Can numbers be infinitely large or small in mathematics?

[Pjpic]

I have two questions about infinity:

1. Can infinity increase? In other words if a number can increase, wasn't it less than infinite before the increase?


2. If a number is finite, must it have a limit?


The folks over in cosmology can't get through to me on how the universe can either be 'finite and unbounded' or "infinite and expanding'. Neither concept sounds logical and I thought there might some (accessible to a novice) literature on the subject in mathematics.

 

[CRGreathouse]

1. Can infinity increase? In other words if a number can increase, wasn't it less than infinite before the increase?

Without knowing what you mean by "infinity" (aleph-null? the greatest extended real? Cantor's Omega? the continuum?), I'm going to say no.

Can 1 increase? How about pi? No, these are just what they are.

2. If a number is finite, must it have a limit?

Every number, finite or infinite, has a limit equal to itself as any variable goes anywhere:

$$\lim_{n\to\infty}7.2=7.2$$

The folks over in cosmology can't get through to me on how the universe can either be 'finite and unbounded' or "infinite and expanding'. Neither concept sounds logical and I thought there might some (accessible to a novice) literature on the subject in mathematics.

"Finite and unbounded" (not in the mathematical sense of unbounded) might refer to a system which loops around. Think of this like modular/clock arithmetic: six hours after 7:00 is 1:00. There's no boundary, but there are only twelve (or twenty-four, or whatever) hours. In physics that might mean that if you peer far enough away you'll see the back of your head.

 

[Pjpic]

Without knowing what you mean by "infinity" (aleph-null? the greatest extended real? Cantor's Omega? the continuum?), I'm going to say no.

Can 1 increase? How about pi? No, these are just what they are.



Every number, finite or infinite, has a limit equal to itself as any variable goes anywhere:

$$\lim_{n\to\infty}7.2=7.2$$



"Finite and unbounded" (not in the mathematical sense of unbounded) might refer to a system which loops around. Think of this like modular/clock arithmetic: six hours after 7:00 is 1:00. There's no boundary, but there are only twelve (or twenty-four, or whatever) hours. In physics that might mean that if you peer far enough away you'll see the back of your head.

Thanks for your help on this. I'm afraid "types of infinity" is beyond me. But it sounds like the only quainity with that has infinity as a limit is infinity. And that a circular system's boundry would be perpendicular to the circle.

 

[CRGreathouse]

But it sounds like the only quainity with that has infinity as a limit is infinity.

Well, certainly it is true that for any constant x in $\mathbb{R},$
$$\lim_{n\to\alpha}x=x\neq\infty$$

For example,
$$\lim_{n\to\infty}7=7\neq\infty$$

But functions can still have infinite limits even if they're real for real arguments:
$$\lim_{x\to\infty}x^2=\infty$$
$$\lim_{x\to\infty}x=\infty$$
$$\lim_{x\to\infty}\log(x)=\infty$$

And that a circular system's boundry would be perpendicular to the circle.

I don't even know what that would mean. If this is in reference to modular/clock arithmetic, there is no such thing as 'perpendicular'.

 

[HallsofIvy]

I have two questions about infinity:

1. Can infinity increase? In other words if a number can increase, wasn't it less than infinite before the increase?

NO number can "increase". I assume you mean "numerical variable". Since infinity is not a number, a numerical variable must be finite and yes, "less than infinite".

2. If a number is finite, must it have a limit?

A number can't have a "limit". Either you mean "upperbound"- which case the answer is "yes" any number, x, has x+1 as an upperbound- or you mean a numerical function. Which may or may not have a limit. For example, f(x)= x is always finite but its limit as x-> infinity does not exist.

The folks over in cosmology can't get through to me on how the universe can either be 'finite and unbounded' or "infinite and expanding'. Neither concept sounds logical and I thought there might some (accessible to a novice) literature on the subject in mathematics.

Well, that's not math but space can be both "finite and unbounded" in the same way the surface of a sphere has finite area but no boundaries or that a circle can have finite circumference but no boundary. Of course that requires that space have positive curvature which is possible in General Relativity. Your questions above were about the number line, not geometric objects. Similarly, a space can be "infinite and expanding" if the distance between objects in the space is increasing.

 

[Pjpic]

NO number can "increase". I assume you mean "numerical variable". Since infinity is not a number, a numerical variable must be finite and yes, "less than infinite".


A number can't have a "limit". Either you mean "upperbound"- which case the answer is "yes" any number, x, has x+1 as an upperbound- or you mean a numerical function. Which may or may not have a limit. For example, f(x)= x is always finite but its limit as x-> infinity does not exist.



Well, that's not math but space can be both "finite and unbounded" in the same way the surface of a sphere has finite area but no boundaries or that a circle can have finite circumference but no boundary. Of course that requires that space have positive curvature which is possible in General Relativity. Your questions above were about the number line, not geometric objects. Similarly, a space can be "infinite and expanding" if the distance between objects in the space is increasing.

I believe I do mean numerial variable and upperbound. It sounds, to me, like you are saying a infintiy can not increase if it is on in a number line but in can if it is a volume?

If a line is curved wouldn't the vertex of the curve be outside the boundry of the line?

 

[HallsofIvy]

I believe I do mean numerial variable and upperbound. It sounds, to me, like you are saying a infintiy can not increase if it is on in a number line but in can if it is a volume?

If a line is curved wouldn't the vertex of the curve be outside the boundry of the line?

I have no idea what you mean by either of those statements! In what sense could "infinity" be a volume? What do you mean by the vertex of a curve? or by the "boundary" of a line?

 

[Pjpic]

I have no idea what you mean by either of those statements! In what sense could "infinity" be a volume? What do you mean by the vertex of a curve? or by the "boundary" of a line?

I'm taking beyond my knowledge here but:

The concept of a volume of infinite size seems to be comparable to that of a line of infinite length.

I thought the arc of a curve is defined by a angle formed by two lines intersecting at a vertex.

I was supposing that a point on a line is within the boundry of the line, but a point not on the line is outside the boundry of that line.

 

[HallsofIvy]

I'm taking beyond my knowledge here but:

The concept of a volume of infinite size seems to be comparable to that of a line of infinite length.

Yes, but neither of those is inherent in just "infinity".

I thought the arc of a curve is defined by a angle formed by two lines intersecting at a vertex.

No, that is not a curve at all.

I was supposing that a point on a line is within the boundry of the line, but a point not on the line is outside the boundry of that line.

A line does not HAVE a "boundary".

 

[CRGreathouse]

I believe I do mean numerial variable and upperbound. It sounds, to me, like you are saying a infintiy can not increase if it is on in a number line but in can if it is a volume?

"Infinity" cannot increase. Period. A space can grow larger, but not "infinity".

A sequence of cardinals can be both increasing and infinite at every point, though:

$$a_1=\aleph_0,a_k=2^{a_{k-1}}=\beth_k$$

 

[granpa]

I'm not a big fan of infinity. I much prefer the idea of 'arbitrarily large'.

the important thing to know about different types of infinities is that infinity isn't the largest number. there is no such thing as the largest number. infinity is probably best thought of as just 'neverending'. consider the 2 series 1 2 3 4 5... and 2 4 6 8 10... both can be neverending yet the second series is, at every point, twice as large as the first. that's basically the idea behind different infinities having different sizes. it also works for 'arbitrarily large' numbers.

 

[Hurkyl]

I'm not a big fan of infinity. I much prefer the idea of 'arbitrarily large'.

A number cannot be arbitrarily large; it is exactly as large as it is, no more, and no less. :tongue: Arbitrarily large is not a useful term for talking about numbers -- it's more appropriate for talking about things like sequences or functions or variables.

the important thing to know about different types of infinities is that infinity isn't the largest number.

Except, of course, when it is. For example, the number called $+\infty$ is the largest number in the set of extended real numbers

there is no such thing as the largest number. infinity is probably best thought of as just 'neverending'.

No it's not. In fact, one of the common problems among students who have trouble with the infinite is precisely that they think the way you advocate: they can imagine a neverending sequences of approximations to an idea (such as an infinite number, or 0.999...), but fail to distinguish between the sequence of approximations and the object being approximated.

consider the 2 series 1 2 3 4 5... and 2 4 6 8 10... both can be neverending yet the second series is, at every point, twice as large as the first. that's basically the idea behind different infinities having different sizes.

Actually, for most of the notions of 'infinite number' a student would be exposed to, both of those sequences converge to the same infinite number, rather than two infinite numbers of different sizes.

 

[mech]

Interesting.. -reads-

 

[phyti]

I have two questions about infinity:

The folks over in cosmology can't get through to me on how the universe can either be 'finite and unbounded' or "infinite and expanding'. Neither concept sounds logical and I thought there might some (accessible to a novice) literature on the subject in mathematics.

Don't objectify infinity, it has no value, think of it as a condition, such as unbounded.

 

[Hurkyl]

Don't objectify infinity, it has no value, think of it as a condition, such as unbounded.

Please don't spread misinformation like this. :grumpy: The elements $-\infty$ and $+\infty$ of the extended real numbers, the element $\infty$ of the projective real numbers, for example, have just as much claim to being an 'object' with a 'value' as other familiar things like zero and one.

The related adjective, incidentally, is 'infinite'.
Correct: "There are infinitely many natural numbers"
Correct: "The size of the set of natural numbers is an infinite cardinal number"
Incorrect: "There is an infinity of natural numbers"

 

[granpa]

A number cannot be arbitrarily large; it is exactly as large as it is, no more, and no less. :tongue: Arbitrarily large is not a useful term for talking about numbers -- it's more appropriate for talking about things like sequences or functions or variables.

an arbitrarily large number can be thought of as the inverse of an insignificant number. its a number so large that it makes any normal number insignificant (for the purpose of the equation at hand).

when one uses calculus to solve a problem like determining the electric field of a charge distribution one thinks of dq as a single electron. this is not infinitesimal but rather insignificantly small.