That's a mathematical infinity, not an actual one. I was asking for something that can be demonstrated to exist that is infinite. Not a concept or an idea, but a thing.
I still think there is a good chance that we are talking past each other. It would help to have a clear idea between what qualifies infinity as a concept or idea and what qualifies infinity as a thing. Without working definitions then miscommunication and confusion will arise. The best thing to do to really get at the issues you raise would be to explain what you mean by "infinity actually existing." What's a clear definition of that? Equivalently, what would qualify as infinity actually existing?
And for the record, there aren't an infinite number of points on a ruler, because there is a smallest possible distance - the Planck Length (1.616199(97)×10−35 metres), and any distance consists of a finite number of them.
As far as I know this is a gross misunderstanding of the Planck Length. If I understand your assertion correctly, it is that there exists a three dimensional grid in space in which everything occurs and the minimum difference between any two points on the grid is the Planck Length. I see no reason to believe (accept) this and reasons not to.
My experience with quantum mechanics (limited mind you, just an undergraduate experience) is that wave functions are defined on a non-discrete typology of space and a completely precise position measurement (impractical by actual humans) could result in any of the infinite number of points even in a confined environment (eg. infinite square well).
Such an assertion also seams to violate relativity in two ways. One is that if a grid existed and I was moving relative to that grid length contraction would shrink the grid and I could measure differences in distances smaller then the Planck length. If I still had to only be able to measure differences in distances to be an integer multiple of a Planck length, then I wouldn't be able to make any distance measurements in a frame that is not moving relative to me as the grid would need to be of infinite length (or very large length if there exists some maximum speed limit less then that of c). One could say that in any rest frame the grid is of Planck length and one can measure the lengths smaller then the Planck length. If this is the case then there would indeed again be an infinite number of points along the ruler because I could just send the ruler into motion.
It also seams to violate relativity in that the laws of physics would indeed be different in different inertial reference frames. It would seam to be the case that a reference frame that is half a Planck length off and travailing with he same velocity would have different physics. Since experiments cannot currently study nature on Planck scale, this effect hasn't been ruled out by experiment but to accept an idea that seams to violate one of the fundamental ideas of relativity would require a good reason to accept that and I fail to see one.
I should also mention that a grid idea would lead to a failure in the conservation of momentum. This isn't a problem because it is my understanding of quantum mechanics that momentum isn't strictly conserved on Planck length scale.
It is my guess that most physicists would reject a Planck length grid idea. If there is serious discussion within the physics community of an idea of such a grid please inform me of such. I should state that the idea is very different then the idea of guantum foam
. Quantum foam is the proposed idea that on Planck length scales, the nature of space-time itself is effected by the uncertainty principle and that specific points in space-time itself have uncertainty. A consequence of this would be that on a Plank length scale, the Planck length itself would be indeterminant. This would again, create an infinite number of points along a ruler even if such a grid exists.
In any case I could be discussing an idea that you did not actually intend to propose. I'm just using my best understanding of of what you've said. In any case, I see reason to believe (accept) that a wave function in position space is non-trivially defined on a domain that is not quantized (thus there are an infinite number of points along a ruler) and no reason to think otherwise.
There aren't an infinite number of colours, because the frequency of light is quantized - there is a minimum amount of difference - and the difference between two colours consists of a finite number of those jumps.
It is true that the frequency of light from any single emission source is quantized but there is no reason to believe that the possible emission sources themselves are quantized. It is clearly possible to create frequencies of light from one blackbody that another blackbody does not make. Even if the emission sources themselves are quantized the differences in the period of allowed photons would be much smaller then Planck time. This is actually further evidence against a Planck length grid as because blackbody radiation would create contractions in such a system. In any case for me to believe (accept) that emission sources are quantized requires a pretty convincing argument.
The gravitational field near a black hole approaches infinity, but doesn't actually reach it because the closest you can get to a singularity is the Planck Length (at which point the gravitational field is extremely high but not infinite), and that is hidden from all exploration by the black hole's event horizon.
I included this example because infinity is a different concept in different contexts. Most of the discussions in this thread are about infinity as a class of cardinal numbers but it has a different meaning when talking about limits. The assertion that the closest one can get to a singularity is the Planck length is problematic as I don't know why I should believe (accept) this. From my discussion above, it seams that I shouldn't. The question about being able to explore this is related to the difference between discussing what exists in nature and the ability of humans to actively explore that nature in experiment.
If you're willing to discuss this et cetera without just glossing over it I'm willing to do so.
I was asking for something that can be demonstrated to exist that is infinite.
This seams to be important. What do you mean by "demonstrated." It is likely that there are elements to nature that humans will never demonstrate in experiment. Some of these elements can be explored as a consequence of what is demonstrable by experiment. It may be the case that humans will never be able to "demonstrate infinity" by experiment while being able to rigorously infer infinity be the consequences of experiments. This also goes back to infinity as it applies to the physical world and infinity as it applies to homo sapien thoughts including how homo sapiens perceive that physical world. The two are vary different things.
Why should one limit oneself to what humans can differentiate between?
I still don't see any reason to impose this limitation is we are talking about infinity in the physical world.
Also, while this (how infinity applies to the physical world) is an interesting tangent (and it is) it is still not necessary to discuss the original question and other tangents more directly related to it. One need not argue that infinity is impossible to argue that human language is finite and seams to be a very expensive path to reach that conclusion.