# Infinitely long piece of string.

## Main Question or Discussion Point

Not sure if this is a topology question but here goes.

Imagine an infinitely long piece of string and I cut it in half with a pair of scissors.

Both pieces are still infinitely long but I'm now stood at the ends of infinitely long strings, how can I be at the end of something with infinite length?

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Both pieces are still infinitely long but I'm now stood at the ends of infinitely long strings, how can I be at the end of something with infinite length?
So what's the problem?

Why do you think you can't 'extend to infinity' from any chosen starting point?

How long is the interval $[0,\infty)$?

So what's the problem?

Why do you think you can't 'extend to infinity' from any chosen starting point?
For the same reason I can't imagine the universe having an edge I can't imagine an infinitely long piece of string having "ends", at least in my mind the moment I imagine the string having ends it has a definite length stops being infinite. I suppose the cut string is like knowing the first few digits of pi but never knowing the last digit. Somehow cutting the string seems to transform it into a different kind of infinity.

How long is the interval $[0,\infty)$?
Sorry my knowledge of math comes to an abrupt end after + - x / , does that represent 0 divided by infinity?

If it represents 0 to infinity then the interval is infinite (I think)

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HallsofIvy
Homework Helper
Yes. There is absolutely no reason why an infinitely long string cannot have one endpoint- it just cannot have two endpoints because then the length of the string would be the distance between those two endpoints.

Similarly, you can have the "half-plane", the set of all points above the line y= 0 in a coordinate plane which has infinite area but has the x-axis as boundary. "Unbounded" does NOT mean "has NO boundary".

Well tinyboss asked you the same question as I did, only more mathematically.

BTW I am not assuming you have to cut your string in exactly half - The effect is the same wherever you cut it.

The whole point of an infinite line is that you can start anywhere on the line, set off along the line in either direction and never come to the end. No cutting involved.

Do you not agree with this?

Sorry my knowledge of math comes to an abrupt end after + - x / , does that represent 0 divided by infinity?

If it represents 0 to infinity then the interval is infinite (I think)
Well it's half-infinite. That is, it has exactly one endpoint, namely zero. It's just like taking the real number line and standing at the point 0. To your left, it's infinite. To your right, it's infinite. But you can still stand at the point 0.

Each half is an infinite ray. Just like standing right where you are and looking up at the universe. Even if the universe is infinite, you are still standing at some particular point.

By the way it's worth noting that math is not the same as physics, so trying to understand mathematical infinity by analogy with the physical universe is generally a bad idea. But if it helps to imagine standing outside your front door and looking up, then the analogy may be useful. Your line of sight may go on forever, but you are at one endpoint of the line of sight.

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Bacle2
Wouldn't you also believe that the set:

{1,2,3,....,n,n+1,...} is infinite?

The difference between maths and physics has already been mentioned and I would add philosophy to that list.

Take for instance the single 'end point'.

Mathematically the whole of the line interval (-∞, a] is available for any point a as well as the line interval [a, ∞).

However this is not so in physics.

Imagine picking a point and starting a train off from that point going onwards towards infinity.
Philisophically we can concieve of this actually happening (well being attempted).

No let us consider the reverse.

Imagine waiting at a point for a train to arrive from infinity.

That scenario will never happen because the train will never arrive.

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Yes. There is absolutely no reason why an infinitely long string cannot have one endpoint- it just cannot have two endpoints because then the length of the string would be the distance between those two endpoints.
This is what I believe intuitively yet it still poses the paradox that I'm now kind of "stuck" at the "end" , once its cut no matter where you stand at that length of string its infinite to the left and finite to the right.

BTW I am not assuming you have to cut your string in exactly half - The effect is the same wherever you cut it.
I'm going to be pedantic and say there is no half way point but yes I see your point perfectly well.

The whole point of an infinite line is that you can start anywhere on the line, set off along the line in either direction and never come to the end. No cutting involved.

Do you not agree with this?
yes and no, if it was uncut then I can never reach an end however it has been cut and I'm now at an end. The two pieces are still both infinitely long and it now contradicts your statement.
you can start anywhere on the line, set off along the line in either direction and never come to the end.
Wouldn't you also believe that the set:

{1,2,3,....,n,n+1,...} is infinite?
Yes in the sense that I can count forever but even if I did count forever n=n+1 the value of "n" will never be infinite.

Well it's half-infinite. That is, it has exactly one endpoint, namely zero. It's just like taking the real number line and standing at the point 0. To your left, it's infinite. To your right, it's infinite. But you can still stand at the point 0.
Point taken but I would still argue that the two cut pieces of string now have properties that the original uncut doesn't have, its like a completely different kind of infinity.

By the way it's worth noting that math is not the same as physics, so trying to understand mathematical infinity by analogy with the physical universe is generally a bad idea.
On this I strongly disagree. The universe as a whole provides an infinite volume of space and is the arena for all these thought experiments to take place in. Some people would argue that the three dimensions of space simply exist in some other higher dimension but that only begs the question what does that dimension exist in. You could go on indefinitely. It puzzles me that science considers a theory to be wrong if it has infinities and yet I cannot imagine a theory that explains something that is intuitively infinite if it didn't include infinities.

Imagine waiting at a point for atrian to arrive from infinity.

That scenario will never happen because the train will never arrive.
omg stop it your scaring my brain , love the analogy :)