Math Is Fun Forum

Laterally Speaking

Real Member

Registered: 2007-05-21

Posts: 356

Infinite lengths, or sizes

I figured I'd bring this up, considering the number of threads that stray towards the question of infinity.

Infinite comes from the Latin finus, which means end. The prefix, in, reverses the meaning to be without end. Thus, something infinite is something that has no end, or boundaries.

How exactly can you prove that a line has no end? You cannot see the end if there is none. You can only practically prove that it isn't infinitely long, but you can't prove that it is. This would seem to suggest that nothing physical is infinite, and that makes sense to people who cannot grasp the concept of infinity. However, there are some things that are infinite yet still physical. If you know any, tell; if not, I will.

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"Knowledge is directly proportional to the amount of equipment ruined."
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Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Infinite lengths, or sizes

How exactly can you prove that a line has no end?

Define what a line is.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Laterally Speaking

Real Member

Registered: 2007-05-21

Posts: 356

Re: Infinite lengths, or sizes

Any direction in space. A physical line.
For example, to "prove" that two lines are parallel, you supposedly extend them to infinite lengths, and if they never cross, they are parallel. If they're infinitely long, isn't there an infinitely large probability of them crossing? also, if they're infinitely long, there would be no end, but it could still be called a VERY long line, because you would not be able to find the end either way, until one day somebody finds the end of the VERY long line, thus proving that it is just that. If the end is not found, both possibilities remain.

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"Knowledge is directly proportional to the amount of equipment ruined."
"This woman painted a picture of me; she was clearly a psychopath"

MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,713

Re: Infinite lengths, or sizes

I like to think of infinity as the "simplest" case: no end defined.

A Line is so simple it has no end. A Ray has one end. A Line Segment has two ends.

If I say "the line y=2x" I have not defined an end, so it has none, and is hence infinite.

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"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Infinite lengths, or sizes

You are confusing “infinity” with “boundedness”. The line, ray and line segment are all infinite in the sense that they contain infinitely many points. If what you mean is that the line can be extended indefinitely in either direction, then what you mean is that the line is unbounded. The ray is bounded at one end, while the line segment is bounded at both ends.

You can see this by considering the real line (i.e. the x-axis). Then interval [0,∞) corresponds to a ray and the interval [0,1] corresponds to a line segment. Clearly it’s not a question which of them is infinite (they all are!) but which of them is bounded – and there already exist unambiguous definitions of what bounded sets are. This applies to all straight lines since all straight lines are congruent to the real line. (In fact, it applies to all continuous unbounded lines that don’t cross themselves in the Cartesian plane since all such lines are topologically homeomorphic to the real line.)

It’s usually the case that what is called “infinite” can be mathematically defined in non-infinity terms. Take the definition of an infinite set. A set X is finite iff it’s empty or there is a bijection between X and the set {1,…,n} for some natural number n; a set is then defined to be infinite iff it is not finite. (Equivalently, a set S is infinite iff there is a bijection between S and a proper subset of S.) The idea of infinite lengths is nothing more or less than the concept of unbounded sets.

Last edited by JaneFairfax (2007-06-14 21:43:46)

Laterally Speaking

Real Member

Registered: 2007-05-21

Posts: 356

Re: Infinite lengths, or sizes

Here's sort of what I was talking about: a single dimension in space. In the case above, the dimension is, well, I guess that, like you said, unbounded. A dimension that has no boundaries or ends.

Of course, in mathematics, there are plenty of ways of proving that a number is infinite, but a spacial dimension is not like a number.

Like I said in my example, a dimension can be considered one of two things until it is found to be the latter one: unbounded, with no ends; or bounded, with the end just out of sight at the moment. You don't have any way of proving that the latter is not true, but if do find the end, then you know that it does have one, thus confirming one theory and disproving the other.

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"Knowledge is directly proportional to the amount of equipment ruined."
"This woman painted a picture of me; she was clearly a psychopath"

MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,713

Re: Infinite lengths, or sizes

The line, ray and line segment are all infinite in the sense that they contain infinitely many points.

True. I should have said Infinitely Long

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"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

Laterally Speaking

Real Member

Registered: 2007-05-21

Posts: 356

Re: Infinite lengths, or sizes

Oops! I said I would mention some things that are physical yet infinite, but I forgot what I was going to say (yes, I did have some in mind when I started the topic).

Here's one that isn't really what I had in mind, but I thought I should say something: the time it takes a non-moving object to move one attometer (if you don't know what that is, see Euler Avenue>Number Giants).

Last edited by Laterally Speaking (2007-06-16 07:52:33)

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"Knowledge is directly proportional to the amount of equipment ruined."
"This woman painted a picture of me; she was clearly a psychopath"