Is infinity a long way?

[uncut1234]

id say its closer than you think, just keep turning right, you will get there eventually

 

[TwasBrillig]

Perhaps we should consider if Zeno ever got over that bridge he was trying to cross.

Nah. We'll burn that bridge when we get to it.

 

[Northland]

Or is it just around the corner

I was going to ask a genius but he doesn't seem to be answering my questions. Maybe one of the other clever people on here will be able to answer this question?

It's where you want it to be-next door or in the next county or even on another planet or it could be as close as your crotch.

 

[Jovial]

Infinity is a concept, not a place or number. It can only be understood in terms of a limit process. To say something is infinite means no matter how big of a finite number we chose, we can always choose something bigger.

For example, take the number of real numbers between zero and one. Give me any amount, say one million, and I can come up with more than one million numbers between zero and one. If you want a billion or a trillion, no problem, I can always show that there are at least that many real numbers between zero and one.

 

[JustAsking]

Infinity is a concept, not a place or number. It can only be understood in terms of a limit process. To say something is infinite means no matter how big of a finite number we chose, we can always choose something bigger.

For example, take the number of real numbers between zero and one. Give me any amount, say one million, and I can come up with more than one million numbers between zero and one. If you want a billion or a trillion, no problem, I can always show that there are at least that many real numbers between zero and one.

Yes, Jovial, you have it right. Infinity is as real and as useful as any other mathematical construct. There are an infinite number of real numbers between 0 and 1, and there are an infinite number of points on a circle.

Infinity is an essential mathematical construct since it is used in limit processes as you so mentioned. Without it, certain areas of useful mathematics would be unworkable, just like the notion of zero enabled entire areas of mathematics.

One might say that infinity is an artificial notion that has no analog in the real world, but you would also have to say that about irrational numbers. Ironically, two irrational numbers "pi", and "e", are about the most useful numbers you can imagine in engineering. You could not do much engineering without those two numbers.

In order to say that infinity is an artificial notion, you have to say that all mathematics is an artificial notion. You can say that, but you will have to explain why it has so much power of prediction in the real world, when mathematics is used to model the real world.

 

[Hugh Mann]

Originally Posted by Jovial
Infinity is a concept, not a place or number. It can only be understood in terms of a limit process. To say something is infinite means no matter how big of a finite number we chose, we can always choose something bigger.

One of the things that amazed me about math when I was a kid was that, even though "infinity" is not a number, infinite sets can be compared in size much like finite sets can. We know S = {1,2,3} is "smaller" than T = {1,2,3,4}. Intuitively, we know this because 3 < 4. Rigorously speaking, this is proved by showing that there is an "injection" from S to T: this is defined simply as a function where no two elements of S are mapped to the same element of T. You can convince yourself that there can be no injection from T to S.

This same idea applies to infinite sets...if S and T are infinite, then S is smaller than T if you can find an injection from S into T. Furthermore (and this also applies to the above), if you can ALSO find an injection from T into S, then S and T are equal in size (cardinality). Using these ideas, you can show that the set of all integers {...,-2,-1,0,1,2,...} has the same cardinality as the naturals {0,1,2,...}. Even weirder, there are just as many natural numbers as there are rationals. What is most amazing is that you can prove that there exists no injection from the set of reals into the set of naturals; in essence, there are more real numbers than naturals (and hence, more reals than rationals!). There are, in fact, different "sizes," or orders, if you will, of infinite quantities. Cantor proved this in the late 1800's with his diagonal argument. It's a proof that can be explained to almost anyone in less than an hour, but whose consequences are deeply disturbing to most people.

In order to say that infinity is an artificial notion, you have to say that all mathematics is an artificial notion. You can say that, but you will have to explain why it has so much power of prediction in the real world, when mathematics is used to model the real world.

Very well said. Whether or not the ambient space in which our fundamental particles exist is continuous or discrete (and this question has not been settled), our space behaves so much like a continuum (at reasonable distances/masses/speeds) that calculus describes it accurately. And of course, calculus is fundamentally based on the infinite divisibility and connectedness of the real line.

 

[JustAsking]

...Very well said. Whether or not the ambient space in which our fundamental particles exist is continuous or discrete (and this question has not been settled), our space behaves so much like a continuum (at reasonable distances/masses/speeds) that calculus describes it accurately. And of course, calculus is fundamentally based on the infinite divisibility and connectedness of the real line.

Yes, Hugh, I thought about quantum physics when I mentioned the number of points on a circle. But I realized I didn't have to go there. My point is this: that mathematics is used everyday to make useful predictions in the real world even if you limit yourself to classical physics. And that mathematics as we know it would fall apart if we did not acknowledge the notion of infinity, just like it would fall apart if we did not acknowledge the notion of zero.

That two infinite sets can be compared to each other is very interesting. Your Cantorian approach reminds me of Hilbert's Grand Hotel Paradox. If I remember correctly, he showed how a hotel with an infinite number of rooms that are all occupied could always accomodate an infinite number of additional guests. When the next guest arrives, you move the guest in room 1 to room 2, the guest in room 2 to room 3, and so forth to infinity, thereby freeing up room 1 for the new guest. Then when the next guest arrives you do the same thing, and so on.

Infinity makes my brain hurt.

 

[deleted213967]

I was somewhat reassured, yet also disappointed to hear from a friend with an advanced degree in Astrophysics that the number of stars in our universe was finite, a mere tens of billions (as I very vaguely recall).

What's up with that? I grew up under the assumption that no one could count all the stars.

Infinity is not what it used to be...

 

[Jovial]

I was somewhat reassured, yet also disappointed to hear from a friend with an advanced degree in Astrophysics that the number of stars in our universe was finite, a mere tens of billions (as I very vaguely recall).

What's up with that? I grew up under the assumption that no one could count all the stars.

Infinity is not what it used to be...

Actually it's estimated to be more like 10^21 stars. That's 10 billion times 100 billion, but still not infinite.

 

[mista geechee]

ESTIMATED.

Let's not forget we're only human and we make mistakes. Even scientist. I doubt anyone can count all the stars in the "uni"verse. As advanced as our telescopes are, they can only see as far as we are capable of programming them.

If this universe is finite, why isn't an edge defined ? What is outside of it? Where did it come from?

 

[crescendo69]

Welcome to LUSG!

 

[Jovial]

If this universe is finite, why isn't an edge defined ?

For the same reason there is no edge of the earth's surface.

 

[D_Thoraxis_Biggulp]

From inside going outward, there certainly is. It's called the crust. In a universal sense, we're inside of it rather than on it surface. Our view of the edge would be likened to the underside of the earth's crust.

 

[mista geechee]

For the same reason there is no edge of the earth's surface.

Actually it's defined at 60.something miles if I remember correctly.

MY question is, If we can't see (or even conceive of) the edge of the universe, how do we know it's there?

 

[D_Thoraxis_Biggulp]

First it was assumptions. Now it's based on the mathematically supported theory that our universe is one of many.

 

[D_Kay_Sarah_Sarah]

Infinity is closer then you think. Everything must come to an end at some point.

 

[Jovial]

From inside going outward, there certainly is. It's called the crust. In a universal sense, we're inside of it rather than on it surface. Our view of the edge would be likened to the underside of the earth's crust.

That is incorrect. It's not like our universe is the inside of a sphere and there is an edge to it, at least not in 3 dimensions.

The 2-dimensional analogy is like the earth's surface or a balloon. If an ant is restricted to walking on a balloon, he can't go inside or outside of it. His world has no edge for as long as he walks. The balloon can be inflated and the surface gets bigger just as our universe is getting bigger. In 3 dimensions you could say the edge of the balloon is the entire surface, and in our universe we would have to say the edge is everywhere. But to think of an edge to our universe it would have to be in 4 dimensions. We could think of time as defining the edge of our universe. We're on the edge of the universe and time is the dimension that it's expanding in like the balloon inflating. But instead of our universe being a 2-dimensional surface like a balloon, it is a 3-dimensional space. The entire space is expanding. There is no center where the big bang was. The big bang was everywhere, just like if you started the balloon expanding from a point, every point on that balloon would have been part of that original point.

Actually it's defined at 60.something miles if I remember correctly.

MY question is, If we can't see (or even conceive of) the edge of the universe, how do we know it's there?

We don't. No one said there is an edge because there isn't an edge.

 

[D_Thoraxis_Biggulp]

The idea of our universe having an edge suggests that we're inside of (what we perceive as) a 3-dimensional object. On Earth, we move along the top of it, and sometimes down into it. Traveling beyond here though, we travel upward and outward, moving along three dimensions.
This idea of a finite universe is supported by evidence of potentially endless multiple universes. Two infinite spaces cannot coexist.

 

[Jovial]

The idea of our universe having an edge suggests that we're inside of (what we perceive as) a 3-dimensional object. On Earth, we move along the top of it, and sometimes down into it. Traveling beyond here though, we travel upward and outward, moving along three dimensions.
This idea of a finite universe is supported by evidence of potentially endless multiple universes. Two infinite spaces cannot coexist.

I don't think that is supported by our observations. If we were inside of some giant sphere, then we could detect which direction was the closest way to the edge. But from our position it looks like the universe is the same in all directions. The background radiation that was left over from the big bang is the same in all directions suggesting, like I said, that the universe is expanding everywhere. The universe can be finite but not have an edge.

 

[D_Thoraxis_Biggulp]

With our technology, no we can't detect the edge of the universe whether there is one or not. Not in our lifetimes. The signal would take too long to reach it and bounce back.
It's not necessarily a perceivable edge in the dimensions we can see, but it does stop at some point. Not just in terms of no more matter, but just not our universe anymore.
Anyway, it sounded like you were saying the universe is infinite.