I think if the universe is infinite in size then the number of particles must be infinite (I'm assuming isotropic and homogeneous, on large scales, over the whole of it)Drakkith said:
That's an excellent question. You already have some excellent answers.LightningInAJar said:
Well, if you go with “anything” then that would also include infinitesimals since the inverse of an infinitesimal is infinite. And our best descriptions of nature involve infinitesimals, so there is a lot of evidence supporting them and not anything strongly opposing them. So I would keep an open mind on the topicLightningInAJar said:
I remember, when I was a lad, this (the surface of a sphere plus others) was brought up as an 'infinity' example. However, it's finite but 'boundless'.Godot_ said:
That may or may not follow, depending on how you define a particle. When really big numbers of 'particles' are involved, the formula for the sum may or may not be convergent. i.e. is 'the number of particles' really an integer?phinds said:
The surface per se is boundless, but finite - it has a finite area, that can be determined. The distance you can travel, however, is infinite.sophiecentaur said:
This is basically the intuition behind the mathematical notion of compactness: it's a more precise answer to the question of whether a sphere's surface is finite or infinite.sophiecentaur said:
I can't make sense of that. If the universe is infinite, and homogenous and isotropic, then it is filled with stars and galaxies everywhere. One way to define the number of particles is the number of atoms. No formula is involved. No sums.sophiecentaur said:
There are two ways to express the idea of an infinite number of particles:anorlunda said:
Yes indeed. AFAIK, our speculations that the universe is or is not finite can never be confirmed. But the cosmological principle is not a product of mathematics, it stems from observations; correct?PeroK said:
Like all of science. Our observations so far are consistent with the hypothesis.anorlunda said:
Yes, the cosmological principle is a physical assumption underlying the current model. Observations show no clear evidence of finiteness and make the infinite model appealing. But, I'm not sure how infiniteness could ever be confirmed.anorlunda said:
According to our best models, the ones with the most experimental verification, there is no limit. These models underly all of physics, from Newtonian physics to quantum field theory and general relativity.LightningInAJar said:
It is more than that. As far as I am aware there isn’t a known model of physics that both entirely avoids calculus and is consistent with current data.LightningInAJar said:
Godot_ said:
I would not call such examples infinite or even unlimited.LightningInAJar said:
That is a theoretical limit.Dale said:
While your arguments about the zero and infinite wavelengths hold some water, ...DaveC426913 said:
Depends on the sphere. Where's the insurmountable barrier on a billard ball?DaveC426913 said:
He didn't give any other than that they have to be there in the real, physical world. All else is your implication only. And that they're easier to prove or explain with math doesn't make them any less real. If you take the time and a sufficiently finely graded (or infinitely long) measuring tool, they can be observed.DaveC426913 said:
Current paradigm has it that the universe keeps expanding, and hence should be infinite in the time-forward-dimension. Yes, there has been some beginning - but there won't be an end. Like with natural numbers: There's a lower limit - 1 or 0, depending on whether it's N or N0 we're talking about, but no upper limit, so it still is infinite.DaveC426913 said:
Slanting the measuring scale yields another set of subdivisions, thanks to trigonometry. And don't tell me that that only exists in math...DaveC426913 said:
You don't need to invoke anything more than the uncountably infinite set of points in a portion of space. But, that (like the EM spectrum) is a mathematical model of the physical universe. Clearly, we have mathematical models that assume space, time and the EM spectrum are infinitely divisible. But, you can never confirm that fully and conclude that there are, for example, an infinite number of possible wavelengths.Godot_ said:
That's explicitly mathematics.Godot_ said:
In mathematics the "ad infinitum" is possible by definition. In physics, you would need to confirm by experiment. And an infinite sequence of experiments is physically impossible.Godot_ said:
I'm not surprised as I didn't get my words together correctly. I've re-thought what I was trying to say and the following may not be so dodgy. The thread is discussing matters like 'how many particles' but at the extremes of conditions our familiar particles are not the same. Theory suggests that particles are created in very low densities and that, under conditions of very high density, the lose their identity so counting them becomes meaningless and any model runs out.anorlunda said:
Yes - the established maths doesn't deal with all situations so we can't rely on it to deal with the Infinity question or even give any meaning to it.PeroK said:
Physically, it is limited by the Plank length.Godot_ said:
There will be a physical end to any sphere and any person or device attempting to traverse it. They will decompose.Godot_ said:
Just because there is no upper limit in theory does not mean that an upper limit can be reached physically.Godot_ said:
It exists physically down to some physical limit. It is at least limited by the Plank length.Godot_ said:
No. Your observing tool is physical, and - whether atomic or photonic - has some lower limit of resolution.Godot_ said:
Really really small is not the same as infinitely small.The OP's question was not 'Are there physical things in the universe that are really, really big or really, really small?' If it had been, this thread would have ended after post 2: "Yes. Lots."Godot_ said:
You shouldn’t have added the [of EMR] to my quote. I was responding with a very general statement about all of modern physics, not limiting my comments to EM at all. That is why I said “These models underly all of physics”.DaveC426913 said:
Calculus doesn't depend on infinitesimals, which technically are part of "non-standard" analysis. Standard calculus is based on limits, which require only the properties of real numbers. One such property is the Archimedean property (which is not compatible with infinitesimals):Dale said:
Which plank are you ralking about?DaveC426913 said:
To me, the salient point is the converse. There are no models that I am aware of that both explain the data and don’t involve a continuum or infinitesimals. So the idea of a physical continuum or physical infinitesimals is definitely plausible and the idea of the converse is very questionable. It is not required by experimental data and it not known to be even theoretically consistent with observation.PeroK said:
That is indeed a valid point but since the word “infinitesimals” is commonly used when describing such limits I felt it was fine. Particularly given the OP’s use of the word “infinite” which is also a limit in standard analysis. In whatever sense the OP intends “infinite” I can use “infinitesimal” in the same sense and my points remain valid without distorting their meaning.PeroK said:
The Real number system is, however, mathematically tricky. There is a countable subset of computable numbers, which can be described with a finite amount of information. "Most" Real numbers, therefore, are literally impossible to describe. If I asked you to give me a Real number that is not computable, then you would simply be unable to describe it. A lottery based on Real numbers, for example, is not possible!Dale said:
The mathematical trickiness is not particularly worrisome to physicists. The models work. And models without those tricky features don’t.PeroK said: