# What exactly is infinite

by MightyKaykoher
Tags: infinite
P: 1,716
 Quote by CompuChip I suggest the OP stop treating infinity as a number, like 2, that you can multiply by another number, like 7, and get a sensible answer. In "standard" mathematics (i.e. up to somewhere in an Analysis 2 course at University level) infinity is not a number, but a concept. It is the result of a limit process that can be well-defined, and that we can give a name, but we shouldn't treat the limit as an element of the sequence that produces that limit. Infinities can be really slippery to work with, and in mathematics you need to resort to e.g. extended reals, as mentioned earlier, to do it properly. In physics they're usually just a big head ache as they don't represent anything physical as far as we are aware, and any theory that yields unbounded observables is in trouble.
Infinity is not a Real number. But neither is i or the unit quaternions. Nevertheess infinite ordinal numbers are well defined numbers.
 Sci Advisor HW Helper P: 4,300 I know lavinia, but they're not part of the real numbers, and the usage of statements like "inf - inf = 0" strongly suggests to me that the OP is not ready to be introduced to the mathematical details without a conceptual understanding of why infinity can not be treated that way without additional tools.
P: 1,716
 Quote by CompuChip I know lavinia, but they're not part of the real numbers, and the usage of statements like "inf - inf = 0" strongly suggests to me that the OP is not ready to be introduced to the mathematical details without a conceptual understanding of why infinity can not be treated that way without additional tools.
Fair enough.
P: 1,290
 Quote by lendav_rott It is never wrong to prove others wrong. Just by saying "you are wrong" is not enough in my case.
OK, my apologies.

 1.) is inf - inf equal to 0? No one could tell. If you assume infinities to be equal then they would have to be finite, but they're not so tough luck. Who is to say that one infinity is equal to/lesser than/bigger than the other? There is just no way you can tell.
inf - inf is undefined but not for the reasons you list.

Assuming infinities to be equal does not mean they have to be finite. I would like to know how you came to that conclusion. For example, we know that there are an equal number of real numbers and irrational numbers, and both of these sets are infinite in cardinality.

inf - inf is undefined because you run into problems defining inf + inf (or subtraction) and retaining distributive/associative laws, and other field axioms (even though an extension to infinity technically isn't a field (afaik) we would still want to keep it as "field-like" as possible and retain these simple properties.)

No operation on infinity would be left undefined just because of some "well infinity is super weird," cop-out explanation, it would be left undefined because we can't find a way to define it without coming up with undesirable consequences, and that is all.

 2.) is inf times inf = inf squared or just inf? How could you add to something or multiply infinity by something if you don't know how much that "infinity" contains. Operations work with finite figures.
Infinity is not some really large, unspecified, growing value. Question: do we know how many "3s" are in the repeating decimal ##0.\overline{3}##? The answer is yes.

Operations work with whatever the hell we want them to as long as our definition of those operations are consistent.

Infinity * Infinity is not like Infinity - Infinity. We can define it just fine and run into absolutely no problems whatsoever.

Real Projective Line

 3.) is 1/infinite 0? Yes and no. In limit calculations you can say it is 0 because limits estimate the boundary, but just having 1/infinity is undefined.
1/infinity can be defined to be 0 in any extension without issue.

The only reason we say "limit as n approached infinty 1/n" in the real numbers is because infinity is not in the set of real numbers. Once we add points for infinity in the extended real line, 1/infinity means exactly the same thing as the limit. We don't even have to do anything to say that 1/infinity = 0 after extending the real line.

Furthermore, there is no reason to confine the discussion to the context of the real number line. The real number line is not the "one, true, number system." Your opinion of the real number line seems almost similar to the opinion of some ancient mathematicians' opinion of the natural numbers. I see a lot of people say that "##1/0## is meaningless because it is not defined in the real number line." No, it's not meaningless, it is meaningless in the context of the real number line because it is not defined in the real number line, in the same way that ##3-27## is meaningless and undefined in the natural numbers.

The moment we start talking about operations on infinity, we are extending the real number line. It is not useful to stay in the real numbers. Your post saying that everything involving infinity is undefined because it is not a real number is no different than me going into the Math Homework forum and telling everyone that involves complex numbers in their work that their entire assignment is undefined and makes no sense because complex numbers are not real numbers.

We can define complex numbers consistently, and so we can work with them. We can define operations on infinity consistently, so we can work with them.

 4.) is inf. - any real number still inf? Undefined. Imagine if you have an expression of 3y - 2z. Try subtracting 10 elephants from 52 motorcycles - you do have the expression, but it means nothing.
Nonsense, again, you insist that the real number line is sacred. I could again apply your same logic to the complex numbers, where elephants are real and motorcycles are imaginary.

Infinity - R = Infinity

In any extension.

 5,) is inf times any real number still inf? again, undefined.
Again, the real number line is somehow sacred.

Infinity * R = Infinity

Where R is not zero in the projective real line. In the extended real line, infinity is signed, and so this operation changes in the expected and natural way in the extended real line.

 Don't think of infinity as some sort of number. It is just a concept - it doesn't obey any rules in maths.
Being closed minded is not how we develop new math.

 Another example of the concept of infinity - how much energy do you need to reach the speed of light? An infinite amount. Wait what? How much is infinite? Exactly.
This is lacking any point.

Yes, an infinite amount of energy is required for a massive particle to reach the speed of light. What do you mean, "wait what? how much is infinite?" You seem to be expressing some sort of problem with the idea that an infinite amount of energy is required for a massive particle to reach the speed of light. That is true (afaik) according to very sound modern physics..
 P: 550 Hey 1mile, I believe what the poster you're replying to was trying to convey the notation of infinite cardinalities coming in different magnitudes and conveying different information, but I could have read too much into it.
 P: 223 It is no problem, I very much appreciate his post.
P: 1,290
 Quote by Student100 Hey 1mile, I believe what the poster you're replying to was trying to convey the notation of infinite cardinalities coming in different magnitudes and conveying different information, but I could have read too much into it.
I'm not sure what you mean, but if I am catching your meaning correctly, the aleph numbers and the ∞ symbol are different things and ∞ does not refer to any aleph number in particular. Aleph numbers are strictly cardinal and "∞" is a point on the extended real number line.

I don't see anything in that user's post reflecting that idea, I took the relevant part his post to mean that I don't know that ∞ = ∞.
 P: 31 I'm still missing something. Infinity can be put into a variable. Let's call it a. a - 4..... It's simplified. If infinity is a concept then why does it appear in real life situations. Inf^2- okay, my understanding of this statement is there is this limitless number ( not a mathmatical limit ) and this strange irrational endless number is multiplied by itself. I believe in no real life situation does infinity^2 EVER happen so no use going any farther. Inf - inf = 0. Okay so, we don't know what infinite is, but infinite = infinite in all cases. Here's an inequality that IS true infinite - (positive C) < infinite. If this paragraph is completely wrong then infinite does not equal infinite. If inf - inf = 0 then inf - C must be less than infinity. Why? Who knows!! Infinite * 2 is 2infinity. If not then infinite^n * Real number (plus or minus c) is infinity. This makes no sense however this isn't how infinity shows up in real life. Infinite/infinite this is in fact 1 1/0 = infinity but, 0 times infinite equals 0. This means you can divide one into as many groups of zero as you want. Until you reach infinity groups.( this will never happen ). Why is zero times infinite 0? Well 0 + 0 + 0... Will never reach the sum of a number > 0. Infinite + C. Just the inverse of inf - C. How about the square root of -infinity. That's a tough answer
Mentor
P: 11,617
 Quote by MightyKaykoher I'm still missing something. Infinity can be put into a variable. Let's call it a. a - 4..... It's simplified.
To do this, you first have to define a set of "objects" where you want such operations take place, and what subtraction and addition means in this set of objects. This is easy to do with natural or real numbers without infinity, but it gets more problematic if you want to include something that looks like infinity.

 If infinity is a concept then why does it appear in real life situations.
3 is another mathematical concept. Does 3 appear in real-life situations?

 Inf^2- okay, my understanding of this statement is there is this limitless number ( not a mathmatical limit ) and this strange irrational endless number is multiplied by itself. I believe in no real life situation does infinity^2 EVER happen so no use going any farther.
If you consider the infinite number of real numbers in an interval (like "between 0 and 1"), an area has something like "infinity^2" points.

 Inf - inf = 0.
Or undefined. Depends on the way you define your inf and subtraction. This is your choice!

 Okay so, we don't know what infinite is, but infinite = infinite in all cases. Here's an inequality that IS true infinite - (positive C) < infinite. If this paragraph is completely wrong then infinite does not equal infinite. If inf - inf = 0 then inf - C must be less than infinity. Why? Who knows!!
As you can see: if you do those definitions wrong, you get contradictions.
PF Gold
P: 6,120
 Quote by MightyKaykoher If infinity is a concept then why does it appear in real life situations.
It doesn't. AGAIN, I ask you to state an example of what you think is infinity in the real world. Stick with demonstrable/observable things. You tried that before and all you did was talk about math.
P: 1,290
 Quote by MightyKaykoher I'm still missing something. Infinity can be put into a variable. Let's call it a. a - 4..... It's simplified. If infinity is a concept then why does it appear in real life situations. Inf^2- okay, my understanding of this statement is there is this limitless number ( not a mathmatical limit ) and this strange irrational endless number is multiplied by itself. I believe in no real life situation does infinity^2 EVER happen so no use going any farther. Inf - inf = 0. Okay so, we don't know what infinite is, but infinite = infinite in all cases. Here's an inequality that IS true infinite - (positive C) < infinite. If this paragraph is completely wrong then infinite does not equal infinite. If inf - inf = 0 then inf - C must be less than infinity. Why? Who knows!! Infinite * 2 is 2infinity. If not then infinite^n * Real number (plus or minus c) is infinity. This makes no sense however this isn't how infinity shows up in real life. Infinite/infinite this is in fact 1 1/0 = infinity but, 0 times infinite equals 0. This means you can divide one into as many groups of zero as you want. Until you reach infinity groups.( this will never happen ). Why is zero times infinite 0? Well 0 + 0 + 0... Will never reach the sum of a number > 0. Infinite + C. Just the inverse of inf - C. How about the square root of -infinity. That's a tough answer
The operations have nothing to do with "real life" they have to do with a consistent definition that preserves field properties. Math can be used to solve real world problems, but math is not about or limited to the real world. As mfb said, you could define these things however you want, as long as your definitions do not produce contradictions.
 P: 31 Infinite occurs in real life. Look up the definition of infinity on Wikipedia. ;
PF Gold
P: 6,120
 Quote by MightyKaykoher Infinite occurs in real life. Look up the definition of infinity on Wikipedia. ;
You must be reading a different Wikipedia definition that the one I saw, since the one I saw does NOT say that.

You keep refusing to actually give any meaningful example of what you think is infinite in the real world. This is getting old.
P: 820
 Quote by MightyKaykoher Infinite occurs in real life. Look up the definition of infinity on Wikipedia. ;
You mean this?

 In physics, approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements (i.e. counting). It is therefore assumed by physicists that no measurable quantity could have an infinite value,[citation needed] for instance by taking an infinite value in an extended real number system, or by requiring the counting of an infinite number of events. It is for example presumed impossible for any body to have infinite mass or infinite energy. Concepts of infinite things such as an infinite plane wave exist, but there are no experimental means to generate them.[citation needed]
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P: 21,215
 Quote by MightyKaykoher Infinite occurs in real life. Look up the definition of infinity on Wikipedia. ;
The noun is infinity. Infinite is an adjective. Please get it straight.

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