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Let's say we've built a machine, that prints a sequence of all natural numbers from 0 to $\infty$. We "could" do this in a Universe with an infinite amount time (if we also make many other assumptions I don't want deal right now with). What if we would like to do the same experiment, but this time we print the real numbers? The two infinities (the amount of all natural and real numbers) are very different in nature. One is bigger than the other. If one says the Universe will last forever, does by "forever" he mean the amount of time that can be counted by the first machine, which prints a number, for example, every second? Or by "forever" he means some continuous line where time is not something you can quantize? Or is the "physical infinity" something very different from mathematical?

I probably do not understand the concept of time whatsoever, because I think it might have something to do with "events occurring at a specific rate relatively to some other events occurring at some other rate". If there are no events (for example the Universe "dies", "cracks" or whatever), there is no time then and no sense of measuring it (either by counting its "seconds" or stating it's a continuum).

But I would really like to know what does a physicist mean, when he uses the word "infinity" when he's talking about time related to the future history of our Universe.

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  • $\begingroup$ "The two infinities (the amount of all natural and real numbers) are very different in nature" No, not in nature. This is a mathematical consequence. "some continuous line" that would be a primitive way to see the concept of time, but it is good enough. Don't worry... time is still a concept that can't be defined in a absolute way. Every theory has its own way to defined it. $\endgroup$ – raul Oct 17 '15 at 8:21
  • $\begingroup$ @raul That's true. By "in nature" I didn't really mean "in The Nature". I would just like to grasp the relation between those mathematical concepts of infinities and the statements made when describing the real properties of the Universe. $\endgroup$ – kuonirat Oct 17 '15 at 8:28
  • $\begingroup$ For example, in $\textbf{classical mechanics}$: Time is absolute for every inertial frame (an observer). If I say "it's 3pm" a person in a rocket would say the same (as long as we synchronize our watches). What it is not absolute, it's our position, that's relative and depends on the observer. Ergo, time is a parameter and our position is described trough dimensions. $\textbf{In relativity}$: time is no longer absolute (space still is), then, time is another dimension. $\endgroup$ – raul Oct 17 '15 at 8:30
  • $\begingroup$ So, all reduce to: "Is the theory working time as something absolute or relative?", then time is a parameter or a dimension. $\endgroup$ – raul Oct 17 '15 at 8:34
  • $\begingroup$ sorry, in the part of relativity I mean to say "(space still isn't)" $\endgroup$ – raul Oct 17 '15 at 8:39
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Well, since your machine prints a finite amount of numbers in a finite time, it always print only a countably many numbers.

You can just divide the time from now to infinity in equally sized chunks of "1" to get a countably infinite set. During each of those chunks you print some amount of numbers, which in total gives you another countably infinite set. So that process "quantises" time.

When physicists say "infinite" they mean more often than not that the quantity is bigger than anything else we compare it to. In case of the universe, no matter what $t$ you pick, the universe still exists.

In calculations, we often write $\lim_{T\to\infty}$, where $T$ is the quantity of interest. That makes it very easy to make manipulations without worrying too much about how big $T$ actually is.

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    $\begingroup$ "When physicists say "infinite" they mean more often than not that the quantity is bigger than anything else we compare it to" - I like this description. It means that the $\infty$ symbol is used as an abstraction not really tied to a particular mathematical concept, but for the usefulness of someone's temporary considerations. $\endgroup$ – kuonirat Oct 17 '15 at 9:01
  • $\begingroup$ @kuonirat Your comment is spot on and reminds me of one my final year highschool maths teacher made to us, telling us to think of the symbol $\infty$ in this way,to recall that $\infty$ is not the symbol of a number and that if you want the number symbols you need to be precise and choose $\aleph_0,\,\beth_1,\,\omega\,\cdots$ as appropriate. That was 35 years ago: he used to give us a potted presentation of an interesting beyond-high-school maths topic every morning for the first 10 to 15 minutes of his class. $\endgroup$ – WetSavannaAnimal Oct 18 '15 at 22:11
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Let's say we've built a machine, that prints a sequence of all natural numbers from 0 to $\infty$. We "could" do this in a Universe with an infinite amount time (if we also make many other assumptions I don't want deal right now with).

No. You can't ever build such a machine. You machine has a finite number of parts that can be arranged a finite number of ways, so there is a finite number of numbers it could print out.

What if we would like to do the same experiment, but this time we print the real numbers?

That doesn't even make sense. Again, you can only print numbers that have a finite description, the describable numbers. And mathematicians would describe that as a mere small (and boring) subset of the real numbers.

The two infinities (the amount of all natural and real numbers) are very different in nature.

The natural numbers and the describable numbers aren't that different, one is an encoding of the other. The alleged undescribable numbers are different.

They are pretend.

One is bigger than the other.

You can't say a pretend thing is larger than something else. And even mathematicians don't say one is larger. They instead talk about an abstract partial order (a reflexive, transitive, antisymmetric relation) that bears similarities to size comparisons and then they use isomorphic (correspondingly similar) language.

And if you tried to claim one was actually bigger, what do you do about the downward Lowenheim-Skolem theorem? Question beg it by assuming larger things in your language or logic? And what's the point if one collection is full of pseudoscience (the undescribable numbers are by definition never going to be seen).

If one says the Universe will last forever, does by "forever" he mean the amount of time that can be counted by the first machine, which prints a number, for example, every second?

We mean that for any region of spacetime and any level of accuracy there are models that agree with observations in that spacetime up to that accuracy and that for any describable number D there regions of spacetime such that the corresponding parts of the model have time like curves that extend from that region to have a proper time that exceeds D.

And we could have used just the naturals or just the square numbers or any unbounded sequence. Or you could just say the proper times that curves extend are not bounded.

In science you never have to bring up fictional alleged undescribable numbers since be definition they don't actually show up in either a theoretical prediction or in an experimental observations.

They can be safely confined to storytelling about the mathematical models. Which is fine, if having them makes it easier to talk about the models, then have then if you want. But they aren't where the important things happen. The can't be, they never will be, and they were merely alleged anyway.

Or by "forever" he means some continuous line where time is not something you can quantize?

When a physicist says continuous and a mathematician says continuous they aren't talking about the same thing. A physicist is contrasting models, and so is distinguishing models with discontinuities from models without such discontinuities. The physicist isn't worried about which numbers exist beyond the ones that are needed to make the predictions.

Or is the "physical infinity" something very different from mathematical?

Yes. The physical one is agnostic at best about alleged numbers that are never actually seen.

I probably do not understand the concept of time whatsoever,

Fundamentally you should learn what an event is. That is the proper unified generalization of a moment and a location. Then you can learn about durations, which is a generalized length of curves made up of events.

because I think it might have something to do with "events occurring at a specific rate relatively to some other events occurring at some other rate".

No, its just a length. Just like a space has a length of a curve, so a 4d spacetime has a generalize length associated to a curve in 4d and that is duration. And in general they don't have a super close relationship to any naive notion of time or duration in particular that everything is comparable. A length of a curve is a length of a curve and it is pretty unrelated to other curves which have their own lengths.

If there are no events (for example the Universe "dies", "cracks" or whatever), there is no time then and no sense of measuring it (either by counting its "seconds" or stating it's a continuum).

If there are no events there are no questions to ask and no universe. Either do physics or do not, there is no try.

But I would really like to know what does a physicist mean, when he uses the word "infinity" when he's talking about time related to the future history of our Universe.

Not bounded.

is it really the case you can't get for example $\sqrt{2}$ in a theoretical prediction or in an experimental observation as an actual value? What about rational numbers?

You can describe the natural numbers. You can describe the integers. You can describe he rationals. You can describe the algebraic numbers (which includes $\sqrt 2$ and every other zero of a polynomial with integer coefficient). And that's all still countable. But you can describe more. You can describe $\pi$ you can describe $e$ you can describe any computable number (one with a finite program for a fixed Turing Machine that takes an arbitrary natural number n as input and then prints the nth digit of the number). You can even describe some uncomputable numbers such as Chaitin's constant, the number between zero and one whose nth binary digit is zero or one depending on whether the nth program for a fixed Turing Machine ever halts or runs forever.

The describable numbers contain any specific fixed number that could ever be described with a finite description. And it can get a bit fussy to nail that down exactly and precisely. But if the number can be described clearly enough that there is an obvious and objective answer to what its digits are, then that roughly means it is describable.

What it does not include would be numbers with random digits. And since you can list the descriptions, a mathematician will say that "most" real numbers are indescribable. But they are also pointless for science. They don't come up in specific predictions, they don't come up in specific measurements or in specific observations. They can't be specified in a procedure so that a specific protocol can be repeated.

What a mathematician calls the real numbers is supposed to be vastly bigger and more more complicated than what can be described. But obviously they can't exhibit a single example concretely, they could try to assert that there is a number between zero and one whose binary digits encodes a belonging relationship on the natural numbers that satisfies the ZFC axioms. But firstly we don't know if such a number exists. Secondly they can't actually prove it exists. And thirdly, even if such a number exists there are many such numbers and they still haven't described a single one of them.

The indescribable numbers are allegedly vast but we've never seen a single one and we never will. And we don't actually need them. If pretending they are there makes things easier or more convenient, maybe they are harmless, like a ghost story. But if you start to worry about them like they matter, then this is a sign of a deep problem or that you have waded into pure nonsense and you should go back and learn from your mistake. Just as if you started to worry about specific ghosts as if that was affecting your experiments. The fiction should have been harmless and convenient or you shouldn't have done it at all.

If your number has digits and you can describe them all and a describable specific order in some finite description then its a perfectly fine number. And there aren't oodles of those numbers.

We don't need numbers to act as Oracles, secretly knowing and pre-encoding mystically unknowable outcomes by an unknowable non procedure.

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  • $\begingroup$ Thank you for the detailed answer. I'm not going to argue with the feasibility of constructing my "machines", because I do know, you can't actually build them. I also had troubles with the idea of printing "all real numbers", since I couldn't imagine an algorithm for this. It seems like the distinction between describable and non-describable numbers is well defined and I wasn't aware of that. But here is another question: is it really the case you can't get for example $\sqrt{2}$ in a theoretical prediction or in an experimental observation as an actual value? What about rational numbers? $\endgroup$ – kuonirat Oct 18 '15 at 9:55
  • $\begingroup$ "you can only print numbers that have a finite description" - by writing $\sqrt{2}$ - did I just describe the number or not? I guess I could also describe any rational. Still I'm pretty confused, since (if I understand You correctly) not every real number is describable, so we could get only some subset of the real numbers as a result of an actual physical experiment. $\endgroup$ – kuonirat Oct 18 '15 at 10:15
  • $\begingroup$ @kuonirat, We could argue about what it means to "print" a number, but since virtually all real numbers are non-computable, (en.wikipedia.org/wiki/Computable_number), that means that virtually all of them are non-printable by any definition of the word "print". $\endgroup$ – Solomon Slow Oct 18 '15 at 17:47
  • $\begingroup$ @jameslarge We could describe numbers that aren't computable, but at least are still falsifiable. Such as constants whose values are determined by procedures that produce fixed and determinate results, such as running a fixed input on machines with larger and larger size for larger and larger times that are designed to report when they run out of memory. Then you can simulate a Turing machine running a fixed code and describe Chaitin's constant even though it isn't computable. So you can even talk about a larger class than the computables, but it still is countable. $\endgroup$ – Timaeus Oct 18 '15 at 18:43
  • $\begingroup$ @kuonirat Edited to respond to your concerns. $\endgroup$ – Timaeus Oct 18 '15 at 18:43
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The only honest answer is we don't really know, and may not know ever. If you think about it by trying to consider properties of $\mathbb{R}$, I'd vouch you are actually taking the maths "too seriously". There is no empirical evidence, and cannot be, that space time really is $\mathbb{R}$ (or well, some expanded/warped version i.e. an $\mathbb{R}$-manifold), something whose mathematical properties will also depend on things like what axioms of set theory (or some other suitable mathematical framework theory but ZFC set theory is the one usually used) you take as valid and what not. $\mathbb{R}$ is used because it's a convenient structure. From an empirical point of view, all measurements are finite in number and limited of precision, so they could be described in a sense as a strict finite subset of $\mathbb{Z}$. But writing down physics math would be hideously difficult and unintuitive in $\mathbb{Z}$. (This is why I just said there "cannot be any empirical evidence that space-time is $\mathbb{R}$" because that would require examining it to infinite resolution, an impossibility.) So we choose $\mathbb{R}$ as a convenience tool for modeling. To then extract "predictions" about the universe based on deep-analyzing $\mathbb{R}$ is like trying to analyze this post by scrutinizing the precise shapes of the letters of the font it is displayed in, and then claiming that tells you something meaningful or relevant about the content of that post. In particular, regarding your argumentation about "printing the real numbers", math says that any list of the real numbers must have ordinal type greater than $\omega$, in fact vastly greater. But here's the cinch -- just what ordinal type will depend on what you assume about your set theory. It is firm that at the least we can have $\mathrm{ordinal\ type}(\mathrm{list}\ \mathbb{R}) = \omega_1$, but it could be much larger. In fact you can arbitrarily jig your set theory with great freedom to choose what it is. This freedom shows you then it's like scrutinizing the deep details of our modeling tools.

That said, we can falsify that the universe is $\mathbb{R}$. Show that there's an absolute minimum increment of space and time, and then it's done. In lieu of that, we won't know. And I'd say it's not necessarily reasonable to consider it "verified to high confidence" like other theories even if we measure extraordinarily fine, because $\mathbb{R}$ as a choice is still a modeling choice to undergird what will always be finite precision anyway.

Pragmatically, you cannot build your infinite machine. Entropy will destroy it or render it privated of energy to keep going, before it can run 'infinitely long'. Nothing lasts forever in the universe. Thus any clock or ticker or whatever we can ever build will only measure as much time as entropy takes to kill it off.

Mathematically though, the longest "ruler" you can construct in $\mathbb{R}$ with regularly-spaced ticks (either a spatial ruler, or a clock ticking at regular intervals) will have ticks with ordinal type $\omega$, that is, the "type of infinity" is that of the natural numbers. But as just discussed, this says nothing about the real universe. Thus a "listing machine" in a hypothetical $\mathbb{R}$-universe will not list all the elements of $\mathbb{R}$. But if it's something bigger than $\mathbb{R}$, that may be possible at some (non-Archimedean, trans-finite) time in the future (but it will have to be a LOT bigger, I believe.).

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  • $\begingroup$ I like the argument about "taking the maths too seriously" and that about axioms. Especially when you consider the fact that there have to exist true statements that cannot be proven - we should probably not always extrapolate everything into the real world. However, sometimes even the weirdest properties (which a continuous set definitely has), might serve as hints or insights into what the reality really is. Thus my question, I guess. $\endgroup$ – kuonirat Jun 10 '17 at 20:24
  • $\begingroup$ I wonder - if there's "an absolute increment of space and time" (as you say), wouldn't that put into question calculus as a tool to describe the reality? I mean, it's very successful up to these days, so it would be weird to say one day: we thought we use calculus to approximate things, but in fact we were over-calculating them. The reality is really discrete... $\endgroup$ – kuonirat Jun 10 '17 at 20:25
  • $\begingroup$ I know my machine cannot be built for real, but you have a really good point in your last paragraph, that it would say nothing about the real universe, which was it's sole purpose. I guess I both can't and don't have to build it. $\endgroup$ – kuonirat Jun 10 '17 at 20:25
  • $\begingroup$ @kuonirat : An absolute minimum increment would not make calculus useless. It would still be a good approximation, because the increment would have to be far smaller than anything we can currently measure. But it would not be a fundamental theory. $\mathbb{R}$ is a model of reality, not reality itself -- that's the whole point. That's all we can get in science though really, is models, not "reality itself". Ideally we want our models to be useful, and they have to explain all existing data, but none are the absolute final truth of "reality". $\endgroup$ – The_Sympathizer Jun 11 '17 at 2:51
  • $\begingroup$ @kuonirat : For what it's worth I'd want to point out also that even if we find a minimum increment, that wouldn't automatically imply the structure must be $\mathbb{N}$ or $\mathbb{Z}$ either -- there exist weird "non standard" models of $\mathbb{N}$ that have transfinite elements. This would not be ruled out, and could not be for an infinitely long time (thus never, empirically, because the machine must inevitably entropize, same as with an $\mathbb{R}$-universe). Or it could even be the universe is finite -- at some point it just stops existing. $\endgroup$ – The_Sympathizer Jun 11 '17 at 2:53

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