What exactly is infinite

  • #1
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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Answers and Replies

  • #2
phinds
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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
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
pwsnafu
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Edit: Before anything else, use "infinity" for the noun. "Infinite" can be used as a noun, but not in the way you want to.

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.


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
Chronos
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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
pwsnafu
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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
lavinia
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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
lavinia
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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
lavinia
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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
lavinia
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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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  • #12
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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
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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
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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
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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
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It is never wrong to prove others wrong. Just by saying "you are wrong" is not enough in my case.
 
  • #18
CompuChip
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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
lavinia
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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
CompuChip
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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
lavinia
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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
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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
Student100
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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
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It is no problem, I very much appreciate his post.
 
  • #25
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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 ∞ = ∞.
 

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