"Finite but unbounded"

[Bikewer]

I think the quote is essentially the one from Stephen Hawking's Brief History Of Time:
"The universe is finite, but without boundary or edge."

That's easy enough to visualize if "the universe" is all the stuff that makes it up, all the matter, energies, particles, etc... And is expanding into "the void", which is essentially nothing.
However, if the universe is essentially "everything", then it's a little more difficult to visualize.

 

[jasonpatterson]

Being bounded to the surface of a sphere, globe or any shape isn't being unbounded.

It is unbound in the two dimensions that the ant can travel in. There is no edge to the surface, in other words. Heck, even if the universe were infinite in extent, we would still be bound to traveling in three spatial dimensions. The circle/sphere comparisons were 1D and 2D versions of the same thing.

 

[spin0]

[qimg]http://www.internationalskeptics.com/forums/imagehosting/thum_1957506feacd5c843.jpg[/qimg]

Happy now?

Thanks ants.

Thants.

 

[Marras]

I probably am not qualified to comment, but that doesn't seem the same as the solid-surface example.

His example is pretty much the standard way of how the expression 'finite but unbounded' is used in the number theory.

Here you assert that the dollars themselves are infinite, but you can only have a finite, but unspecified number of them.

Not necessarily. Suppose that the machine prints the dollars as the lever is pressed. (*) Then there's never more machine-given dollars in existence than the number of times that the lever is pulled.

(*) And that the machine has a robot that runs to buy more dollar-making supplies such as paper and ink whenever it's running low on supplies.

 

[Folly]

Kevin's example fits what I had learned - the rest have to do with boundaries, not boundedness. It still seems a bit indirect to me, but I can't come up with a nice example. Here's a lousy one, instead.

Consider a different machine, which just tells you a number when you pull the lever. If it's generating the numbers according to a bell curve, the numbers you get when you pull the lever are finite but unbounded. Every single time you pull the lever, you'll get a real number, so it's finite. For any number, however, there's a chance you could have gotten a higher number, so it's unbounded.

 

[ynot]

[URL]http://www.internationalskeptics.com/forums/imagehosting/thum_1957506feacd5c843.jpg[/URL]

Happy now?

Yep - thanks - the universe is saved!

 

[Soapy Sam]

I didn't know it had a finite area.

Well, many years ago, I had a poster of it on my kitchen door, which was about 7 ft 6" by 3 ft, so I think it has to be finite, because otherwise I'd still be painting it.

Now I'm struggling to remember, but I was reading about examples of infinities where the incremental slope of a line on a graph perpetually declines, but the line is not asymptotic to a finite y-value. That hurt my head. I think Zeno's paradox is so familiar that, ironically, it has become "common sense" and to find something with a similar set-up but a different result is counter-intuitive.

Infinity is like the Loch Ness Monster: A nice idea. I've yet to see any physical evidence of one.
I keep wondering, if one physical infinity of anything exists, then what is everything else made of?

 

[Acleron]

Area of the Mandelbrot set

Thanks. Interesting but slightly disappointing it can only be approximated but certainly indicates that it is well bounded.

 

[ddt]

I didn't know it had a finite area.

Now I'm struggling to remember, but I was reading about examples of infinities where the incremental slope of a line on a graph perpetually declines, but the line is not asymptotic to a finite y-value. That hurt my head. I think Zeno's paradox is so familiar that, ironically, it has become "common sense" and to find something with a similar set-up but a different result is counter-intuitive.

The logarithm is what you're looking for.

Hint: the typical example of a series whose members go to zero but which itself does not have a finite sum is the series of reciprocals 1, 1/2, 1/3, ... The logarithm is the integral of 1/x

 

[Badly Shaved Monkey]

The logarithm is what you're looking for.

Hint: the typical example of a series whose members go to zero but which itself does not have a finite sum is the series of reciprocals 1, 1/2, 1/3, ... The logarithm is the integral of 1/x

Thanks. I think that is what I was remembering. Now I need to remember which book it was that raised the topic.

 

[TubbaBlubba]

Thanks. I think that is what I was remembering. Now I need to remember which book it was that raised the topic.

Michael Spivak covers it in "Calculus" as an example of a function that is not bounded above and also quite visually unintuitive.