On 7/19/2013 1:54 PM, Stephen wrote:
> 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
>

Haha, that seems to be the answer to a lot of math questions of this 
sort!  I did what I could to make my answer not too painful though.

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On 7/19/2013 1:56 PM, Stephen wrote:
> 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
>

You know, if you wrote that as "2 * infinity" the equation would 
actually be correct as far as the major definitions of infinity go. 
Ain't math grand?

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>> 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.

Then my last paragraph contains all you need to know. ;-)

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On 19/07/2013 09:34 PM, Kevin Wampler wrote:
> 2) What is addition?
>
> And in addition, for your question:

I SEE WHAT YOU DID THERE! ;-)

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On 19/07/2013 09:37 PM, Nekar Xenos wrote:
> This is all very interesting. :)
>
> I was thinking of adding any particular infinity to itself.

I may also have glossed over the fact that the results you get from set 
theory vary slightly depending on WHICH VERSION OF SET THEORY you happen 
to be talking about. There are several slight variations of it.

Think of it like playing chess; different chess tournaments have very 
slightly different rules. Usually it doesn't matter, but sometimes the 
outcome of a particular match can depend on exactly which rules you 
choose to play by.

(One of the fun things about mathematics is simply INVENTING new rules 
out of thin air and seeing where that takes you...)

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On 19/07/2013 11:04 PM, Kevin Wampler wrote:
> On 7/19/2013 1:54 PM, Stephen wrote:
>> 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
>>
>
> Haha, that seems to be the answer to a lot of math questions of this
> sort!  I did what I could to make my answer not too painful though.


Indeed I understood it while I was reading it. ;-)

Seriously there a some pretty good minds around here but Nekar hasn't 
mentioned to what standard his maths is. No offence Nekar, I only 
studded maths, no every thing, to the end of highschool. (I think that 
is what you call it. To just before university about 19 in the UK.)
And we all know that it is different answers for different folk.

-- 
Regards
     Stephen

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On 19/07/2013 11:19 PM, Orchid Win7 v1 wrote:
> On 19/07/2013 09:34 PM, Kevin Wampler wrote:
>> 2) What is addition?
>>
>> And in addition, for your question:
>
> I SEE WHAT YOU DID THERE! ;-)
>

+1

-- 
Regards
     Stephen

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On Fri, 19 Jul 2013 23:26:02 +0200, Stephen <mca### [at] aolcom> wrote:

> On 19/07/2013 11:04 PM, Kevin Wampler wrote:
>> On 7/19/2013 1:54 PM, Stephen wrote:
>>> 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
>>>
>>
>> Haha, that seems to be the answer to a lot of math questions of this
>> sort!  I did what I could to make my answer not too painful though.
>
>
> Indeed I understood it while I was reading it. ;-)
>
> Seriously there a some pretty good minds around here but Nekar hasn't  
> mentioned to what standard his maths is. No offence Nekar, I only  
> studded maths, no every thing, to the end of highschool. (I think that  
> is what you call it. To just before university about 19 in the UK.)
> And we all know that it is different answers for different folk.
>

High school maths...

-- 
-Nekar Xenos-

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On 19/07/2013 11:07 PM, Kevin Wampler wrote:
> On 7/19/2013 1:56 PM, Stephen wrote:
>> 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
>>
>
> You know, if you wrote that as "2 * infinity" the equation would
> actually be correct as far as the major definitions of infinity go.
> Ain't math grand?
>

Damn! that was what I was going to do but thought that it was too big a 
leap to say that infinity * infinity = infinity. From what I had written 
before.
So out of (a passing) interest what is the difference?

PS the passing is my little joke. ;-)


-- 
Regards
     Stephen

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On 19/07/2013 11:16 PM, Orchid Win7 v1 wrote:
>>> 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.
>
> Then my last paragraph contains all you need to know. ;-)

Is that a RTFM comment?

-- 
Regards
     Stephen

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On Fri, 19 Jul 2013 23:02:04 +0200, Kevin Wampler <nob### [at] nowherenet>  
wrote:

> 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).
>
I imagined the complex set would have a larger "density" if you could call  
it that. (Real numbers would have a larger "density" than integers)

So if I have this right
Infinity + Infinity =  Infinity
is correct for real numbers and not for complex numbers.

-- 
-Nekar Xenos-

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On 19/07/2013 11:31 PM, Nekar Xenos wrote:

>
> High school maths...
>

Like me then. What everyone is saying is correct but they are saying it 
with the understanding of the language "mathematics". It is not what we 
mortals understand as the same language that we speak. There is no 
option but to learn it or consign it to "all very well but does it pay 
the bills". (A topic I will leave to the tabloids. :-) )

I've read a little about it and to be truthful if the threat goes any 
deeper then it goes over my head. Infinity is a great thought but after 
a while. My eyes cross when I try focusing on it.

-- 
Regards
     Stephen

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>>> 2) What is addition?
>>>
>>> And in addition, for your question:
>>
>> I SEE WHAT YOU DID THERE! ;-)
>
> +1

"Add one"?

I SEE WHAT YOU DID THERE! :-D

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On 19/07/2013 11:38 PM, Nekar Xenos wrote:

> I imagined the complex set would have a larger "density" if you could
> call it that. (Real numbers would have a larger "density" than integers)
>

Why don’t I mix it a little. :-D

Imagine infinity * infinity.
Which is infinity ^2

Now think about infinity ^ infinity.

Night night. ;-)

(Sorry I'm in that sort of mood. I've had a long day.)

-- 
Regards
     Stephen

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On 19/07/2013 11:46 PM, Orchid Win7 v1 wrote:
>>>> 2) What is addition?
>>>>
>>>> And in addition, for your question:
>>>
>>> I SEE WHAT YOU DID THERE! ;-)
>>
>> +1
>
> "Add one"?
>
> I SEE WHAT YOU DID THERE! :-D

And I saw what YOU did there.

Unfortunately I wrote plus one.
Which is different to plus four's

-- 
Regards
     Stephen

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On Fri, 19 Jul 2013 23:49:26 +0200, Stephen <mca### [at] aolcom> wrote:

> On 19/07/2013 11:38 PM, Nekar Xenos wrote:
>
>> I imagined the complex set would have a larger "density" if you could

>> call it that. (Real numbers would have a larger "density" than intege
rs)
>>
>
> Why don’t I mix it a little. :-D
>
> Imagine infinity * infinity.
> Which is infinity ^2
>
> Now think about infinity ^ infinity.
>
> Night night. ;-)
>
> (Sorry I'm in that sort of mood. I've had a long day.)
>

How about:

Complex Infinity ^ Complex Infinity

But I suppose if you can't add them, you can't power them...

-- 

-Nekar Xenos-

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On 7/19/2013 2:32 PM, Stephen wrote:
>
> Damn! that was what I was going to do but thought that it was too big a
> leap to say that infinity * infinity = infinity. From what I had written
> before.
> So out of (a passing) interest what is the difference?
>
> PS the passing is my little joke. ;-)
>

I'm trying to think of an answer which won't make you regret asking that 
question :)

For some definitions of infinity there isn't a difference and 2*infinity 
= infinity*2 = infinity

The most common definition of infinity for which there *is* a difference 
can be thought of as reasoning about (possibly infinitely long) 
sequences of things.  In order to not get into the confusing details 
about this, let me just say that it turns out that if you follow this 
logic through you find that you're allowed to take a sequence, A, and 
create a new sequence by replacing each entry in A with another sequence B.

Let me give an example.  I'll denote each element in a sequence tieh the 
character @.  You can represent a finite number by a sequence of a 
certain length, so 1 is "@", 2 is "@@", 3 is "@@@", 7 is "@@@@@@@" etc. 
  So let's say we take the sequence for 3:

@@@

and replace each element in it by the sequence for 2, the we get:

(@@)(@@)(@@)

Where I've put parentheses in for readability.  Writing it without the 
parens you get:

@@@@@@

Which is the sequence of 6.  And what a coincidence 2*3 = 6!  In fact, 
this strange process works exactly like the multiplication you've come 
to know and love for all finite numbers.  Since it works just like 
multiplication, we'll just say that this is what we mean by multiplying 
two of these sequences together (if it walks like a duck and it quacks 
like a duck).  Here's a slightly more precise definition:

If A and B are sequences, then by "A*B" we mean the sequence which would 
result by replacing each element in B by the sequence A.

By this definition:

2*3 = (@@)*(@@@) = (@@)(@@)(@@) = @@@@@@ = 6
3*2 = (@@@)*(@@) = (@@@)(@@@) = @@@@@@ = 6

So we get 2*3 = 3*2 = 6 like you'd expect.  But notice that the 
parentheses were different in that middle step.  This is where 
everything goes to hell when you allow infinite sequences.

Let me give an example.  Say you've got your basic infinite sequence 
(which I'll write as "w" since writing "infinity" get confusing when you 
have more than one infinity).  So:

w = @@@@@...

Where the "..." just means "goes on forever".  I won't force you to 
endure a mathematically precise definition of that.  So with our 
previous definition of multiplication:

2*w = (@@)(@@)(@@)... = @@@@@@... = w
w*2 = (@@@@...)(@@@@...) = @@@@...@@@@... = now you may be confused

I don't have a way to make that w*2 feel like it's not a little crazy, 
but the best I can do is restate it in plain English:

2*w = "count in forever in multiple of two"
w*2 = "count forever, then when you're done, do it again."

So while counting forever by multiples of two is basically the same as 
counting forever normally, counting forever and then doing it again is 
another thing entirely.

Provided you're willing to accept that any of this makes sense at all, 
that's what the difference is (for that definition of "number" of 
course, which surprisingly is a relatively common one).

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On Sat, 20 Jul 2013 00:02:11 +0200, Kevin Wampler <nob### [at] nowherenet>  
wrote:

> On 7/19/2013 2:32 PM, Stephen wrote:

> 2*w = "count in forever in multiple of two"
> w*2 = "count forever, then when you're done, do it again."
>

I see a light bulb! ;)


-- 
-Nekar Xenos-

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On 19/07/2013 11:59 PM, Nekar Xenos wrote:

>
> How about:
>
> Complex Infinity ^ Complex Infinity
>

Good question. What do you think?

> But I suppose if you can't add them, you can't power them...
>
Who said that? I can't find it in the thread as it is getting long.
I thought you could, if it makes any in ordinary language. Is it a 
property of them having one more dimension?

-- 
Regards
     Stephen

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On 20/07/2013 12:12 AM, Stephen wrote:
> if it makes any in ordinary language

if it makes any sense in ordinary language

-- 
Regards
     Stephen

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On 20/07/2013 12:02 AM, Kevin Wampler wrote:
> I'm trying to think of an answer which won't make you regret asking that
> question :)

In that case I'll leave it 'till the morning to read it. As I said 
earlier. "I've had a long day" and I've crossed a timezone. The wrong 
way. :-(


-- 
Regards
     Stephen

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On 20/07/2013 12:07 AM, Nekar Xenos wrote:
> On Sat, 20 Jul 2013 00:02:11 +0200, Kevin Wampler <nob### [at] nowherenet>
> wrote:
>
>> On 7/19/2013 2:32 PM, Stephen wrote:
>
>> 2*w = "count in forever in multiple of two"
>> w*2 = "count forever, then when you're done, do it again."
>>
>
> I see a light bulb! ;)
>
>

Eureka! :-D

But from what you quoted.
Don’t they have a BODMAS? (well that is what I was taught the order of 
operations, was called). And the meanings.


-- 
Regards
     Stephen

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On 7/19/2013 2:38 PM, Nekar Xenos wrote:
> I imagined the complex set would have a larger "density" if you could
> call it that. (Real numbers would have a larger "density" than integers)

What you mean be "density" is a whole 'nother issue which is 
surprisingly complex when you get into it.  When talking about questions 
like "Infinity+Infinity" it's best not to worry about things like "density".

 > So if I have this right
 > Infinity + Infinity =  Infinity
 > is correct for real numbers and not for complex numbers.

That's basically correct.  More precisely, for the definition of 
infinity which turns out to be most useful for the reals 
Infinity+Infinity=Infinity.  For the definition of Infinity which turns 
out to be the most useful for the complex numbers Infinity+Infinity is 
undefined.

People can (and occasionally do) use different definitions of Infinity 
for the reals/complexes than these, depending on what's useful for the 
problem they want to solve.  But you've got the right idea for the most 
common choices.

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On 7/19/2013 3:07 PM, Nekar Xenos wrote:
> On Sat, 20 Jul 2013 00:02:11 +0200, Kevin Wampler <nob### [at] nowherenet>
> wrote:
>
>> On 7/19/2013 2:32 PM, Stephen wrote:
>
>> 2*w = "count in forever in multiple of two"
>> w*2 = "count forever, then when you're done, do it again."
>>
>
> I see a light bulb! ;)
>


Excellent!  I should point out that this is an entirely different 
definition of "number" than Le_Forgeron/Orchid have been using.  They 
were treating numbers as reasoning about quantities, whereas which I 
wrote treats numbers as reasoning about sequences/orders.  Both are 
equally valid, but they give different answers to your question.

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Am 19.07.2013 23:38, schrieb Nekar Xenos:

>> 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).
>>
> I imagined the complex set would have a larger "density" if you could
> call it that. (Real numbers would have a larger "density" than integers)

No, the set of complex numbers is just "as infinite" as the set of real 
numbers; you can use a digit-interleaving technique similar to that 
which gets you from the natural numbers to all of the integers.

> So if I have this right
> Infinity + Infinity =  Infinity
> is correct for real numbers and not for complex numbers.

If you mean

   Infinity := (number of elements in the set of complex numbers)

then no, Infinity + Infinity = Infinity still holds there.

I guess what he meant was that some

   ComplexInfinity := an infinite that behaves much like a
                      complex number, and possibly even has
                      a representation of the form (a + bi)
                      for some presumably weird choice of a,b

Gives you

   ComplexInfinity + ComplexInfinity = <undefined>

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On 7/19/2013 4:06 PM, clipka wrote:
>
> I guess what he meant was that some
>
>    ComplexInfinity := an infinite that behaves much like a
>                       complex number, and possibly even has
>                       a representation of the form (a + bi)
>                       for some presumably weird choice of a,b
>
> Gives you
>
>    ComplexInfinity + ComplexInfinity = <undefined>
>

This is indeed what I meant by "complex infinity".  As far as I'm aware, 
it's not decomposable into the form a+bi where "a" and "b" are real, 
since doing so would require a "real infinity" which is distinct from 
the complex infinity (I mean, you probably *could* define such a thing, 
but I've never seen it done).

Good catch that Nekar was using "complex infinity" in a different way 
than I was thinking he was.

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On 19/07/2013 11:02 PM, Kevin Wampler wrote:
> So while counting forever by multiples of two is basically the same as
> counting forever normally, counting forever and then doing it again is
> another thing entirely.
>
> Provided you're willing to accept that any of this makes sense at all,
> that's what the difference is (for that definition of "number" of
> course, which surprisingly is a relatively common one).

That all made sense and no new ideas. Knowing the syntax and putting it 
in context, helps. Thanks.

It was well explained, so do you lecture in mathematics or is it just 
part of your job?

-- 
Regards
     Stephen

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On 7/20/2013 3:34 AM, Stephen wrote:
>
> It was well explained, so do you lecture in mathematics or is it just
> part of your job?
>

Thanks!  I'm glad to hear that you liked the explanation.  I don't 
lecture in mathematics or use this sort of math for my job.  I just 
enjoy mathematics and have studied various bits of it more or less for fun.

I do use a lot of math in my job though, but it's just applied math and 
not set theory stuff like this.

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Nekar Xenos <nek### [at] gmailcom> wrote:
> I was thinking of adding any particular infinity to itself.

In a sense, adding infinity to itself doesn't make any sense because
infinity is not a number. It's a concept used to describe an abstraction.

There are certain situations where you can use mathematical operators
on infinities as a kind of special notation.

Calculating limits is one situation where infinity-as-a-notation can be
useful.

For example, if lim(x->a) f(x) is infinity, and lim(x->a) g(x) is
infinity, then you can use the notation that lim(x->a) f(x)+g(x) =
infinity + infinity = infinity. In other words, since both f(x) and
g(x) approach infinity when x approaches a, you know that their sum
will also do so.

This doesn't mean that "infinity" is some kind of "special number" that
you can add or do special operations. It's not a number. As said, this
is just a notation you can use to make such calculations easier.

The same is true for multiplication. lim(x->a) f(x)*g(x) is also infinity.

This is also the reason why it's said that eg. "infinity - infinity" is
indetermined. It basically means that the notation system fails and
cannot be used to determine the correct result.

In other words, if it were lim(x->a) f(x)-g(x), the result depends on
what f(x) and g(x) are. You cannot know the answer only from knowing
that their individual limits go to infinity.

-- 
                                                          - Warp

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Nekar Xenos <nek### [at] gmailcom> wrote:
> I imagined the complex set would have a larger "density" if you could call  
> it that. (Real numbers would have a larger "density" than integers)

You can uniquely map each complex number to each real number on a
one-to-one basis, which means that the set of complex numbers is as
big as the set of reals.

Likewise you can map each rational number to each natural number on
a one-to-one basis, which means that the set of rational numbers is as
big as the set of natural numbers. (There's actually a rather easy way
to do this enumeration. See if you can figure it out.)

Perhaps a bit surprisingly, there is no one-to-one mapping between
rational numbers and real numbers, which means that the set of real
numbers is larger than the set of rational numbers.

This last thing is quite curious given that, for example, between any
two real numbers there is an infinite amount of rational numbers, and
vice-versa. It feels like there should be the same amount...

-- 
                                                          - Warp

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On 7/20/2013 11:28 AM, Warp wrote:
> Nekar Xenos <nek### [at] gmailcom> wrote:
>> I was thinking of adding any particular infinity to itself.
>
> In a sense, adding infinity to itself doesn't make any sense because
> infinity is not a number. It's a concept used to describe an abstraction.

Huh?  Certainly infinity isn't an integer/real/rational/etc, but there 
are consistent and sensible definitions for "number" which include 
infinity(s) (I'm pretty sure you're aware of this, so maybe I'm missing 
your point?).

Of course infinity is *also* used as an abstraction (as in the case with 
limits you mention), but this doesn't mean that in other cases you can't 
use it as a perfectly fine number.

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Kevin Wampler <nob### [at] nowherenet> wrote:
> Huh?  Certainly infinity isn't an integer/real/rational/etc, but there 
> are consistent and sensible definitions for "number" which include 
> infinity(s) (I'm pretty sure you're aware of this, so maybe I'm missing 
> your point?).

So it's a number that's not any kind of number? It's not an integer,
it's not rational, it's not irrational, it's not transcendental, and it
doesn't follow any of the rules of any of the other sets (eg. if you
add 1 to any number, you get a new number that's larger than the
original; or if you multiply any number by 0 you get 0.) All mathematical
operators and functions work completely different for it than for any
other numbers (moreover, most of them aren't even well-defined for
infinity.)

Infinity is not a number. It's just a concept that's used to describe
a more abstract notion. You can use it a bit like if it were a number
when dealing with things like limits, but even then it's just a shortcut
notation.

-- 
                                                          - Warp

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Am 21.07.2013 01:27, schrieb Warp:
> Kevin Wampler <nob### [at] nowherenet> wrote:
>> Huh?  Certainly infinity isn't an integer/real/rational/etc, but there
>> are consistent and sensible definitions for "number" which include
>> infinity(s) (I'm pretty sure you're aware of this, so maybe I'm missing
>> your point?).
>
> So it's a number that's not any kind of number? It's not an integer,
> it's not rational, it's not irrational, it's not transcendental, and it
> doesn't follow any of the rules of any of the other sets (eg. if you
> add 1 to any number, you get a new number that's larger than the
> original; or if you multiply any number by 0 you get 0.) All mathematical
> operators and functions work completely different for it than for any
> other numbers (moreover, most of them aren't even well-defined for
> infinity.)
>
> Infinity is not a number. It's just a concept that's used to describe
> a more abstract notion. You can use it a bit like if it were a number
> when dealing with things like limits, but even then it's just a shortcut
> notation.

Well, I guess it simply boils down to definitions again: What, actually, 
/is/ a "number" (in the system you're currently examining)?

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On 7/20/2013 4:27 PM, Warp wrote:
> So it's a number that's not any kind of number? It's not an integer,
> it's not rational, it's not irrational, it's not transcendental, and it
> doesn't follow any of the rules of any of the other sets (eg. if you
> add 1 to any number, you get a new number that's larger than the
> original; or if you multiply any number by 0 you get 0.)  All mathematical
> operators and functions work completely different for it than for any
> other numbers (moreover, most of them aren't even well-defined for
> infinity.)

Before I say anything, it's important to point out that I don't find the 
term "number" to be particularly mathematically meaningful in the first 
place.  It's probably best to just be explicit to say "integer", "real", 
"cardinal number" etc., and not bother too much about what counts as a 
"true number" and what doesn't.  That said, if you're going to take some 
sort of definition as to what counts as a "number", then there are some 
good reasons that you might want to include some notions of infinity as 
numbers.

What you say above is true for the particular type of infinity you seem 
to be thinking of, but it is *not* true in general for other definitions 
of infinity.  I'll address it point by point:

Q) It's not an integer, it's not rational, it's not irrational, it's not 
transcendental

A) If this is your criteria then you've basically said from the 
beginning that infinity isn't a number by definition -- no matter how 
much like a number it behaves.  You can certainly take this approach, 
but you'll need to accept the existence of "number-like things which are 
not numbers".  This is completely valid, in which case I don't think we 
really disagree in any substantial way but just draw the line between 
"number" and "number-like thing" is different places.

Q) it doesn't follow any of the rules of any of the other sets (eg. if 
you add 1 to any number, you get a new number that's larger than the 
original; or if you multiply any number by 0 you get 0.)

A) This is not true in general.  There are definitions of infinity for 
which infinity+1 > infinity, and for which infinity*0 = 0.  And besides, 
there are similar situations with "normal" numbers anyway.  For instance 
in the reals you can calculate 1/x for every number x except 0, and 
something like x+1 > x doesn't makes sense with complex numbers.  Heck, 
in the integers x/y only makes sense if y if a factor of x.  "Normal" 
numbers have arithmetic exceptions too, and it doesn't prevent you from 
thinking of them as genuine numbers.

Q) All mathematical operators and functions work completely different 
for it than for any other numbers (moreover, most of them aren't even 
well-defined for infinity.)

A) I actually find the change in how things work by including infinite 
numbers to be *less* than the difference between the integers and the 
reals.  Saying "All mathematical operators and functions work completely 
different for it than for any other numbers" simply isn't true in 
general.  It's certainly true for *some* definitions of infinity, but 
for other definitions you get perfectly well defined addition and 
multiplication, and for a few you even get commutative addition and 
multiplication, as well as division, subtraction, etc.  Heck there are 
some with these properties where you even maintain that either a <= b or 
b <= a, which isn't even true of the complex numbers!

I mean, obviously you need to draw the line somewhere between what you 
call a number, and what behaves differently enough that the term 
"number" no longer makes sense, but I think it's pretty reasonable to 
consider some (but all) definitions of infinity as sufficiently 
number-like to be genuine numbers.  I also think that it's fine to call 
complex numbers, quaternions, elements in cyclic groups etc. as numbers 
too, so maybe I just take a pretty broad stance as to what I think of as 
a number.


> Infinity is not a number. It's just a concept that's used to describe
> a more abstract notion. You can use it a bit like if it were a number
> when dealing with things like limits, but even then it's just a shortcut
> notation.

This is math we're talking about, everything is abstract in some sense. 
  I agree with limits that it's a notational shortcut, but there are 
many many other ways of defining infinity quantities, and some of them 
behave quite a bit like numbers.

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Kevin Wampler <nob### [at] nowherenet> wrote:
>   I agree with limits that it's a notational shortcut, but there are 
> many many other ways of defining infinity quantities, and some of them 
> behave quite a bit like numbers.

So how do you define infinity so that it's a number?

-- 
                                                          - Warp

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On Sun, 21 Jul 2013 03:26:02 +0200, Kevin Wampler <nob### [at] nowherenet>  
wrote:

> On 7/20/2013 4:27 PM, Warp wrote:

> Q) All mathematical operators and functions work completely different  
> for it than for any other numbers (moreover, most of them aren't even  
> well-defined for infinity.)
>
> A) I actually find the change in how things work by including infinite  
> numbers to be *less* than the difference between the integers and the  
> reals.  Saying "All mathematical operators and functions work completely  
> different for it than for any other numbers" simply isn't true in  
> general.  It's certainly true for *some* definitions of infinity, but  
> for other definitions you get perfectly well defined addition and  
> multiplication, and for a few you even get commutative addition and  
> multiplication, as well as division, subtraction, etc.  Heck there are  
> some with these properties where you even maintain that either a <= b or  
> b <= a, which isn't even true of the complex numbers!
>

It almost seems that any mathematical operation using infinity has the  
same answer: Infinity
:)

-- 
-Nekar Xenos-

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On 21/07/2013 8:37 AM, Nekar Xenos wrote:
> It almost seems that any mathematical operation using infinity has the
> same answer: Infinity
> :)


And some answer only to themselves. O.o

-- 
Regards
     Stephen

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On 7/21/2013 12:37 AM, Nekar Xenos wrote:
> On Sun, 21 Jul 2013 03:26:02 +0200, Kevin Wampler <nob### [at] nowherenet>
>
> It almost seems that any mathematical operation using infinity has the
> same answer: Infinity
> :)
>

Well, for some of these you get a whole slew of different "infinite 
numbers", so the result of many mathematical operations using infinity 
just give you different infinities as their answers (which is pretty 
much what you'd expect if you count these infinite quantities as actual 
numbers).

In situations where such things are allowed, dividing by infinity can be 
useful, in which case it gives you either zero or an infinitesimal, 
depending on which system you're using.

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On 7/21/2013 12:34 AM, Warp wrote:
> Kevin Wampler <nob### [at] nowherenet> wrote:
>>    I agree with limits that it's a notational shortcut, but there are
>> many many other ways of defining infinity quantities, and some of them
>> behave quite a bit like numbers.
>
> So how do you define infinity so that it's a number?
>

I would count any of the following, but you may find the last two to be 
better examples than the first two:

1) cardinal numbers: http://en.wikipedia.org/wiki/Cardinal_number
2) ordinal numbers: http://en.wikipedia.org/wiki/Ordinal_number
3) extended complex plane: http://en.wikipedia.org/wiki/Riemann_sphere
4) surreal numbers: http://en.wikipedia.org/wiki/Surreal_number

There are probably plenty of other ways too, but these are the ones that 
come to mind for me.

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I wanted to post about this earlier but couldn't remember the name. 
Anyway stumbled across it again today, possibly relevant to this thread:

http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox

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