Infinity

[psiji]

This is a concept that has bothered the hell out of me for years. How can anything be infinite? It seems that everything should logically have an end, yet so many of the concepts used in math and science involve thinking that there are certain things which will never end. Zeno's paradox is a prime example.

 

I'm sure everyone is familiar with Zeno's paradox and the many forms it has taken, but I'll give a brief summary of it if anyone is unfamiliar with it. You start walking towards a wall, and stop at the halfway point. From that point you walk halfway to the wall again, and stop. From that point you walk halfway to the wall and stop again...I think you can see where I'm going with this. You end up with a division of space that goes on ad infinitum.

 

You can of course reverse this to extend in the opposite direction, since it's generally assumed by the lay-person (I don't currently know physicists point of view on this) that space goes on ad infinitum as well. This creates a new set of logical paradoxes, such as; if space goes on for infinity, will there be an infinite number of earths existing simultaneously? This has always troubled me, because it it was true, it seems every event that can happen, will happen, and will happen an infinite number of times...I'm sure you've all spent time banging your head on a philosophical wall with these kinds of problems as well.

 

So is the concept of infinity actually a real concept? Or do we use infinity to take the place of hidden variables we can't yet calculate?

[Sisyphus]

How can anything be infinite? It seems that everything should logically have an end,

 

"Not having an end" and "being infinite" are not quite the same thing. If I start counting integers, there will never be a point at which I reach the end. However, at any point, the number that I've counted is still finite.

 

Infinites are things which can only be taken "all at once," not "over time." There are an infinite number of integers. However, counting them, it is impossible to reach an infinite number.

 

That paradox is a good example. Space is not infinitely divided there, it is just indefinitely divisible. The same for the time it takes you to travel. You can't ever reach a non-finite distance or time, but you won't ever reach a point where it can't be divided smaller, either. It's not a paradox because the divisions are only potential.

 

Most "infinites" in mathematics are actually "indefinites." Calculus deals in indefinites. "Smaller than any division you arbitrarily could make." "The value towards which this series will get closer and closer forever but never actually reach."

 

Actual "infinites" are much rarer, and are "all at once" kind of things. Take 0.9 repeating. It is not "approaching" one, because the nines are already there. It is EQUAL to one.

 

Similarly, the universe, if infinite, is simply THERE, not something which "goes on," as if you were travelling through it. Oftentimes seemingly unimportant semantic differences like that make the difference between absurd and perfectly natural.

 

You can of course reverse this to extend in the opposite direction, since it's generally assumed by the lay-person (I don't currently know physicists point of view on this) that space goes on ad infinitum as well. This creates a new set of logical paradoxes, such as; if space goes on for infinity, will there be an infinite number of earths existing simultaneously? This has always troubled me, because it it was true, it seems every event that can happen, will happen, and will happen an infinite number of times...I'm sure you've all spent time banging your head on a philosophical wall with these kinds of problems as well.

 

There are several assumptions behind those supposed paradoxes. You're assuming that space has to repeat itself in order to be infinite. You're also assuming that it has to exhaust all possibilities in order to have an infinite number of variations. Both of these are false assumptions.

 

Look at a numerical analogy. How many even numbers are there? An infinite amount. So look at "the set of even numbers" as an analogy for "the infinite universe." Now, no two even numbers are alike. There aren't "an infinite number of fours," analogous to "an infinite number of Earths." Also, not "everything which could happen does happen." Three could happen. But it doesn't.

[Royston]

Psiji, you may be interested in the Hilbert's Hotel metaphor...

 

http://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel

[psiji]

"Not having an end" and "being infinite" are not quite the same thing. If I start counting integers, there will never be a point at which I reach the end. However, at any point, the number that I've counted is still finite.

 

Infinites are things which can only be taken "all at once," not "over time." There are an infinite number of integers. However, counting them, it is impossible to reach an infinite number.

 

That paradox is a good example. Space is not infinitely divided there, it is just indefinitely divisible. The same for the time it takes you to travel. You can't ever reach a non-finite distance or time, but you won't ever reach a point where it can't be divided smaller, either. It's not a paradox because the divisions are only potential.

 

Most "infinites" in mathematics are actually "indefinites." Calculus deals in indefinites. "Smaller than any division you arbitrarily could make." "The value towards which this series will get closer and closer forever but never actually reach."

 

Actual "infinites" are much rarer, and are "all at once" kind of things. Take 0.9 repeating. It is not "approaching" one, because the nines are already there. It is EQUAL to one.

 

Similarly, the universe, if infinite, is simply THERE, not something which "goes on," as if you were travelling through it. Oftentimes seemingly unimportant semantic differences like that make the difference between absurd and perfectly natural.

 

 

 

There are several assumptions behind those supposed paradoxes. You're assuming that space has to repeat itself in order to be infinite. You're also assuming that it has to exhaust all possibilities in order to have an infinite number of variations. Both of these are false assumptions.

 

Look at a numerical analogy. How many even numbers are there? An infinite amount. So look at "the set of even numbers" as an analogy for "the infinite universe." Now, no two even numbers are alike. There aren't "an infinite number of fours," analogous to "an infinite number of Earths." Also, not "everything which could happen does happen." Three could happen. But it doesn't.

 

Thanks for the clarification on this. This definetely straightens out alot of questions I've had about the concept of infinity. So apparently I'm just confusing infinity with other similar concepts.

[psiji]

Psiji, you may be interested in the Hilbert's Hotel metaphor...

 

http://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel

 

Good stuff - thank you.

[Genecks]

So

 

infinity = constant

to count to infinity, you must = constant of infinity

 

What's to say we aren't already constant, though. I mean, matter doesn't get created nor destroyed...

 

'Twould be odd to say you've already counted to infinity, so you don't have to now. Hmm.

 

I wonder if a number system, however, is a good way of saying you can't count to infinity. I mean, counting up is not a constant. But staying at one is a constant.

[Gypsy Cake]

∞ is a stubborn constant. It's impossible to change it or get rid of; except by multiplying by 1/∞.

 

∞-1=∞

ANS+5=∞

ANS^2=(∞+5)^2=∞^2+10∞+25=∞

 

grrrrrrrrr!!!!!!!!!!!!!

 

∞ x 1/∞=1

 

ahhhh. relief from the incesent infinties.

 

But of course ∞ isn't the only troublesome constant. I put forward the obstinate zero. Very similar, though not quite so uncooperative. It's interesting how the two extremes cause such problems amoungst scientists/philosophers/mathematicians...

 

∞/0=∞

0/∞=0

[VikingF]

∞ is a stubborn constant.

 

∞ is not a constant at all, it's more like a direction. A constant, let's say n, can move towards ∞ (n->∞), or towards -∞ (n->-∞), but it can never BE ∞.

[Sisyphus]

 

∞ x 1/∞=1

 

 

That is not true. The answer would be undefined.

 

EDIT: I think it would be most accurate to say that "infinity" is not a number, or a constant, or even a direction. It is a property of quantities.

[the tree]

I like the idea of it being a direction, it seems quite a simple explanation and it kind of covers it.

[Gypsy Cake]

I accept all the corrections made to my comment except that I thought infinity divided by infinity would be 1. No?.

 

I'm with the tree, in that infinity being a direction feels quite nice. But is it actually incorrect to call it a constant? on the same point, is it incorrect to say a number is a constant? (could someone clear that up for me?cheers)

[Sisyphus]

No. Arithmetic operations can never be performed on infinity with meaningful results, because it isn't a number. Infinity /= infinity.

 

To see what is meant by this, consider:

 

Divide the number of integers by the number of even integers.

Divide the number of irrational numbers by the number of rational numbers.

 

You see where I'm going with this?