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- Is anything in the physical universe unlimited or just in math?

Is anything in the physical universe unlimited or infinite or does Infinity only exist to make some math work? Is Infinity a real thing outside of math?

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- #1

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- TL;DR Summary
- Is anything in the physical universe unlimited or just in math?

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The distance you can travel on the surface of a sphere, Moebius band, ...

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It is still unknown whether the size of the universe is finite or infinite. It could be infinite.

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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)If the universe is infinite in size then the number of particles is likely infinite as well,

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That's an excellent question. You already have some excellent answers.Is anything in the physical universe unlimited or infinite or does Infinity only exist to make some math work?

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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 topicSummary::Is anything in the physical universe unlimited or just in math?

Is anything in the physical universe unlimited or infinite or does Infinity only exist to make some math work? Is Infinity a real thing outside of math?

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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'.The distance you can travel on the surface of a sphere, Moebius band, ...

That's just a mathematical idea and doesn't prove anything ' physical' about the Universe.

Interestingly, the heroic and nerdy Mathematicians sorted out a lot about what infinity can mean, hundreds of years ago. But that's all based on Axioms and some of the results don't necessarily 'make sense' to our intuition.

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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?if the universe is infinite in size then the number of particles must be infinite

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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.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'.

That's just a mathematical idea and doesn't prove anything ' physical' about the Universe.

Interestingly, the heroic and nerdy Mathematicians sorted out a lot about what infinity can mean, hundreds of years ago. But that's all based on Axioms and some of the results don't necessarily 'make sense' to our intuition.

Easy way to look at it would be e.g.:

You can "count" the area by painting it with exactly one layer of paint. You start somewhere... ...and eventually you run out of unpainted places. => Finite.

The number of steps you can take in any direction - that is infinite: After each step, you add one to the count, yet you still can take another, uncounted step... ...ad infinitum. Each step is a new one - even though you've been there before...

/nitpicking

[Edited for better understanding]

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If position or time or any other quantity were infinitely divisible, how could we distinguish that from discrete-but-smaller-than-we-can-slice?

If it turns out that these things are truly finite or discrete, we might eventually figure that out. If they are truly infinite or infinitesimal, we may never know.

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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.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'.

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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.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?

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There are two ways to express the idea of an infinite number of particles: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.

Given any number ##N##, you can find at least ##N## particles in the universe.

The set of all particles in the universe is infinite - in the sense that it has a countably infinite cardinality.

That said, you can prove or disprove these properties for abstract mathematical objects but you can't prove them for physical objects in the same way. You can have a model that predicts an infinite number of particles, but it's not clear how you could ever confirm that the model is correct in that respect.

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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?but it's not clear how you could ever confirm that the model is correct in that respect.

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Like all of science. Our observations so far are consistent with the hypothesis.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?

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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.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?

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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.Can you really not cease from dividing the electromagnetic spectrum into smaller slices or space into smaller lengths?

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.Or can we simply not prove otherwise because we are so much larger and measuring things larger than us affords us better tools?

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The OP has equated unlimited with infinite. That's tricky.

There seems to be limits to pretty much everything in the universe, preventing infinites. Here's what I mean:

*infinite or even unlimited*.

The OP is asking about the*physical* world.

You can*not* actually travel an infinite distance on a sphere in reality. There are real barriers that *will* ultimately prevent it.

While it can be said to be*theoretically* infinite (which - by definition - means *ignoring* certain physical, real-world constraints), that doesn't meet the OP's explicit condition of infinite, IMO. I guess "can't occur in the finite lifespan of the universe" is a boundary.

Likewise - while there are an infinite number of "real numbers" - this can only occur in the*abstract* world of math. The *physical* world (as far we we understand) has real constraints on infinite subdivision. I guess "the Planck length" is a boundary.

There seems to be limits to pretty much everything in the universe, preventing infinites. Here's what I mean:

The distance you can travel on the surface of a sphere, Moebius band, ...

I would not call such examplesI received the walking around the Earth's equator example

The OP is asking about the

You can

While it can be said to be

Likewise - while there are an infinite number of "real numbers" - this can only occur in the

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That is aAccording to our best models [of EMR], 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.

A

An infinitely short wavelength would have other problems. I think, including - but not limited to -

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While your arguments about the zero and infinite wavelengths hold some water, ...[...]

Aphysicallyinfinite EM emission would have an infinitely long wavelength. [...]

...it completely ignores what, for simplicity's sake, I'd call the "inner" infinity: You can find an infinity of wavelengths between any two arbitrarily chosen wavelengths. Black body radiation and doppler shift take care of that...

To learn more, try looking at the Bolzano-Weierstrass-Theorem e.g. @wikipedia or try googling Bolzano Weierstrass Epsilontics (or epsilon-delta) ...

While it's way easier to use math to prove it, it still is observable in the physical world - down to the best precision your measuring instrument permits. And when you then build a more finely graded instrument, you can open the next can... ...ad infinitum.

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Depends on the sphere. Where's the insurmountable barrier on a billard ball?[...]

You cannotactually travel an infinite distance on a sphere in reality. There are real barriers thatwillultimately prevent it.

[...]

The only barrier there would be the lifespan of the traveler... ..and, meh, he'll just pass the baton to the next traveler...

Also, if an insurmountable barrier exists, it's actually, by definition, not a sphere...

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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.[...] that doesn't meet the OP's explicit condition of infinite [...]

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 N[...]IMO. I guess "can't occur in the finite lifespan of the universe" is a boundary.

[...]

Slanting the measuring scale yields another set of subdivisions, thanks to trigonometry. And don't tell me that that only exists in math...Likewise - while there are an infinite number of "real numbers" - this can only occur in theabstractworld of math. Thephysicalworld (as far we we understand) has real constraints on infinite subdivision. I guess "the Planck length" is a boundary.

Also, there definitely are infinite angles. You can always nudge your observing tool a bit to the right or left...

To give a physical example:

All those faraway galaxies in the recently published JWST image... ...their apparent width is in the milliarcsecond range. But they're composed of individual stars, some of which will be observed when they go supernova, eventually... ...which opens up the micro- and nano-arcsecond scale. Also, they're composed of atoms. Those atoms' apparent width would then be in the ... I won't bother to look up the appropriate prefix right now. You get the image, I suppose.

Are those atoms not real and physical, just because the distance is too great to observe them?

As I said above: Using math as a convenient tool to explain / extra- / interpolate stuff doesn't mean that that stuff isn't real or not physical.

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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 aYou can find an infinity of wavelengths between any two arbitrarily chosen wavelengths. Black body radiation and doppler shift take care of that...

That's explicitly mathematics.To learn more, try looking at the Bolzano-Weierstrass-Theorem e.g. @wikipedia or try googling Bolzano Weierstrass Epsilontics (or epsilon-delta) ...

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.While it's way easier to use math to prove it, it still is observable in the physical world - down to the best precision your measuring instrument permits. And when you then build a more finely graded instrument, you can open the next can... ...ad infinitum.

Note that you can plan an infinite sequence of experiments, but you'll never finish them all. You will only ever have conducted a finite sequence of experiments, regardless of how much time you have.

Note also that even if the universe will exist for infinite time, physically that means that time never ends. But, not that at some point you have experienced an infinite amount of time.

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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.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.

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.That said, you can prove or disprove these properties for abstract mathematical objects but you can't prove them for physical objects in the same way.

This could be interpreted as a personal theory so I will stop now.

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Physically, it is limited by the Plank length.If you take the time and a sufficiently finely graded (or infinitely long) measuring tool, they can be observed.

There will be aCurrent 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.

Just because there is no upper limitLike with natural numbers: There's a lower limit - 1 or 0, depending on whether it's N or N_{0}we're talking about, but no upper limit, so it still is infinite.

Take the example of the unlimited frequency of an EMR emanation. At some point, it has more energy than the universe itself. And that's self-contradictory, since it's

It exists physically down to some physical limit. It is at least limited by the Plank length.Slanting the measuring scale yields another set of subdivisions, thanks to trigonometry. And don't tell me that that only exists in math...

No. Your observing tool is physical, and - whether atomic or photonic - has some lower limit of resolution.Also, there definitely are infinite angles. You can always nudge your observing tool a bit to the right or left...

Even a science fiction tool of fabulous precision will still be limited by the Planck length.

ReallyAll those faraway galaxies in the recently published JWST image... ...their apparent width is in the milliarcsecond range. But they're composed of individual stars, some of which will be observed when they go supernova, eventually... ...which opens up the micro- and nano-arcsecond scale. Also, they're composed of atoms. Those atoms' apparent width would then be in the ... I won't bother to look up the appropriate prefix right now. You get the image, I suppose.

Are those atoms not real and physical, just because the distance is too great to observe them?

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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”.That is atheoreticallimit.

Aphysicallyinfinite EM emission would have an infinitely long wavelength. Not only would it never oscillate; it would never evenstartit first oscillate. I guess "an EM wave that is too long to oscillate" is a boundary.

An infinitely short wavelength would have other problems. I think, including - but not limited to -~~having~~beingan infinite amount of energy. I guess "would have more energy than the universe" is a boundary.

To clarify, I was talking about calculus. All of our scientific models are based on calculus, and all of calculus is based on infinitesimals. So all of our physical models presume that infinitesimals are physically meaningful. To my knowledge we don’t have any valid physical models that lack this feature.

@LightningInAJar made a very broad original claim, and then brought in the EMR spectrum, but I was still responding to the broader context.

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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):All of our scientific models are based on calculus, and all of calculus is based on infinitesimals. So all of our physical models presume that infinitesimals are physically meaningful.

https://en.wikipedia.org/wiki/Infinitesimal

https://en.wikipedia.org/wiki/Archimedean_property

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Which plank are you ralking about?Physically, it is limited by the Plank length.

If you want to talk about the Planck length, nothing magical happens there. I wish people would stop saying otherwise. The Planck resistance is 30 ohms. Nothing magic happens there.

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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.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.

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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.Standard calculus is based on limits

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The Real number system is, however, mathematically tricky. There is a countable subset ofTo 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.

So, when we say something as seemingly innocuous as "Let ##x \in \mathbb R##", we are already doing something that is purely mathematical and cannot be represented physically or represent experimental data.

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The mathematical trickiness is not particularly worrisome to physicists. The models work. And models without those tricky features don’t.The Real number system is, however, mathematically tricky. There is a countable subset ofcomputablenumbers, 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!

So, when we say something as seemingly innocuous as "Let ##x \in \mathbb R##", we are already doing something that is purely mathematical and cannot be represented physically or represent experimental data.

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