Ideas on Infinity.

  • #26
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We can imagine infinity as something that never stops, but to me at least, this is something that is always moving out and out in all directions.
However infinity can't move, because it is infinite, if something moves it has an edge, so basically an infinite universe must be everywhere.
The word everywhere is not even the correct use of the word, there is no english word to describe the location of infinity, because it has no location.

As you can see it's impossible to describe or talk about infinity in any purposeful way.
This was an addendum to Lifter's post.

Timespace must be finite, because the mere concept of infinity is such that everything that can possibly ever happen, has already happened, and will happen infinitely into the future.
Once again, the words "future" and "everything" are meaningless when talking about infinity, but it's all I got with the english language.
 
  • #27
Hurkyl
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I would agree, is the universe we live in not generally accepted fact that it has a finite amount of atoms?
No, it is not generally accepted, which is why I objected! There are two basic possibilities for the universe:

(1) You can keep going in one direction, and eventually "wrap around" and come back where you started (more or less). Think of Pac-Man or Asteroids, if you remember way back then. This is a closed universe.

(2) You can keep going in any direction, and never return. (so it is infinite in extent) This is either flat or open.

There can be mixes of these -- open in one direction, but closed in another. I get the impression that current theories of gravity rule out these mixed possibilities.

According to the WMAP Mission page, the data seems to indicate that the universe is flat, and thus infinite in extent. (Flat means that spatially, it is very similar to Euclidean space that we learned in geometry)


But, it's still possible for the universe to be flat, but with a finite amount of matter in it -- but as I understand, the evidence seems to indicate a homogeneous universe, which means on the largest scales, it looks more or less uniform. In particular, it is not an island universe where we have a finite "island" of matter, and then nothingness.


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Okay, now I've gotten the physics out of the way, I can speak about the infinite!

There are lots of nebulous thoughts you can toss around, but "infinite" does has a simple, easy to understand definition: not finite.

So, the question is what is finite? Well, the natural numbers seem to be the prototypical finite things -- when we say something is finite, we mean that there is some sense in which we can compare that thing to a natural number, and find a natural number that is bigger.

For example, we might say the number of people on Earth is finite, because we can find a natural number bigger than the number of people. We might say that a line segment is finite, because we can find a natural number bigger than its length.

However, we would not say that a line (from good ol' Euclidean geometry) is finite. One would often say that the line is unlimited in extent, and I posit this sort of thing is from where the notion that the infinite is something "unlimited" stems.

Of course, in another sense, the line is clearly limited in its extent -- it cannot go outside of itself! Also, it can't go outside of the Euclidean plane. I assert that this sort of "shifting" of the meaning of the word "limited" is one of the major sources of confusion people have when trying to understand the term "infinite".


Now, the word infinity. It is a rather unfortunate word, because it seems to get people to identify all things infinite into a mangled mess. But, let's look at some of the phrases in which the word "infinity" is used:

A Euclidean line just keeps going in both directions off to infinity.
The size of the Euclidean plane is infinity.

In both of these cases, and many other similar cases, the word "infinity" is simply referring to some "place" further away than any finite distance, or some "value" larger than any finite value.

These notions are very easy to capture in a rigorous way. I will define something called the extended real numbers, as follows:

The extended real numbers consists of all the ordinary real numbers, and two additional things which I will write as +&infin; and -&infin;. I will will then extend the definition of "<=" as follows:

-&infin; <= a <= +&infin;

for any extended real number a. I will also define |±&infin;| = +&infin;.

For a more thorough exposition (including the arithmetic operations) see:
http://en.wikipedia.org/wiki/Extended_real_line
http://mathworld.wolfram.com/AffinelyExtendedRealNumbers.html

And *poof*, we now have a line that contains the old real number line, but has added two "endpoints", whose magnitude is greater than any natural number -- in other words, infinite.

(In some sense, this is directly analogous to starting with the interval (0, 1) and adding its two endpoints to get the interval [0, 1])

So, here we have a precisely defined notion of something that is bigger than any natural number. It is also a useful notion, as evidenced by the fact that the extended real number line is what is used in calculus.


This isn't some modern abstract nonsense either: it's been around since at least the early 1800s. One of the great successes of analytic geometry was the projective plane which adds a whole line "at infinity".



Does any of this capture the notion of "infinite"? I have to say yes:

First off, it satisfies the definition of the infinite in the fact that it involves things "beyond" the finite.

Secondly, it accurately captures the things people are trying to express when using the words "infinite" or "infinity". e.g.

If we extend the Euclidean plane to get the Projective plane, then the every Euclidean line does, in fact, contine to infinity, and contains a single point of the line at infinity.

If we use the extended real numbers for measuring things, then the area of the Euclidean plane really is +&infin;.


There are, of course, other places where "infinity" or "infinite" are used, but I think I've written enough, so I will not go into detail on things like:

Cardinals and ordinals -- mathematical things generalizing the notions of quantity and counting to infinite sets.
Hyperreals -- a number system consisting of infinite numbers and infinitessimal numbers, which is "internally" indistingushable from the ordinary real numbers.
 
  • #28
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I'm not into the whole math thing, more the philosophy, so if what I'm about to say is incorrect please correct me so I can learn something.

Regardless, if you have a universe that you can map with cartesian coordinates, each coordinate is unique.
The problem is, does the universe contain an unlimited amount of coordinates, or is there a highest number?
And if so how can you prove that it does from earth?
How can we trust that the mathematical model is portraying 100% of the universe?
Is that even possible?

I dunno, enlighten me.
 
  • #29
Hurkyl
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Well, the modern philosophical viewpoint on infinity (at least according to Wikipedia) is that it's a topic to be addressed mathematically, so :tongue:.

if you have a universe that you can map with cartesian coordinates, each coordinate is unique.
The problem is, does the universe contain an unlimited amount of coordinates, or is there a highest number?
Based on the hypothesis, the answer is that there are infinitely many coordinate values, because there are infinitely many cartesian coordinates.

(Though, I suspect that you meant to say that you're mapping the universe to some subset of cartesian coordinates)

In the mathematics currently used to model the universe, a coordinate chart consists of all possible combinations real number values that can be assigned to the components -- this is true whether you have a coordinate chart that covers the entire universe (if such a thing is possible), or a little tiny portion of space-time.

How can we trust that the mathematical model is portraying 100% of the universe? Is that even possible?
I'll have to leave that question to the scientists, but I would presume the answer is that it's not pragmatic to assume that the universe hoodwinking us into thinking its infinite and more or less homogeneous and isotropic.
 

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