I Is anything in the physical world infinite?

[Vanadium 50]

Physically, it is limited by the Plank length.

Which plank are you ralking about?

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.

 

[Dale]

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.

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.

 

[Dale]

Standard calculus is based on limits

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]

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.

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!

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.

 

[Dale]

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!

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.

The mathematical trickiness is not particularly worrisome to physicists. The models work. And models without those tricky features don’t.

 

[PeroK]

That is indeed a valid point but since the word “infinitesimals” is commonly used when describing such limits ...

Not by mathematics students. A major breakthrough in 19th mathematics was to develop calculus on a rigorous basis without recourse to "infinities" or "infinitesimals".

"Infinity" appears as a term in standard real analysis, but only in terms of the properties of real numbers. For example, in the construction $n \rightarrow \infty$, there is in fact to appeal to a mathematical object called $\infty$.

 

[weirdoguy]

It is at least limited by the Plank length.

No it is not: https://www.physicsforums.com/insights/hand-wavy-discussion-planck-length/

 

[Dale]

For example, in the construction n→∞,

This is still a limit. I understand your point but I don’t think it is important for this thread specifically nor to physics generally.

A major breakthrough in 19th mathematics was to develop calculus on a rigorous basis without recourse to "infinities" or "infinitesimals".

You may consider it a breakthrough. I consider it a step backwards that mathematical pedagogy has still not overcome.

 

[DaveC426913]

Which plank are you ralking about?

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.

.

That was a typo, I swear. I know how to spell Planck.

And mea culpa re: hand waving Planck length. That was sloppy of me.

But my point remains, in the physical world there are limits to precision. At the very least, atoms and wavelengths will limit actual measurements.
It doesn't mean we won't refine our tools but that's not the same as saying things are infinitesimal or that measurements can be infinitely precise.

 

[jbriggs444]

in the physical world there are limits to precision

If for no other reason, because it would take infinitely long to write down the decimal expansion of an infinitely precise result.

 

[Klystron]

Remaining in the physical realm, the cosmic microwave background radiation (CMBR) can be offered as an example of an infinite unlimited, as far as our instruments detect, electromagnetic system.

[edit: corrected language to match the original post.]

 

[PeroK]

Remaining in the physical realm, the cosmic microwave background radiation (CMBR) can be offered as an example of an infinite, as far as our instruments detect, electromagnetic system.

Infinite in what way?

 

[Klystron]

Infinite in what way?

See correction. I meant to use unlimited as in the OP.

We detect the CMBR In every direction as far as we can see.

 

[DaveC426913]

In every direction. As far as we detect.

At the risk of being glib, how is that any more significant than simply saying everything within my personal field of view (be it sky, ground, or my own feet) is an unlimited EM system?

 

[Vanadium 50]

But my point remains

In my view, it is either wrong or at best confusing and unhelpful. One can measure things smaller than atoms - which, by the way, is a completely different claim than your original one about Planck units.

 

[Vanadium 50]

As a PS, Dave, you have well and truly hijacked this thread.

 

[DaveC426913]

In my view, it is either wrong or at best confusing and unhelpful. One can measure things smaller than atoms - which, by the way, is a completely different claim than your original one about Planck units.

I wasn't suggesting one can't measure things smaller than atoms. Sorry if it seemed that way.

It sort of turns on what one means by an "infinity in the real world". Everyone seems to keep coming up with theoretical ideas, none of which sound like they would survive an actual demonstration (especially the ones that require infinite time to execute).

As a PS, Dave, you have well and truly hijacked this thread.

I don't see how that is possible. It am directly, literally addressing the OP's question.

Now, if you mean I am dominating the thread with my views, that's something different from hijacking.

 

[jbriggs444]

I am dominating the thread

The PF way -- everyone has to make sure that their two cents worth is taken seriously.

 

[Klystron]

At the risk of being glib, how is that any more significant than simply saying everything within my personal field of view (be it sky, ground, or my own feet) is an unlimited EM system?

Sorry, you lost me. The CMBR, first detected IMS in 1963 using a modified radar receiver, has no significant relation to your examples. Within our technological limits we detect the CMBR in all directions as far as we can see.

If you mean to argue that all EMR emissions in free space propagate unlimitedly, then I tend to agree except that EMR eventually dissipates below threshold of detection. According to Weinberg, Wilson and Dicke, among others, the CMBR persists since shortly after the Big Bang.

I also considered offering 'cosmic rays' as an example but disqualified them since theoretically cosmic rays have preferred directions and identifiable sources such as stellar objects, supernovae, galactic cores, etc. While pervasive, cosmic rays have distinct limits compared to CMBR.

 

[DaveC426913]

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.

It is only ad infinitum in theory. This is explicitly about what exists.

Again, the OP is not asking about theoretical infinities or near infinities; that is a trivial question to answer.

He is asking if there is anything in the physical world that is infinite. "can" and "might" don't cut it here.
That is entire crux of this thread (interpreting the opening post).

 

[Hornbein]

There is no neccessity to believe that the approximations of calculus hold in the domain of the extremely small. Physics is based on the "close enough for jazz" principle. If this is ever found to be wrong it will be celebrated as a fertile source of PhD topics.

I used to think that "infinity" was naught but a useful shorthand until observations combined with theoretical considerations showed that the Universe itself might very well be infinite. I wouldn't give up on a proof. Maybe someone will come up with a way of showing that if the Universe is not flat then a contradiction results.

Real numbers aren't at all real. They are useful and traditional shorthand for successive approximations. On the other hand, imaginary numbers are fundamental and not at all imaginary. In physics one learns to place no weight on names. In physics names evolve randomly and often have a meaning unrelated to or even opposite their original purpose. This is an endless source of confusion to outsiders. You just have to get used to it.

When explaining such things to outsiders I often begin with telling them they have to clear their mind of any meaning of things like "particle" and start with a blank slate. Their original conception is an obstacle that will block all learning.

 

[Dale]

There is no neccessity to believe that the approximations of calculus hold in the domain of the extremely small.

There is also no necessity to disbelieve. And since the calculus models do work and non-calculus models don’t, the belief has merit.

 

[PeroK]

Within our technological limits we detect the CMBR in all directions as far as we can see.

The detected CMBR is a finite number of photons per second. It can't be otherwise is the point.

 

[Klystron]

I want to 'like' the last post but cannot find an emoji for "Duh".

 

[PeroK]

Here's an example of the difference between maths and physics.

If we imagine one event at time $\frac 1 2$ second. Then another event at time $\frac 3 4$ second. A third event at time $\frac 7 8$ second etc. Then, mathematically we have an infinite sequence of events within $1$ second.

But, you cannot do this physically. You might argue that there is no limit to how quickly the events can happen one after the other. Whatever limit anyone specifies you might be able to do better. But, you cannot go on indefinitely making the increment smaller and smaller. If you try to generate an infinite number of events in finite time, you must hit a limit somewhere.

One example might be how fast a computer can count. It's impossible to put an absolute limit on it, but any given computer must have a limit.

 

[Klystron]

Concur. Another non-electromagnetic physical example to answer the OP occurred to me while falling asleep: the fractal length of the coast of England*.

This measurement proceeds merrily using smaller then tinier 'fractal angles' to increase the measured length but eventually encounters some practical limit. Dale's first post succinctly addresses this contretemps, but this thread has been instructive.

*Tradition uses England or Madagascar in geometry textbooks. Any island makes an example.

 

[anorlunda]

If you try to generate an infinite number of events in finite time, you must hit a limit somewhere.

Is that true if the events are not at the same place in space? I'm thinking spacetime rather than time.

If there are an infinite number of stars, isn't there collectively an infinite number of scattering events within those stars at each second of time? (Dodging the question of how to define a second in that context.)

So I'm thinking that you must hit a limit only if you constrain the events to be contained at the same place.

 

[PeroK]

If there are an infinite number of stars, isn't there collectively an infinite number of scattering events within those stars at each second of time? (Dodging the question of how to define a second in that context.)

That's a big if. Let's assume it's true. Describe your experiment. How do you organise such as experiment. It's not enough, IMO, to wave your hands at the infinite cosmos and say it's happening!

 

[DaveC426913]

Is that true if the events are not at the same place in space? I'm thinking spacetime rather than time.

If there are an infinite number of stars, isn't there collectively an infinite number of scattering events within those stars at each second of time? (Dodging the question of how to define a second in that context.)

Simpler. If we accept the overarching premise that the universe is infinite in extent, then infinite phenomena in space (or in space time) are as trivial to find as snowflakes underfoot in winter.

 

[PeroK]

Here's a thought. If we assume that classical mechanics were correct, then a particle in motion would represent an uncountably infinite sequence of events every second - simply by virtue of being at every point $x(t)$.

That implies that nature is generating an infinite amount of information (re that particle) and it's only our experimental constraints that pick out a finite sample.

However, if QM is correct, then it is meaningless to talk about that classical trajectory and the only information that nature generates is the finite information that we obtain from our experiments.

And, if we apply QM principles to an infinite universe, then it may be equally meaningless to talk about those events as happening unless we obtain information about them by experiment.

In a way, the underlying principles of QM put a finite limit on what we can say about even an infinite universe.