Thinking about infinity

[LastPoet]

One uses  irrational numbers alot. Like pi, the most famous one. To better use them, it pays to know their properties.

This has to do with countable sets vs non countable sets, if you want to know more, by the web. Lunchtime here, gotta get back to woook woooook.

[RoadRunner79]

I am confused by your statement, about the abstract of real numbers, not being infinite. Could you clarify?   I do agree about the line though....  In any case you are going to have a finite number of points. You can redefine what the point means, but then you still have a finite number.

How exactly do you think that real numbers are infinite?

If you are talking about the abstract set including all real numbers, then it has no end.  Any real number that you can think of; is in that set.  You also do not have to reach any particular number by successive addition.  All you have to do is verify that it meets the criteria of a  real number.

[robvalue]

Yeah, an infinity is countable if each element can be mapped to a unique natural number.

So obviously the natural numbers are countable. Basically, it means you can put them in a straight line and count them.

The integers are also countable. You can just alternate between positive and negative to count them all.

Rational numbers (fractions) are countable too. (A/B) <=> (2^A) * (3^A) gives unique natural numbers due to unique factorisation into primes.

But real numbers are not countable. They are a "bigger infinity". It's impossible to form such a map between these and the natural numbers. I can show you why if you like.

As for whether a finite object being represented by an infinite number of subsections is "real", I'm not really sure how to answer it. A line is a continuum anyway, and consists of an infinite number of points along the way. So you move through infinitely many iterations to move a finite distance. This is just a different way of "counting" the distance, rather than adding it continuously. So yes, I'd say it's a logically possible way of viewing an object. Obviously it doesn't become split up into all this bits just because it's possible; but it still is the sum of all those bits. If you removed any, you'd no longer have the whole length.

[RoadRunner79]

As for whether a finite object being represented by an infinite number of subsections is "real", I'm not really sure how to answer it. A line is a continuum anyway, and consists of an infinite number of points along the way. So you move through infinitely many iterations to move a finite distance. This is just a different way of "counting" the distance, rather than adding it continuously. So yes, I'd say it's a logically possible way of viewing an object. Obviously it doesn't become split up into all this bits just because it's possible; but it still is the sum of all those bits. If you removed any, you'd no longer have the whole length.

I have been thinking about this saying that "a line is made up of an infinite number of points" lately.  The point doesn't seem to be very well defined.   It seems that to me, that the only way you could get an infinite number of points, is if you have your point defined with zero size or length.  In which case, if you add them, you are not going to get anywhere.  Any number greater than zero, will get you to the overall length, but in a finite number of steps.  The same with dividing the line.    It is potentially infinite provided, that you can keep dividing into smaller segments.  But, at each step, it is still finite.  You are talking about an unending process which is potentially infinite but finite at each point. 

There is also the law of non-contradiction, which states that contradictory statements cannot be both true in the same way, at the same time.  A line cannot be both finite in length and infinite. Therefore, in order for the statement that it is made up of an infinite number of points, the definition of points cannot be related to length.   If this is the case, then wouldn't it be incorrect to make a correspondence between the points and the length afterwards?

[TheRealJoeFish]

...
I have been thinking about this saying that "a line is made up of an infinite number of points" lately.  The point doesn't seem to be very well defined.   It seems that to me, that the only way you could get an infinite number of points, is if you have your point defined with zero size or length.  In which case, if you add them, you are not going to get anywhere. 
...

Well, that's the thing. A point is defined as having zero size and length. When we talk about lines and points, we're not talking about drawing them (when you draw a line with a pencil, it's actually a really thin rectangle, and when you draw a point it's actually a really tiny circle); we're rather talking about the "ideal point", which we can approximate. This is sort of related to what I said earlier about "arbitrary largeness"; the smaller your "tiny circle" gets when you draw a pencil (that is, the closer it gets to an ideal "point"), the more of them you'll be able to find on your line-that's-actually-a-thin-rectangle.

[SteveII]

Some sets of infinities certainly seem logically possible. 

For example, the infinite set of natural real numbers seems to exist, which is to say that there is an infinity of natural real numbers existing even now, and there is none of them which do not yet exist.

There are an infinity of points on a line segment.

It is logically possible that reality has an infinite history of cause and effect (i.e. extending "backward" through the big-bang), and that it will have an infinite future.

In what sense could an infinity of simultaneously existing things exist together as a finite thing?

Regarding this portion of the OP, it is not logical that there are an infinite history of causes and effects. The future is potentially infinite but never will be.

[RoadRunner79]

...
I have been thinking about this saying that "a line is made up of an infinite number of points" lately.  The point doesn't seem to be very well defined.   It seems that to me, that the only way you could get an infinite number of points, is if you have your point defined with zero size or length.  In which case, if you add them, you are not going to get anywhere. 
...

Well, that's the thing.  A point is defined as having zero size and length.  When we talk about lines and points, we're not talking about drawing them (when you draw a line with a pencil, it's actually a really thin rectangle, and when you draw a point it's actually a really tiny circle); we're rather talking about the "ideal point", which we can approximate.  This is sort of related to what I said earlier about "arbitrary largeness"; the smaller your "tiny circle" gets when you draw a pencil (that is, the closer it gets to an ideal "point"), the more of them you'll be able to find on your line-that's-actually-a-thin-rectangle.

Ok... so we have an abstract non-real  point of zero size.  But what differentiates these points from one another?   How does that difference correlate to the length of the line?

Theoretically, you could place an infinity of these points on a single point (since they have no size).   However, if there is nothing to differentiate these points, then wouldn't it be more correct to say that there is one point rather than an infinity?

[Time Traveler]

Yes, you're correct, it can be to do with Zeno's paradox.

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It's the problem of insisting we only take finitely many steps and thus never reach our target. But (abstractly at least) we take infinitely many "steps"; each of smaller and smaller lengths.

In reality, it may or may not work like this. Is there a minimum distance something can move? If there is, then you can't take infinitely many steps. There is only a finite number you can take, no matter how small, between two points a finite distance apart. Your first step must be at least this amount (it would likely we would be dealing with multiples of this smallest amount I think). We would in effect be "jumping" from one point to another, without ever occupying the space inbetween.

But if there is no minimum amount, then you have moved through infinitely many. Disallowing this would, as you correctly observe, prevent you from even starting to move at all. But you are essentially stopping the "time taken" from ever reaching a certain point, which is not valid.

It's a weird concept to be sure. But there is no problem at all with a finite number being an infinite sum, abstractly. The only question is whether reality can be subdivided the same way. If there is no limit to how small I can cut something up, I could go on forever. Obviously, if I'm only allowed to make finitely many cuts, I can't make them all. But that's the essence of Zeno's paradox. Insisting on finitely many is the equivalent of insisting I cannot move in the first place. It's assuming the conclusion by disallowing infinities as a premise.

So, for example, let's say an object is of length 1 metre. Let's say there is no minimum length that a subdivision can be. Then, if we ask how many subdivisions of it are there if we use these lengths (I used this example before):

1/2 + 1/4 + 1/8 + ...

then the answer is infinitely many. It doesn't matter if I actually go in and mark them all, obviously that's impossible in a finite amount of time if it takes me a fixed amount of time for each cut. But if the time taken is proportional to the distance I have to move instead to make the cut, then I can theoretically make all the cuts.

It does consist of an infinite amount of them, whether or not they are actually "marked" by someone, or cut into pieces. However, if we reach a minimum distance we can cut, this no longer works. Were confined to (length/minimum distance) numbers of partitions.

Let X = 1/2 + 1/4 + 1/8 + ...

Then 2X = 1 + 1/2 + 1/4 + 1/8 + ...

2X - X = 1  [the rest of the terms all cancel out]

X = 1

This isn't a trick, it's a genuine mathematical method.

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Another way to perhaps think about the method of solving this paradox is to consider that in order to cross an infinite number of spatial segments, you need an infinite number of time segments to offset them. To cross the first distance (lets say 1 meter), it takes Bob 1 second. However, to cross the next 1/2 meter, it only takes Bob 1/2 second, and so on, so that to cross a two meter distance divided in infinite halves, it essentially takes "0 time" to cross the last segment, at which point 2 actual seconds have passed and the 2 meters have been crossed. As viewed in only one dimension (well, the three spatial dimensions, but only one for the purpose of this argument), the solution seems impossible. But by stepping beyond this limitation and factoring in time, we move quite easily through what may be an actual infinite set of spatial segments.

However, this brings up an old joke... The professor puts a math major and a physics major on one side of the room and a pretty girl on the other. The professor says that the two male students can only walk 1/2 the distance to the girl, then 1/2 the next distance in the same time as they covered the first distance, and so on. The pretty girl will reward the winner with a kiss at the end. Both students think about it, and then the physicist takes off. The mathematician just shakes his head and says, "Hey, you idiot, it's infinite in space and time... you'll never get there!" The physicist smiles back, "No... but I'll get close enough!"

[Alex K]

...  Expressed a different way - and, as always when I'm talking about physics, I ask Alex K to elaborate if what I'm saying makes sense or correct me if it doesn't - if there's a finite amount of energy in the universe, and a lower limit on the smallest "amount" of energy that can be transferred at a given time, that may necessarily constrain the number of "things" that can exist...

...  Consider the coastline problem.  ...  What this means is, you'll never measure the coastline to be "infinity", but, if you pick any length, there's a level of precision at which the measurement of the coastline will be greater than that length...

...Sure, you may never be able to count an "infinite" number of points on a line.  But, if I give you any number at all (like, 52 billion), with enough time you could count more points on the line than that number.

This is very helpful, and I do hope Alex K can help out at his convenience. I see a few different aspects in play here. Please correct any of the following:

1) The current models of the universe [may?] exclude the possibility of an actual infinity of things, but as models, these are subject to being replaced by better models and new evidence.

2) I found the coastline problem very helpful. Even though the accuracy with which you can measure the coastline's length can be improved ad infinitum, the length itself being measured is approached (and perhaps never reached) as a finite boundary. The process of increasing accuracy may be considered notionally infinite (like a geometric series?), while the coastline's length remains finite.

3) The infinite divisibility of a line is similar to the coastline problem. The "process" of identifying (counting) more and more points on that line may proceed ad infinitum. The infinity of points on a line are notionally infinite, rather than an infinity of points all simultaneously being the length of the line? <= This one is still a bit fuzzy.

Someone left the Bat signal on?

1) We have currently no evidence that the universe is spatially finite, or vice versa. I don't think it is in principle possible to scientifically rule out that it is finite.

2) The coastline problem as I know it is that it looks like a fractal at intermediate scale, and therefore its measured length depends on a certain power of the resolution of the measuring apparatus. Formally, the mathematical model for fractal coastlines yields an arbitrary large length for finer and finer resolution. In reality, the fractal is cut off when stuff is made from quantumç

3) I don't understand the question

[dyresand]

There is bigger numbers than infinity