What exactly is infinite

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In summary: Infinity is a mathematical concept that extends the idea of finite numbers. It has different definitions and properties depending on the mathematical space it is being used in. While it may appear in real life scenarios, it does not exist as a real number. The concept of infinity has been used in various scientific models, but it is not always relevant or necessary for understanding the real world. Ultimately, the use of infinity in mathematics and science depends on the context and the specific problem being studied.
  • #1
MightyKaykoher
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Infinite is more of a concept than a number, but it sometimes apears in mathematical equations and also real life. It almost always destroys anything its in.

First, I would like to ask some questions about infinite...
1.) is inf - inf equal to 0?
2.) is inf times inf = inf squared or just inf?
3.) is 1/infinite 0?
4.) is inf. - any real number still inf?
5,) is inf times any real number still inf?

Also I have noticed whenever infinite comes into advanced equations or problems that the answer always leads to a paradox or null.

Real life example 1
Is their an infinite amount of time?
If their is, does this mean theirs an infinite amount of space?
If their is an infinite amount of time, every weird thing you can imagine will happen since infinite times 1/infinite times infinite = 1

Then again this depends on the six equations I listed above. Please help. Thanks all
 
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  • #2
MightyKaykoher said:
Infinite is more of a concept than a number, but it sometimes apears in mathematical equations and also real life.

In real life? Really? Where? Can you think of even one example?
 
  • #3
phinds said:
In real life? Really? Where? Can you think of even one example?

I did...
There's the zenos paradox
There's an infinite amount of time (possibly$
There could be an infinite amount of space

There are more, but I was asking the answer to the six equations not the answer to the examples
 
  • #4
Edit: Before anything else, use "infinity" for the noun. "Infinite" can be used as a noun, but not in the way you want to.

MightyKaykoher said:
I did...
There's the zenos paradox

That is a thought experiment and irrelevant to real life: you can disprove it by walking and touching a wall. It's also irrelevant mathematically: i.e. infinite series completely resolves it.

There's an infinite amount of time (possibly$

Unlikely due to the big bang.

There could be an infinite amount of space

"Could be" is not good enough, you need observations.


MightyKaykoher said:
Infinite is more of a concept than a number, but it sometimes apears in mathematical equations and also real life.

Infinity in mathematics has multiple definitions, each with different properties and are non-equivalent. For example in the extended real numbers infinity is a number.

It almost always destroys anything its in.

Infinity does not exist as a real number. It does exist in extended reals.

This is like saying 1/2 does not exist in the integers, hence it destroys the integer mathematics. It does exist in the rationals, hence if you want to work with 1/2 you work in the rationals.

1.) is inf - inf equal to 0?
In the extended reals it is undefined.

2.) is inf times inf = inf squared or just inf?
In the extended reals it is equal to infinity.
In the ordinal numbers it is square of the infinity (note that in the ordinals there are multiple numbers which correspond to infinity).

3.) is 1/infinite 0?
On the extended reals, yes.

4.) is inf. - any real number still inf?
On the extended reals, yes.

5,) is inf times any real number still inf?
In the extended reals, yes.

Real life example 1
Is their an infinite amount of time?
If their is, does this mean theirs an infinite amount of space?
If their is an infinite amount of time, every weird thing you can imagine will happen since infinite times 1/infinite times infinite = 1

Then again this depends on the six equations I listed above. Please help. Thanks all��

None of this is relevant in a mathematics thread because mathematics doesn't deal with the real world.

Edit: Also there not their. Please learn the difference between there, their and they're.
 
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  • #5
Infinity is a mathematical artifact. It has always proven to be an illusion under careful scientific scrutiny [e.g., quantum physics]. Even the universe, while really big, is observationally finite. Mathematics is a logical extension of reality. Division by zero, the classic definition of infinity, is, however, proven to yield illogical results. When you run into infinities in science, you should question the model.
 
  • #6
Chronos said:
Division by zero, the classic definition of infinity, is, however, proven to yield illogical results.

If you are saying division by zero leads to contradictions on the real numbers, this is correct.
If you are saying division by zero leads to contradiction in any mathematical space, that is wrong. See wheel theory.
 
  • #7
pwsnafu said:
That is a thought experiment and irrelevant to real life: you can disprove it by walking and touching a wall. It's also irrelevant mathematically: i.e. infinite series completely resolves it.

So are you saying that Zeno had no idea that you can walk up to a wall and touch it?
 
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  • #8
Chronos said:
Infinity is a mathematical artifact. It has always proven to be an illusion under careful scientific scrutiny [e.g., quantum physics].

The Shroedinger equation uses continuous time and therefore the concept of infinity. It also uses continuous space.

Mathematics is a logical extension of reality.

This is a philosophical statement and does not belong in the Mathematics Forums - in fact not in any Physics forum.
 
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  • #9
pwsnafu said:
None of this is relevant in a mathematics thread because mathematics doesn't deal with the real world.

This is a philosophical statement and does not belong in the Mathematics forum. I'm glad that you know what the real world is though.
 
  • #10
In Mathematics infinite magnitudes extend the idea of the finite. There are different infinite magnitudes and just like counting numbers, they are ordered by size. The smallest infinite magnitude is the size of the integers. The size of the real numbers is a larger magnitude.

The idea of infinity is not an "artifact" whatever that means. It has the same rigor as the idea of one or zero or five thousand. To say that the Universe is finite is no more mathematically rigorous than to say that is has the cardinality of the Continuum and in fact that is what is said in Classical Physics including the General and Special Theories of Relativity and in Quantum mechanics and in Quantum Field Theory and in String Theory ...
 
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  • #11
MightyKaykoher said:
I did...

No, as others have already pointed out, you did not.
 
  • #12
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.
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.
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.
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.
5,) is inf times any real number still inf? again, undefined.
Don't think of infinity as some sort of number. It is just a concept - it doesn't obey any rules in maths.

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.
 
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  • #13
If you say infinite doesn't appear in mathematics and real life we have to agree to disagree. It seems "there" (not their, turning off autocorrect helps my grammar a lot) is no straightforward answer to the equations I have asked. I didn't quite feel anyone have a direct explanation.

Whats the definition of infinite? Endless number, limitless(not a mathmatical limit), or is infinite a number eg 10^10^10^10^10?

One reply said their is not an infinite amount of time because of the Big Bang. Well this depends on what time is and your personal opinion is. Assuming the Big Bang is real; If you can say "before the Big Bang" you can say time has no beggining or end.

The same reply I believe said/suggested their is not an infinite amount of space. This is beyond me. Thats like saying our Earth is flat and you can not go beyond the edge.
 
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  • #14
MightyKaykoher said:
The same reply I believe said/suggested their is not an infinite amount of space. This is beyond me. Thats like saying our Earth is flat and you can not go beyond the edge.
The surface of Earth is a good example of a finite area without a boundary. The universe could have the same shape (just with one more dimension of course), we don't know.

Whats the definition of infinite? Endless number, limitless(not a mathmatical limit), or is infinite a number eg 10^10^10^10^10?
Nothing of that.
Infinity (note the spelling).
 
  • #15
10n is perfectly finite as long as n is finite add however many powers of finite figures as you want. You can't make an infinite figure out of finite material.
 
  • #16
lendav_rott said:
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.
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.
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.
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.
5,) is inf times any real number still inf? again, undefined.
Don't think of infinity as some sort of number. It is just a concept - it doesn't obey any rules in maths.

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.

I suggest the OP disregard this post, I struggle to find one correct statement.
 
  • #17
It is never wrong to prove others wrong. Just by saying "you are wrong" is not enough in my case.
 
  • #18
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.
 
  • #19
CompuChip said:
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.
 
  • #20
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.
 
  • #21
CompuChip said:
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.
 
  • #22
lendav_rott said:
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..
 
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  • #23
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.
 
  • #24
It is no problem, I very much appreciate his post.
 
  • #25
Student100 said:
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 ∞ = ∞.
 
  • #26
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
 
  • #27
MightyKaykoher said:
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.
 
  • #28
MightyKaykoher said:
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.
 
  • #29
MightyKaykoher said:
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.
 
  • #30
Infinite occurs in real life. Look up the definition of infinity on Wikipedia. ;
 
  • #31
MightyKaykoher said:
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.
 
  • #32
MightyKaykoher said:
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 anybody 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]
Emphasis added.
 
  • #33
MightyKaykoher said:
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.

As the original question has been asked and answered I am closing this thread.
 

1. What is the concept of infinity?

The concept of infinity refers to something that has no limit or end. It is a theoretical concept that is used in mathematics, physics, and philosophy to describe something that is boundless and never-ending.

2. Is infinity a number?

No, infinity is not a number. It is a concept that represents something without a limit or end. In mathematics, infinity is often used as a symbol to represent a quantity that is larger than any real number.

3. Can infinity be measured or calculated?

No, infinity cannot be measured or calculated. It is a theoretical concept that cannot be quantified in the same way that numbers can. However, mathematicians and scientists use the concept of infinity in equations and theories to solve problems and understand the universe.

4. Does infinity exist in the physical world?

It is debatable whether infinity exists in the physical world. Some argue that it is a purely theoretical concept, while others believe that it can be found in certain aspects of the universe, such as the infinite nature of space and time.

5. What is the difference between potential and actual infinity?

Potential infinity refers to a process or sequence that can continue indefinitely, but it is not actually infinite. For example, counting numbers can go on forever, but there is no actual infinite number. Actual infinity, on the other hand, refers to something that is truly infinite and unbounded, such as the concept of the infinite universe.

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