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Is infinity a number?

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EvilToothpaste

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Is infinity a number?
« on: May 03, 2008, 04:08:53 PM »
This is continued from another thread, linked in the quote below:

Quote from: Username on May 03, 2008, 02:22:45 PM
Quote from: EvilToothpaste on May 03, 2008, 01:57:19 PM
Quote from: Username on May 03, 2008, 09:21:39 AM
No, actually it can be a number in some systems.
Maybe you mean it can be treated like a number in some systems.  It is a limit, though, not a number. 
No, it *is* a number in some systems.

Furthermore, you could argue it is a number in all system - depending on whose definition of number you use.  Frege for example has a compatible definition for infinity and number.

Not to mention the work of Georg Cantor.
That's funny, I've always heard and understood quite the opposite, even after reading of Georg Cantor's work. 

In what systems is infinity a number?  And, please tell me how the two are given equality in Frege's definition. 

(Isn't Frege the guy whose life work was scuttled by Bertrand Russel shorty after it was printed?  By scuttled I mean his system of axioms was found to be contradictory.)
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narcberry

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Re: Is infinity a number?
« Reply #1 on: May 03, 2008, 04:36:24 PM »
The problem with considering infinity as a number is that it infinity has more attributes than a canonical number. Due to that fact, it is difficult to compare infinities with canonical numbers, yet much simpler with other infinities.

For example.
Consider three infinite sets of pennies.
Set1: A penny exists every 5 centimeters in a line.
Set2: A penny exists every 10 centimeters in a line.
Set3: A penny exists every 2x centimeters in a line, where x = the distance between the previous two pennies.

If you were to take a canonical number, lets say '4', your comparisons become rather repetitive. Sets in below algebra represent the number of elements in that set.

Typical misuse of Infinity
4 - Set1 = -infinity
4 - Set2 = -infinity
4 - Set3 = -infinity

4 * Set1 = infinity
4 * Set2 = infinity
4 * Set3 = infinity

4 / Set1 = 0
4 / Set2 = 0
4 / Set3 = 0

Set1 - Set2 = -infinity
Set3 - Set2 = infinity		               Correct use of infinities
4 - Set1 = -Set1
4 - Set2 = -Set2
4 - Set3 = -Set3

4 * Set1 = 4Set1 (ie pennies every 4/5 cm apart)
4 * Set2 = 4Set2
4 * Set3 = 4Set3

4 / Set1 = 0
4 / Set2 = 0
4 / Set3 = 0

Set1 - Set2 = -Set1
Set3 - Set2 = Set3

The problem isn't considering infinity, it's considering infinity as a single number. Infinity is actually an uncountably infinite set of numbers, like the real numbers. To properly compare them with canonical numbers, their attributes must be fully extrapolated to be accurate. Comparison without extrapolation can give some useful information, but is actually not fully accurate. That would be like saying 4 - 5 = a number. It's true, but not fully accurate, there is more available information.

Integrals and derivatives are helpful concepts to the subject at hand.
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narcberry

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Re: Is infinity a number?
« Reply #2 on: May 03, 2008, 05:01:46 PM »
I still don't know what my point was, yet I posted all that...
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EvilToothpaste

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Re: Is infinity a number?
« Reply #3 on: May 03, 2008, 08:21:23 PM »
Nah, that's good. 

I was going to use the same idea to show how infinites do not follow the same rules as numbers. 
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Mr. Ireland

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Re: Is infinity a number?
« Reply #4 on: May 04, 2008, 11:39:54 AM »
I was taught it's a concept.
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Beeper

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Re: Is infinity a number?
« Reply #5 on: May 04, 2008, 02:36:49 PM »
no
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Easy as 3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679
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Username

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Re: Is infinity a number?
« Reply #6 on: May 04, 2008, 03:50:06 PM »
Well, ignoring some applied concepts like IEEE floating point which border on engineering rather than pure math (yet still fit the criteria), transfinite numbers come to mind which are often called infinite (and rightfully so.)  Also comes to mind is [overline]R[/overline].
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cbarnett97

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Re: Is infinity a number?
« Reply #7 on: May 06, 2008, 06:44:54 PM »
Quote from: Mr. Ireland on May 04, 2008, 11:39:54 AM
I was taught it's a concept.
That is what I was taught as well
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CyborgJesus

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Re: Is infinity a number?
« Reply #8 on: May 06, 2008, 06:48:04 PM »
Quote from: Mr. Ireland on May 04, 2008, 11:39:54 AM
I was taught it's a concept.
Ah, but I was also taught that the Earth is round.
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narcberry

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Re: Is infinity a number?
« Reply #9 on: May 06, 2008, 06:49:27 PM »
Quote from: cbarnett97 on May 06, 2008, 06:44:54 PM
Quote from: Mr. Ireland on May 04, 2008, 11:39:54 AM
I was taught it's a concept.
That is what I was taught as well

It's a set. "Concept" is the teachers, "I dunno, I just work here" crap. Tell your teacher to start teaching or get a real job.
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Username

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Re: Is infinity a number?
« Reply #10 on: May 07, 2008, 02:31:12 AM »
It should also be noted that numbers themselves are "concepts".
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Longy

Re: Is infinity a number?
« Reply #11 on: May 07, 2008, 06:07:18 AM »
Infinity just means something never stops or runs out. If you get an infinite ammount of apples, you will never run out of apples. Simple.
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Username

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Re: Is infinity a number?
« Reply #12 on: May 07, 2008, 09:15:02 AM »
Quote from: Longy on May 07, 2008, 06:07:18 AM
Infinity just means something never stops or runs out. If you get an infinite ammount of apples, you will never run out of apples. Simple.
Not always true.
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Tom Bishop

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Re: Is infinity a number?
« Reply #13 on: May 07, 2008, 11:10:37 AM »
Quote from: Longy on May 07, 2008, 06:07:18 AM
Infinity just means something never stops or runs out. If you get an infinite ammount of apples, you will never run out of apples. Simple.

If we have an infinite number of apple trees with each tree supporting three apples from their branches, how many apples do we have?
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Username

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Re: Is infinity a number?
« Reply #14 on: May 07, 2008, 02:29:55 PM »
Quote from: Tom Bishop on May 07, 2008, 11:10:37 AM
Quote from: Longy on May 07, 2008, 06:07:18 AM
Infinity just means something never stops or runs out. If you get an infinite ammount of apples, you will never run out of apples. Simple.

If we have an infinite number of apple trees with each tree supporting three apples from their branches, how many apples do we have?
And what if each of those trees has a troll named eric underneath it that can eat apples faster than they grow?
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Colonel Gaydafi

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Re: Is infinity a number?
« Reply #15 on: May 07, 2008, 03:31:13 PM »
Quote from: Username on May 07, 2008, 02:29:55 PM
Quote from: Tom Bishop on May 07, 2008, 11:10:37 AM
Quote from: Longy on May 07, 2008, 06:07:18 AM
Infinity just means something never stops or runs out. If you get an infinite ammount of apples, you will never run out of apples. Simple.

If we have an infinite number of apple trees with each tree supporting three apples from their branches, how many apples do we have?
And what if each of those trees has a troll named eric underneath it that can eat apples faster than they grow?

A very gassy troll
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Taters343

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Re: Is infinity a number?
« Reply #16 on: May 07, 2008, 03:52:53 PM »
Quote from: Professor Gaypenguin on May 07, 2008, 03:31:13 PM
Quote from: Username on May 07, 2008, 02:29:55 PM
Quote from: Tom Bishop on May 07, 2008, 11:10:37 AM
Quote from: Longy on May 07, 2008, 06:07:18 AM
Infinity just means something never stops or runs out. If you get an infinite ammount of apples, you will never run out of apples. Simple.

If we have an infinite number of apple trees with each tree supporting three apples from their branches, how many apples do we have?
And what if each of those trees has a troll named eric underneath it that can eat apples faster than they grow?

A very gassy troll

Ugly too.
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Bushido

Re: Is infinity a number?
« Reply #17 on: May 07, 2008, 09:46:26 PM »
Infinity is a number in some number systems, such as affinely extended real numbers and projectively extended real numbers.
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EvilToothpaste

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Re: Is infinity a number?
« Reply #18 on: May 08, 2008, 12:27:27 AM »
Quote from: Bushido on May 07, 2008, 09:46:26 PM
Infinity is a number in some number systems, such as affinely extended real numbers and projectively extended real numbers.

I've never heard of those, therefore infinity is not a number.   :P

But seriously, what are these number systems? 
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EvilToothpaste

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Re: Is infinity a number?
« Reply #19 on: May 08, 2008, 01:29:26 AM »
Quote from: Username on May 04, 2008, 03:50:06 PM
Well, ignoring some applied concepts like IEEE floating point which border on engineering rather than pure math (yet still fit the criteria), transfinite numbers come to mind which are often called infinite (and rightfully so.)  Also comes to mind is [overline]R[/overline].

Infinity in Floating Point is used as error handling, not as an actual infinite.  When overflow occurs, the IEEE convention results in a special string called INF, which is not infinite; it is just too many digits to be stored in the floating point format.  The number that causes some overflow could be infinite, but you'll have a hard time convincing an electrical engineer of that.  It is just a very large number that is indistinguishable from infinity in this specific circumstance ... which is exactly what I was saying about very large numbers in the other thread. 

Transfinite sets are infinite, even if they are not continuous.  Rightfully so indeed. 

I remain unconvinced that an infinite set is a number. 
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Username

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Re: Is infinity a number?
« Reply #20 on: May 08, 2008, 01:38:53 AM »
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Username

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Re: Is infinity a number?
« Reply #21 on: May 08, 2008, 01:41:52 AM »
Also, numbers *are* sets.  Or at least, thats been argued ;-)
« Last Edit: May 08, 2008, 01:45:01 AM by Username »
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Username

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Re: Is infinity a number?
« Reply #22 on: May 08, 2008, 01:46:33 AM »
And transfinite numbers are numbers, at least in context.
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EvilToothpaste

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Re: Is infinity a number?
« Reply #23 on: May 08, 2008, 09:52:49 AM »
Quote from: Username on May 08, 2008, 01:41:52 AM
Also, numbers *are* sets.  Or at least, thats been argued ;-)
Ah, I see, that does make sense. 

So the number 4 is the set {1,2,3,4}, the number 103 is the set {1,2,3,...,101,102,103}. 

So number are sets, I can dig.  But are all sets numbers?  One can't say the set {Bob, Mary, Jane} is the number 3.  It has three elements, sure, but said set is not something one would count with, in all cases. 

All numbers are sets, but not all sets are numbers.  But is a set full of numbers a number?  I suppose if they are random, then no.  If they are sequential I could believe it true.  But then there is the whole endless thing ... really, one cannot construct a set full of all natural numbers; that's what snags me up.  I need to build it to understand it! 
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EvilToothpaste

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Re: Is infinity a number?
« Reply #24 on: May 08, 2008, 09:57:15 AM »
Here is a slightly contradictory statement I found on a professors webpage:

Quote
In my mathematics pages, I have repeatedly warned that infinity is not a number, and that treating infinity as a number leads to trouble (mostly wrong answers). Some of my readers have probably shaken their heads at my ignorance, because they know that Georg Cantor showed that infinity is a number, and that there are several infinities of varying sizes. Let me repeat that infinity is not a number, and that treating infinity as a number leads to trouble (mostly wrong answers). But, it is extremely productive to consider the idea of infinity as a number, and see where it leads. We still need to be careful to never treat infinity as a number in most branches of mathematics. In this one branch of mathematics, the study of transfinite numbers, we will treat infinity as a number. And we will get some interesting and valid results, which we can apply to other branches of mathematics.
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Username

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Re: Is infinity a number?
« Reply #25 on: May 08, 2008, 12:20:34 PM »
That sounds familiar, I may have read that somewhere.

Anyways the point still stands, even with his admission.  In some branches of mathematics, infinity is a number.
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cmdshft

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Re: Is infinity a number?
« Reply #26 on: May 09, 2008, 08:30:04 AM »
The key word being some. The problem is that mathematics must be treated as a universal system in order to create consistent results. You can't just go tossing certain methods out the window.

Mathematics != FET
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cmdshft

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Re: Is infinity a number?
« Reply #27 on: May 09, 2008, 08:33:09 AM »
Quote from: EvilToothpaste on May 08, 2008, 09:52:49 AM
one cannot construct a set full of all natural numbers

Not true. If one had an infinite amount of time, one could construct a set full of all natural numbers.

Ah, the joys of paradox.
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EvilToothpaste

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Re: Is infinity a number?
« Reply #28 on: May 09, 2008, 08:38:07 AM »
Nonsense.  How would one ever live that long? 
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cmdshft

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Re: Is infinity a number?
« Reply #29 on: May 09, 2008, 08:52:14 AM »
The keyword was if.
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