How can one determine, and justify, that something is infinite?

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Can infinite things be measured and known?

Imagine you stood on a long rope, and in the distance, you see that the rope continues past your vision. The further you walk on it, the more rope seems to appear. As far as you can see, it is infinite in both directions. How would you know, and justify claiming, that the rope is, in fact, infinite?

Or, instead imagine before you, what appears to be a simple well, yet one which is claimed to be infinite in depth. How would you determine, measure, or in some way prove, that the well is in fact infinite, given infinite time, life, or any other relevant, conceivable power?

The answer concerning a finite amount of atoms is fair. A material object cannot be infinite. But this question doesn't really ask about vacuums. Consider the depth of a well that has no end or consider infinite time, if not in the past, in the future. If you could travel infinitely in the future could you determine time is infinite?

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I think this question comes down to your mathematical orientation.

If your are not mathematical, then infinity is very hard to justify and indeed I could support the claim that there is no "real" infinity. I say this as a mathematician: it is hard to see how a "real" infinity exists in nature. In particular, I cannot conceive of an infinite rope, or any other object composed of atoms.

But if you are a mathematician, then infinity is no sweat. It's simply a well-defined set of rules in (e.g.) set theory that we agree are the "correct" generalization of finite math to the infinite. You should consult any good book on set theory to learn more, like Jech (2000). The rules for infinity initially seem weird, but ultimately are consistent and work. And this is the key: you need to work with the rules of infinity as defined by set theory, and strongly avoid "intuition", which will fail you mightily with respect to infinity.

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

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Let's imagine for the sake of argument that you do encounter an infinite rope, but you do not know it's infinite. How would you know? Well, any act of measurement takes time. Even the fastest measuring device (e.g.: a laser beam traveling along the length at the speed of light) still travels at finite speed, which means it will never reach the end, not that there's an end to reach.

Ok, so you have an infinite amount of time. But what does that mean? To finish measuring some object, the measurement must complete. That is, you must reach the end of the object. However, if you have an infinite amount of time to measure an infinite object, you will still never reach the end of it; you simply will always be in the act of measuring.

That's the thing with infinity; it's never-ending. So to ask about physically measuring an infinite object is to ask when you'll measure the last bit which is to ask when you would reach the end of a never-ending object, which is a contradiction.

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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.

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Coming from physics I've always seen Infinity as an approximation.

A symbol to describe something much larger or much smaller than the other elements in the equation.

So to me infinity is just a conceptual tool.

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"Can infinite things be measured and known?" If we are talking about physical things and measurements about them, we must have a "theory" (or lets call it paradigm) providing a framework for the "properties" of the physical thing that is subject to the measurement. Take, for instance, mass and remember that the "mass" means different things in Newtonian physics and relativistic physics.Time and distance have shared the same destiny with the mass as we proceeded towards the relativity theory development. This means that physical world theories bring by themselves what and how to measure properties of things whether the result be infinite or finite.

Consider a static universe theory where, by one of its hypothesis, universe should have infinite number of stars and such a universe has no beginning. Now, we would like to measure (by whatever means) the average distance of stars in such a universe. Just measure the amount of light/radiation at a certain point in such a universe. If it is finite, we measured the average distance of stars already, it must be INFINITE! Why? If it would not be infinite, then, given the infinite time elapsed since the beginning of that universe, the light from infinite number of stars (that was a hypothesis!) would eventually have reached to the measurement point (even if the speed of light would be finite) giving rise to the radiation level of infinite. But we measured a finite radiation. Then other stars, on the average, must be infinitely far away from the measurement point.

This is an example of measuring (indirectly) "infinite" average distance among the stars of the universe that this "particular" physical world theory brings by.

We may never say that our (getting ever and ever more successful) future physical world theories and paradigms will never allow non-measurable physical properties of infinite in measured value.

Another way of looking at the matter would be like this: If your theory allows infinite things and if there are more than one infinite things, then you may use (possibly indirectly) one of them as a basis/unit for the measurement that can be done in finite time and with finite effort.

For the rope example, one day in future we may have a theory of physical world which would allow us to conduct a simple experiment on the rope and get a result without travelling all along the rope which would say the rope is fine or infinite. We saw examples of this in history (as in one variation of "static" universe model).

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Although it is hard to come up with a physical infinity that could be "measured," I am aware of one that sort of fits the definition. It is called an Electrical Transmission Line (ETL). The impedance of an ETL is defined as the resistance value obtained for a line that is infinitely long. We can, and do measure the "equivalent" impedance on shorter lengths by the use of "terminating resistors." For example, if we use a short piece of 75 ohm coax line, and we terminate it with a 75 ohm resistor, the line acts as if it were infinitely long. So, if we tested both, a short & terminated piece ETL, and an infinitely long & unterminated ETL, we would not be able to detect any difference. The conclusion is that if we can simulate the infinite something, then we can (indirectly) measure it. Otherwise, we can't.

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Certainly in the context of some physical theories, it is possible to prove that some things are infinite. Given general relativity and given enough observations (say about the distribution of mass and the rate at which the Universe is expanding), one tell whether the Universe is or is not infinite.

Of course, that determination depends on the ambient theory. But then, everything we know about everything depends on some ambient theory. I believe the sun exists because I have a theory that says there's some relationship between my perceptions and external reality.

So to say that we can determine the volume of the Universe is no more or less dicy in principle than to say we can determine the existence of the sun. Different theories of physics might lead to different answers about the Universe's volume, but then different theories of perception lead to different answers about the existence of the sun.

As long as you have a physical theory you're pretty comfortable with, it's perfectly conceivable that observations can prove that some things are infinite.