trb456 has the first part of the answer: you have to have a definition for the word "infinity" before you can determine and justify that something is infinite. Mathematics provides the most agreed upon definitions. In fact, it provides *several* infinities, the most prominent being the countable infinity (the end point of the series 1, 2, 3, 4...) and uncountable infinities (such as the continuum, which is the number of distinct number on the real number line).

However, when it comes to things in physics, such as your rope, you can never "determine" that anything is infinite. Unless your rope is of such a mathematically regular structure that you can prove its structure repeats forever, you never know if at some point the rope ends.

What you can determine is that the length of the rope is indistinguishable from infinity. If you have no test which can determine the length of the rope with an error bound that excludes the possibility of an infinitely long rope, you can often simplify your world on the *assumption* that the rope is infinite, whether it is infinite or not, but you cannot "prove" it.

This can work perfectly up until the day a test becomes able to determine the length of the rope with sufficient precision as to exclude the possibility of an infinite rope. At that time, you have to re-analyze all of your work which assumed the rope was infinite, and see if that work is still valid under this new very-long rope.

As an interesting real life example, consider quantum gravity. We have a model of relativistic gravity that worked very well for large, slow things (large as in: larger than atomic size; slow as in: not traveling at nearly light speed). We have a model of quantum mechanics that worked very well for small, fast things. Our QM model includes an infinity: there is a way to have a particle interact with itself, and interact with the result of that interaction, and so forth on its way towards what mathematics would call "countable infinity."

We had no reason to believe this infinity was not real. Of course it was only our QM model, models can have infinities. Physicists spent a great deal of time proving that they could do an operation called "normalizaion" which tucks the tail of this infinity up inside the model and makes everything valid. Normalization assumes these interactions interactions become weaker as they progress.

Think of it akin to compound interest. If you earn 10% on $100, you might say you have $110, but if you compound it twice, for the first half you get $5, and on the second half you get $5 plus 10% of $5 because you get interest on your interest. If you keep this path going, the faster you compound, the more money you make, but there is a limit. There are equations which tuck that infinity back into the model, and give you "continuously compounded interest," which says how much money you could make with infinite compounding.

Back to QM, we had no reason to believe this infinity would cause trouble, until we tried to match it to relativity to model large fast things. As it turns out, after a lot of math that is beyond my paygrade, relativity's "space-time stretching" behavior causes gravity to behave poorly on the quantum scale. It causes the interactions on the interactions to not peter out quite as quickly, and it turns out that it slows down this effect enough to "prove" particles have infinite energy.

QM people obviously see that this is a modeling issue, not an actual suggestion that particles have infinite energy, because there is too much macroscopic evidence disagreeing with that result. So physics is in a conundrum today: relativity seems to model the universe correctly as we stretch off towards infinity, QM seems to model the universe correctly as we dive towards the infinitesimal. They simply don't agree.

So what are physicists doing? They're continuing to use the same old assumptions of infinity for now, but they recognize that at some point a test is going to show a limit to their infinity. At that time, they will have to dig through everything they have done, and see what is invalidated by this newfound test, and what is not.

1Your answer doesn't address the OP's concern about ropes. The mathematical real line is not a rope. The real line is like a purple unicorn. It exists in my mind but not in the physical world. OP asked about ropes. It's a very good question, since nobody knows if the physical universe is finite or infinite, or even if the question is meaningful. As far as I know, Jech does not talk about ropes. He talks about sets. Surely you agree that ropes have physical existence and sets have mental existence. "It's hard to see X" does not constitute a proof of the nonexistence of X, I hope you agree! – user4894 – 2015-01-08T04:42:40.183