to this message
|
|
| |
| |
|
|
|
|
| |
| |
|
|
On 7/19/2013 11:54 AM, Nekar Xenos wrote:
>
> I don't know.
>
> Maybe I should start with something simpler.
>
> Infinity + Infinity = ?
>
> Is there any other answer than 2(Infinity)?
>
> If so please explain.
>
> -Unlike Warp I don't know the answer and really want to know :)
>
If you've even hung around people majoring in math in undergrad, you may
have heard stories of a set theory class in which a they actually spend
a day's class proving that 1+1 = 2. On the face of it this seems like a
huge waste of time, why would you bother to construct a proof of such an
obvious fact? Well, it's in answering questions such as yours in which
such effort pays off.
There are a few rather subtle points to a simple question like "what is
1+1":
1) What is a number?
2) What is addition?
And in addition, for your question:
3) What do we mean by infinity?
For the most part your answers to these questions don't matter for
ordinary finite numbers, but as soon as you start treating infinity like
a number these subtle points start to matter a great deal.
In terms of your question "Infinity + Infinity = ??" you most commonly
see one of two answers:
a) Infinity * 2 (note: not 2 * Infinity, the order often matters)
b) Infinity
And this entirely glosses over issues that for some answers to questions
1-3 you can get more than one infinite number, at which point it matters
which infinities you were adding!
Sorry I don't have a simpler answer. But basically the only real way to
answer it is "tell me what you really mean by Infinity, and only then
can I tell you what Infinity+Infinity is". The answers given by clipka
and Le_Forgeron/Orchid are but two of many possibilities (although they
both get the same result, they are using different interpretations of
"infinity" to do it).
Post a reply to this message
|
|
| |
| |
|
|
|
|
| |
| |
|
|
On Fri, 19 Jul 2013 22:10:11 +0200, Orchid Win7 v1 <voi### [at] devnull> wrote:
> On 19/07/2013 07:54 PM, Nekar Xenos wrote:
>> I don't know.
>>
>> Maybe I should start with something simpler.
>>
>> Infinity + Infinity = ?
>>
>> Is there any other answer than 2(Infinity)?
>>
>> If so please explain.
>>
>> -Unlike Warp I don't know the answer and really want to know :)
>
> "Infinity" is rather vague.
>
> Let's talk specifics. Suppose we have the set of all natural numbers
> (i.e., all positive whole numbers). There are infinity of these.
>
> This set is commonly denoted by the latter N. I'm going to refer to the
> size of this set as being A0.
>
> Now, suppose we compare N, the set of all *positive* integers, to Z, the
> set of *all* integers, negative and positive. Clearly there are twice as
> many of these. For every item in N, there are two corresponding items in
> Z, one negative and one positive. So the size of Z is A0 + A0.
>
> But wait...
>
> I can map every odd number in N to a negative number in Z, and every
> even number in N to a positive number in Z. In this way, every element
> of N is mapped to *one* element of Z, and every element of Z is mapped
> to *one* element of N.
>
> The existence of this mapping _proves_ that the sets N and Z are
> ACTUALLY THE SAME SIZE! O_O
>
> In other words, A0 + A0 = A0, exactly unchanged.
>
>
>
> This seems utterly bizarre. But it's a consequence of infinitely large
> quantities. To understand, try doing the same trick to prove that the
> set {1 .. 5} is the same size as the set {-5 .. +5}. If you try this,
> you'll discover the following:
>
> {1 .. 5} {-5 .. +5}
> 1 <-> -1
> 2 <-> +1
> 3 <-> -2
> 4 <-> +2
> 5 <-> -3
> +3
> -4
> Oh bugger! +4
> -5
> +5
>
> It doesn't work. You can't match them up. Because - AS IS FRIGGING
> OBVIOUS - the sets aren't the same size. Basically you run out of
> elements of the left set before you've matched up all the elements of
> the right set.
>
> The ONLY reason this works with infinite sets is that, because they're
> infinite, you will never "run out" of stuff to match up. Because of
> this, even "doubling" the size of an infinite set actually leaves its
> size unchanged - bizarre as that is!
>
>
>
> If adding a set to itself once does nothing, doing this twice also does
> nothing. That is,
>
> A0 + A0 + A0 = A0
>
> In general,
>
> x * A0 = A0
>
> This is clearly true for any finite x. But what if x ISN'T FINITE? What
> happens if we do this:
>
> A0 * A0 = ???
>
> Remember that A0 is the size of N, the set of all integers. So if we
> construct the set of all PAIRS of integers, like this...
>
> (1, 1)
> (1, 2)
> (1, 3)
> .
> .
> .
> infinity
> (2, 1)
> (2, 2)
> (2, 3)
> .
> .
> .
> infinity
> (3, 1)
> (3, 2)
> (3, 3)
> .
> .
> .
> infinity
>
> Surely, the size of this set is CLEARLY equal to A0 * A0.
>
> But wait...
>
> If I take both numbers, add zeros to pad them out to the same length,
> and then interleave their digits, I get this:
>
> ... ...
> ... ...
> (123, 456) <-> 142536
> (123, 457) <-> 142537
> (123, 458) <-> 142538
> ... ...
> ... ...
>
> Every number on the right matches exactly one pair on the left. Every
> pair matches on number on the right. THE SETS HAVE THE SAME SIZE! We
> just proved that A0 * A0 = A0.
>
>
>
> Just when it's starting to look like A0 is A0 no matter what you do to
> it, something interesting happens.
>
> The size of N is A0. If we construct all possible subsets of N, then for
> each element of N, a given subset of N either DOES or DOES NOT contain
> that element. That's 2 possibilities, independently for each element of
> N. So that's a total of 2 ^ A0 possible subsets.
>
> It turns out that (2 ^ A0) > A0.
>
> Yes sir, there are actually DIFFERENT SIZES of infinity. Let us write
>
> A1 = 2 ^ A0
>
> A1 > A0
>
> But, by the same algorithm, 2 ^ A1 > A1, so we actually have
>
> A(n+1) = 2 ^ A(n)
>
> A(n+1) > A(n)
>
> In summary, infinity comes in different sizes. (Now you see why the
> original question "what is infinity plus infinity?" is vague; there's
> one than one infinity you may ask about.)
>
> In general,
>
> A(x) + A(y) = A(y) iff y >= x
>
> That is, if you add two infinities, the result is the size of whichever
> is the largest infinity. (Thus, when you add a particular infinity to
> itself, nothing happens.)
This is all very interesting. :)
I was thinking of adding any particular infinity to itself.
--
-Nekar Xenos-
Post a reply to this message
|
|
| |
| |
|
|
|
|
| |
| |
|
|
On Fri, 19 Jul 2013 22:34:29 +0200, Kevin Wampler <nob### [at] nowherenet>
wrote:
> On 7/19/2013 11:54 AM, Nekar Xenos wrote:
>>
>> I don't know.
>>
>> Maybe I should start with something simpler.
>>
>> Infinity + Infinity = ?
>>
>> Is there any other answer than 2(Infinity)?
>>
>> If so please explain.
>>
>> -Unlike Warp I don't know the answer and really want to know :)
>>
>
> If you've even hung around people majoring in math in undergrad, you may
> have heard stories of a set theory class in which a they actually spend
> a day's class proving that 1+1 = 2. On the face of it this seems like a
> huge waste of time, why would you bother to construct a proof of such an
> obvious fact? Well, it's in answering questions such as yours in which
> such effort pays off.
>
> There are a few rather subtle points to a simple question like "what is
> 1+1":
>
> 1) What is a number?
> 2) What is addition?
>
> And in addition, for your question:
>
> 3) What do we mean by infinity?
>
> For the most part your answers to these questions don't matter for
> ordinary finite numbers, but as soon as you start treating infinity like
> a number these subtle points start to matter a great deal.
>
> In terms of your question "Infinity + Infinity = ??" you most commonly
> see one of two answers:
>
> a) Infinity * 2 (note: not 2 * Infinity, the order often matters)
> b) Infinity
>
> And this entirely glosses over issues that for some answers to questions
> 1-3 you can get more than one infinite number, at which point it matters
> which infinities you were adding!
>
> Sorry I don't have a simpler answer. But basically the only real way to
> answer it is "tell me what you really mean by Infinity, and only then
> can I tell you what Infinity+Infinity is". The answers given by clipka
> and Le_Forgeron/Orchid are but two of many possibilities (although they
> both get the same result, they are using different interpretations of
> "infinity" to do it).
>
I think I could say the specific infinity I'm thinking of would be the
biggest type of infinity.
What would that be?
Complex Infinity? (if that could be considered)
--
-Nekar Xenos-
Post a reply to this message
|
|
| |
| |
|
|
|
|
| |
| |
|
|
On 19/07/2013 10:48 PM, Nekar Xenos wrote:
> I think I could say the specific infinity I'm thinking of would be the
> biggest type of infinity.
> What would that be?
You will be sorry you asked that. :-P
--
Regards
Stephen
Post a reply to this message
|
|
| |
| |
|
|
|
|
| |
| |
|
|
On 19/07/2013 8:54 PM, Nekar Xenos wrote:
> Is there any other answer
How about and we may have an infinity of authors spinning in their graves:
The sum of all the odd numbers is infinity.
The sum of all even numbers is infinity.
The sum of all numbers odd and even is infinity.
So by doing ordinary sums:
infinity + infinity = infinity
and
infinity = infinity * 2
Can't be right. And that is because infinity is not a number but an idea.
But if you make a mathematics using infinity as an object it opens a
whole new way for mathematicians to think.
It was at this point that I thought that I would leave it for minds that
want to see things with out drugs.
Give the book that I mentioned at the beginning of this thread "White
Light" a try if you can find it.
--
Regards
Stephen
Post a reply to this message
|
|
| |
| |
|
|
|
|
| |
| |
|
|
On 7/19/2013 1:48 PM, Nekar Xenos wrote:
>
> I think I could say the specific infinity I'm thinking of would be the
> biggest type of infinity.
> What would that be?
> Complex Infinity? (if that could be considered)
>
There are two different ways I'm tempted to interpret your question, but
only one of them makes sense. I'll try to answer them both anyway:
Q1) "Out of all the different ways you can define Infinity, what's the
biggest?
A1) Because the different definitions of Infinity use different
definitions of what a "number" is, there is no way to compare them at
all to say which is bigger -- they are just completely different things.
It's like asking "which is bigger, 4 or fish?".
Q2) "You mentioned that for some definitions you get multiple different
types of infinity, what's the biggest of those?"
A2) The answers depends on what particular definition of Infinity you're
talking about, but the most common answers is that there is no biggest
Infinity -- just like there's no biggest finite integer. Sometimes
people will try to add a "biggest infinity" to things, but you don't
generally allow addition with it anyway.
---
As I mentioned in (A1), it doesn't make sense to ask if "complex
Infinity" is bigger than another definition of infinity.
As an aside, a notion of "complex Infinity" is actually extremely useful
in some areas mathematics. Arguably much more useful than "real
Infinity" is. The standard definition of complex Infinity does not
allow Infinity+Infinity though (it treats it as undefined, much like 1/0
is commonly treated as undefined for the reals).
Post a reply to this message
|
|
| |
| |
|
|
|
|
| |
|
|
