Does it make sense to say that something is almost infinite? If yes, then why?

I remember hearing someone say "almost infinite" in this YouTube video. At 1:23, he says that "almost infinite" pieces of vertical lines are placed along $$X$$ length.

As someone who hasn't studied very much math, "almost infinite" sounds like nonsense. Either something ends or it doesn't, there really isn't a spectrum of unending-ness.

Why not infinite?

• The curvature of a sphere decreases with its radius (it appears flatter if you're standing on it). Earth is large enough that for most everyday purposes, it might as well be a flat surface (i.e., have an infinite radius). – chrylis -cautiouslyoptimistic- Jun 11 '20 at 12:19
• The related concern/concept is "much greater/smaller than" (and the conclusions are the same as in the answers) – WoJ Jun 11 '20 at 13:45
• @JoshEller That's not actually true all the time-- there are circumstances in which those two infinities are actually equally infinite: most notably, they have the same cardinality. – Please stop being evil Jun 11 '20 at 16:16
• Had the same question a bit ago, but I think the answers here are much better math.stackexchange.com/questions/443099/… – mowwwalker Jun 11 '20 at 17:11
• @user76284, "almost zero" is just as meaningless as "almost infinite." For any non-zero number, $N$ I can say how close some other $n$ is to $N$ by expressing the difference as a fraction of $N$ (e.g., "My number $n$ is within 0.001% of $N$.", or $|N-n| < 0.00001N$), and then maybe we can have a discussion about what fraction counts as "almost." But, there's no equivalent way to say how close my number $z$ is to zero. The only way I can express it is to give the difference between $z$ and zero, which of course, is just me telling you the value of $z$. – Solomon Slow Jun 12 '20 at 14:06

Almost infinite can make a lot of sense in physics. There isn't a precise definition but I would interpret it as the following: when something is 'almost infinite' the properties we are considering will barely change when we make the system actually infinite.

Examples:

• In thermodynamics the particle number is often of the order of Avogadro's number $$N\approx 6.022\cdot10^{23}$$. For most properties considered this is basically infinite.
• Let's say we have a gaussian distribution $$f(x)=e^{-\pi x^2}$$. The integral over the whole number line is $$\int_{-\infty}^{\infty}e^{-\pi x^2}\text d x=1$$, but most of the area is in a small portion centered around zero. If we take instead $$\int_{-L}^{L}e^{-\pi x^2}\text d x$$ then this will approximate 1 to many decimal places even if $$L$$ is as small as 5. If you take $$L=100$$ then, as far as $$f$$ is considered, $$L$$ is infinite. In quantum mechanics this $$f$$ could be the wavefunction of a particle for example.
• I wonder, would "effectively" better convey this than "almost"? For a concrete example, a conductor with conductance that is effectively infinite means that, in the context and to the precision one is working, taking $\sigma\rightarrow\infty$ does not change the result. – Alfred Centauri Jun 10 '20 at 15:57
• @AlfredCentauri That's a good point. I would say that 'effectively' is more accurate in this context, so which one you use depends on how much you value accurateness over ease of language. – AccidentalTaylorExpansion Jun 10 '20 at 16:04
• @AlfredCentauri Instead of saying something is "almost" or "effectively" infinitely long/large/far/etc., I like the term "arbitrarily" large/long/far/etc.. It conveys the idea that you can make the number as big as you want, and if you're not convinced, you can make it even bigger. There should be no appreciable difference between using a very large number, and that number times a billion. – Nuclear Hoagie Jun 11 '20 at 13:14
• "[Statistical mechanics] works because Avogadro's number is closer to infinity than to 10." — Ralph Baierlein – Michael Seifert Jun 11 '20 at 19:08
• @eagle275 one should distinguish the case when we take the limit of $x\to\infty$ for some quantity $x$ to simplify calculations from the case when we get infinity as the result of calculations (or as an inconvenient intermediate value, as is often the case in QFT). – Ruslan Jun 11 '20 at 21:32

"Almost infinite" is a sloppy term that might be used to mean "effectively infinite", in a given context. For example, if a large value of $$x$$ in $$y = 1/x$$ produces a value smaller than the accuracy of measurement of $$y$$, then it's often reasonable to set the value of $$y$$ to zero, which is equivalent to setting the value of $$x$$ to infinity.

• I suggest comparing to a function with some scale. Coulomb is famously interesting because it is long-range in a way exponentials couldn't be. – Orion Yeung Jun 12 '20 at 21:05
• That's a good idea. You should write it up as an answer. – S. McGrew Jun 12 '20 at 22:16

Your observation that "either something ends or it doesn't" is correct, but at the same time not particularly useful.

Mathematical language exists to convey ideas; and sometimes slightly sloppy language, which does not correspond to any well-defined mathematical object or property, can help convey those ideas.

Suppose that for some application we are interested in the function $$f(x) = \frac{6}{1-1/x}$$. As $$x\to\infty$$, this function goes to $$6$$; and furthermore, we can even extend the domain of the function to include $$\infty$$, and say that $$f(\infty)=6$$.

But suppose $$x$$ isn't quite infinity, but it's big, say $$x=10^{12}$$. So $$f(x)$$ will be quite close, but not equal, to $$6$$. Suppose that the tiny difference is not meaningful for our application. Then we very well may say that $$x$$ is almost infinite, and that therefore $$f(x)$$ is almost $$f(\infty)$$, which is 6.

So even though $$x$$ isn't actually infinite, the distinction is not essential for our application, and we can communicate this observation by saying it is almost infinite.

In general, as a student advances in her studies of math, first she learns how to do things rigorously, and later she learns how to not do things rigorously. That is, she understands the underlying ideas well enough that she knows when she can sacrifice the accuracy of the language - when it is safe to do so without sacrificing the accuracy of the underlying ideas - in order to facilitate communication.

That said, I don't know if the specific phrase "almost infinite" would be commonly used for this purpose. Among other reasons, because the word "almost" is used in many contexts for properties which do have a specific rigorous meaning.

I'll also note that I didn't watch the linked video, so I can't comment on how it used the term.

• TL;DR: it's short for "One needs a limit argument here but we won't make it formally; we all know how to do that, right?". – Federico Poloni Jun 11 '20 at 10:41
• @FedericoPoloni: For the usage in the video, maybe. For the arithmetic example I gave, I disagree that we need a limit argument. We can use the Real projective line as a bona-fide algebraic structure, which includes infinity as an element. – Meni Rosenfeld Jun 11 '20 at 12:11

In layman's terms, something is “almost infinite” if it is so large that it would make no difference if it was any larger. This can be formalized with the mathematical notion of limit, as shown in previous answers. Here, I would just like to add a simple illustration. Here is a picture of my 35 mm lens:

See the infinity marking I highlighted on the focusing distance scale? This indicates the correct focus for photographing a subject that is infinitely far. Whether it is a mountain range a few kilometers away or a star field a few parsecs away makes no difference. As far as the lens is concerned, anything further than 50 m or so may be considered “at infinity”.

This can be understood by looking at the lens equation: a subject at infinity would produce an image at the lens’ image-side focal point (in the sense of a mathematical limit). If the distance to the subject is much larger than the focal length, then the position of the image is also, to a very good approximation, at that focal point.

How far is infinity obviously depends on the context. A greater film or sensor resolution, a better lens quality, a longer focal lens, or a larger aperture, all push “infinity” further away. It could be argued that the hyperfocal distance is the shortest distance that could be considered at infinity.

In physics if a quantity, call it $$\lambda$$, in a theory was said to be "almost infinite", I would interpret this as stating the effective theory obtained by taking the limit $$\lambda \to \infty$$ is accurate up until some very long length scale or time scale after which it breaks down.

Crucially this breakdown length/time scale is much greater than the intrinsic scales of the effective theory (at least in some useful regimes), so there is a very small approximation error induced by using the $$\lambda \to \infty$$ effective theory on its own intrinsic timescales.

I think a lot of the answers here have missed the key point that $$\lambda$$ is only meaningfully close to $$\infty$$, if the theory's predictions are close to those of the $$\lambda \to \infty$$ effective theory.

Obvious examples are

• the the speed of light $$c$$ in classical mechanics
• the inverse planck constant $$\hbar^{-1}$$ in general relativity
• the Heisenberg time in many body quantum physics
• the stiffness of a billiard ball when playing pool/snooker/billiards

More boringly I would just note that "almost infinite" is just the reciprocal of "almost zero".

Even (and especially) someone who has studied math a great deal would concur with your second paragraph

As someone who hasn't studied very much math, "almost infinite" sounds like nonsense. Either something ends or it doesn't, there really isn't a spectrum of unending-ness.

The intended meaning of the offending phrase “almost infinite” is that the quantity $$x$$ in question is so big that the system concerned is well modelled by the theoretical limiting case $$x\to\infty$$ (which is often mathematically simpler). As others have remarked here, a better shorthand for this description is “effectively infinite”.

• So, it is used as a relative term? Infinity as a measure of bigness with respect to something? Could this sentence one could almost say that it is infinite'' more meaningful than saying it is almost infinite''? – Ishika_96_sparkle Jun 11 '20 at 9:48
• @Ishika_96_sparkle : (1) The estimate obtained from the limiting case is good relative to what the exact (but impractical) calculation based on some defined, very big, number would yield . (2) Bigness is always relative to something. Infinity isn't a measure of bigness; it's a useful mathematical abstraction that only makes sense in certain contexts. (3) In physics, it's problematic to say that anything is infinite. Sometimes, a convenient infinite mathematical model can portray a finite, sufficiently big, physical system to acceptable accuracy. – John Bentin Jun 11 '20 at 12:00
• For (3), the example could be a delta function? – Ishika_96_sparkle Jun 11 '20 at 13:35
• @Ishika_96_sparkle : Well, I didn't have such a thing in mind. From a mathematical perspective, there is no such thing as a delta function. Rather, it is a kind of operator. But that is all too much pedantic fuss for physicists, and they go happily charging ahead with such “functions”. Mathematicians too play fast and loose with such things informally and in private, but they excuse it as a kind of shorthand, and know that it has to be reset in a rigorous form in publications. – John Bentin Jun 11 '20 at 16:05

The video is being sloppy: to form a two-dimensional square out of one-dimensional lines, where both are ideal mathematical objects, not things that exist in the physical world, requires an actual infinite number of lines. An uncountable number, even.

Similar concepts can be made mathematically rigorous, though. If "almost all" elements of an infinite set have some property, depending on context, this means something like "there are only a finite number of exceptions" or "the size of the set of exceptions is a smaller infinite cardinal than the size of the whole set." For instance, almost all prime numbers are odd, almost all integers are either positive or negative, and almost all real numbers are transcendental. Related terms are "almost everywhere" and "almost surely".

"Almost infinite" makes sense as a description of something so extremely large that it cannot be comprehended, and for all intents and purposes it is infinite.

It's like any other description where it can be used if it makes sense within the context. One might say a car is big relative to other cars, but not to other trucks.

However, "almost infinite" mathematically is arbitrary. If a number, say 1e100, is very large, one could say it's "almost infinite", however, any non-infinite number when divided by infinity is zero, so any number you take is going to be 0% of the way to infinity, which is certainly not "almost".

Technically, it makes no sense to say, as nothing can be almost infinite, unless perhaps you're comparing a function that almost approaches infinity but doesn't, however the proximity to the function that would approach infinity is much greater than the proximity of other functions being discussed.

However, it can be used as a description to describe something as incomprehensibly large, such as 1e100 where the number has almost no meaning. It can be used in this context and I think that's a fair usage and way of conveying the number, however from a technical perspective it's wrong.

As many have remarked, things can be treated as if being infinite without making a noticeable difference for all intents and purposes.

That being said and well understood, the OP is right to be uneasy about the word "almost" since it connotes "getting there" and "close to" and well, even the biggest number you can think of falls just as short of infinity as any other one.

There are even different kinds of infinity, and none of them is almost like any other one.

No. All numbers are equally far from infinity---because by definition infinity is a quantity you can never ever reach. You can't reach it from one, from ten, from a gazillion.

If you tried, it would take you an infinity of time to reach it, no matter your starting point. If a gazillion was any closer to infinity, you would reach infinity in a shorter measure of time---except such a thing is possible only for finite numbers.

Since no number can be any closer to infinity than any other, it is nonsense to say that a number is so large that it is practically infinity. When people say this, they are using a figure of speech---a hyperbole---for emphasis. You can do this, because spoken language isn't quantitative and its rules are flexible enough to allow it---and in fact it can be good practice to use figures of speech like this when writing persuasive papers.

Figures of speech come natural to us, and you should use them. But you should also recognize them for all they are---not accurate expressions of how things are, just a means to highlight things of interest to us that we want others to remember.

But mathematics and physics and many other disciplines are quantitative, and there you have to be careful with such statements. You can make them for emphasis in technical papers if you know your audience will understand them to be figures of speech, and if such figures of speech aren't frowned upon (which they can be in strict scientific research papers).

So can a number be almost infinite? No. But can a number be so large that we struggle to comprehend it, so that it feels to us almost like... it's infinite? Sure. I know the age of the universe but I can't really truly comprehend that number the way I do my own age, say, so 13 billion years and infinity feel to me almost the same.

It's just that to feel is not to know, and to know measures we rely on mathematics, which tells us unequivocally that all numbers are equally far from infinity and therefore that no one number can truly be "almost infinite." It either is or it isn't, the same way the clock either says noon or it doesn't---there is no middle ground.

• I agree with your sentiment, but, IMHO, you are too categorical. Can a number, in Physics, be nearly zero? Sure, if there are other numbers with the same units and they are much larger. For example, we often ignore friction/drag in simple 'falling ball' problems, since it is 'nearly zero'. Well, take this number $x$, and number $1/x$ will be nearly infinite. By the same logic. The point is that in some cases numbers can be so large/small that they make no meaningful impact on the solution of the problem at hand. That's when we can call them 'nearly ...' – Cryo Jun 12 '20 at 22:25
• @Cryo, you're talking about the significance of numbers. Numbers can be significant or negligible in mathematical model. If you ignore friction because it is small, you're not saying it is almost zero---you're saying its value is too small to matter on the scale of the mathematical model. Likewise, if you have a fraction 1/x, and x is very large on the scale of the mathematical model, you can take it to be infinite in order to approximately solve that problem by replacing 1/x with 0. This in no way implies the very large number x is in fact almost infinite---only that it is very large. – alex Jun 12 '20 at 22:36
• Infinite for all practical purposes (i.e. to get the solution) is usually enough for me :-). I think, to say that something is not 'almost infinite', you need to define what 'almost infinite' is. I provided my definition. If your definition is 'nothing is almost infinite', would it then mean you are posting tautologies? – Cryo Jun 12 '20 at 23:41
• Well, sure, for practical purposes and for emphasis, yes, it's perfectly sensible to use figures of speech to exaggerate a fact. I do the same. In fact, I can't stand it when people start splitting hairs in normal conversation. But the fact remains that no number can be any closer to infinity than any other. And that's by definition. I answered as I did because the OP asked. Now, If the OP had written a different post in which he or she wrote "my god, the distance to Pluto is infinite" I would have chuckled and agreed because I'd know that's a figure of speech and yes, Pluto is freaking far. – alex Jun 13 '20 at 4:20
• And if I were back in my physics or engineering classes, and I was solving an equation with a fraction in which the denominator is much much larger than the nominator, then I would most definitely approximate that fraction as zero to simplify the solution, so long as an approximate answer is all I need---and in engineering approximate answers are practical answers and usually as good or better than precise answers because approximate answers are also quicker answers :D – alex Jun 13 '20 at 4:24

In the actual real physical world, there are many quantities which are very large, e.g. the speed of light (certainly compared the speed of everyday objects), size of the observable Universe, the age of the Universe, but I cannot think of a quantity which actually is infinite. Arguably infinite density of the singularity at the centre of a black hole may not actually exist because the laws of physics (general relativity) break down at this point and we do not have a full theory which explains what happens. Infinity is a mathematical construct