When does it make sense to say that something is almost infinite?

I remember hearing someone say "almost infinite" on one of the science-esque youtube channels. I can't remember which video exactly, but if I do, I'll include it for reference.

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. Since there are different sized infinities, I knows there's more to the story than what I understand, so I was wondering if there's some context where it makes sense to say that something was almost infinite.

edit: It may have been on the "smarter every day" channel.

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It does sound strange, but without proper context it's impossible to answer. – Asaf Karagila Jul 14 '13 at 0:18
It's just a way some people say "a lot." – Qiaochu Yuan Jul 14 '13 at 0:21
I suspect they mean needlessly or meaninglessly large. – mixedmath Jul 14 '13 at 0:22
Within the context of mathematics, I don't think hat it is ever appropriate. However, it would make sense to say that for all intents and purposes, the universe is infinite regardless of whether or not it genuinely is. – Gamma Function Jul 14 '13 at 0:23
@AsafKaragila, Actually, I'm almost positive that it was used incorrectly in the context that I heard it. I believe the narrator may have been talking about the number of species of a certain family of insects. I was still curious to know if it was ever a correct thing to say. Based on the comments added since I've started typing the comment, it appears otherwise! – mowwwalker Jul 14 '13 at 0:24

Based on the comments suggesting that this was used to describe the number of species of a certain family of insects, or something similar, I would say that this is a perfectly correct prosaic use of the term infinite.

This misuse of the word "infinite" alludes to the fact that there are many many many insects in that family. Much more than we can imagine. This is very similar to how we say that solar power is unlimited power, and that the internet has an infinite supply of pictures of cats.

But note that this is indeed not a mathematical context. In a mathematical context something is finite or infinite, but not both. Especially since the modern definition of infinite is "not finite".

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The number of twin primes is in a quantum superposition of finite and infinite. Only when we observe a proof will it collapse to one or the other. – Gerry Myerson Jul 14 '13 at 0:33
@GerryMyerson: No, it isn't. There's absolutely nothing quantum in twin primes. Quantum superposition is a well-defined concept which is not identical with "we do not know" (indeed, it is in some sense the antithesis of it). – celtschk Jul 14 '13 at 0:34

In standard mathematics, this is indeed a meaningless concept. Some people have attempted, apparently unsuccessfully as yet, to develop a framework of ultrafinitism, which would give this concept some meaning.

The notion has more potential to make a vague sort of sense in a scientific framework, where numbers more than a few orders beyond the number of atoms in the observable universe have very little to do with anything in the "real world".

In computer science, an algorithm could reasonably be said to require an almost infinite amount of memory if any conceivable physical memory (a few bits per atom) would be unworkably large (at the extreme end, large enough to become a black hole).

An algorithm may be reasonably said to require an almost infinite amount of time to run if it would be physically impossible to run it to completion within the lifespan of the universe.

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Ultrafinitism doesn't give "almost infinite" meaning, it makes "infinite" meaningless, therefore rendering "almost infinite" meaningless as well. – Asaf Karagila Jul 14 '13 at 0:33
I should perhaps have explained that better. If there were a sensible ultrafinitist framework, then any natural number that exists in normal mathematics but not in that framework could be termed "almost infinite". – dfeuer Jul 14 '13 at 0:35
But the problem with ultrafinitism is that it doesn't have a concrete, well-defined and sensible framework; and that those who do believe that mathematics should be ultrafinitistics don't know what is the "largest number", and they deny the existence of numbers beyond it - so those cannot be called "almost infinite" by someone who calls them "crazy nonsense" to begin with. – Asaf Karagila Jul 14 '13 at 0:37
I was only pointing out that there are/have been people who have looked at this concept, and given it a name. I was not suggesting it was sensible. – dfeuer Jul 14 '13 at 0:39

If "almost infinite" makes any sense in any context, it must mean "so large that the difference to infinity doesn't matter."

One example where this could be meaningful is if you have parallel resistors and one is so large compared to the others that it doesn't measurably affect the total resistance. Then you could say the resistance of that single resistor is "almost infinite" in the sense that while it is actually finite, it wouldn't make any difference if it were infinite. In other words, for all practical purposes you can treat the resistor as an open connection.

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Saying that something massively big is 'almost infinite' is no different from saying that 1 is almost infinite, since the difference between infinity and 1 and between infinity and massively big is exactly the same - namely, infinite.

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Here is a practical example. Suppose that $A$ and $B$ are two genes, each of which is found in exactly $50\%$ of the population. Suppose also that we have established that there is zero correlation between the occurrence of one gene and another.

So what percentage of the population has both genes? $25\%$, right?

Well, no—in fact, it is impossible that the answer be $25\%$ unless the number of people on Earth is a multiple of $4$. And even if it were, we should still expect a deviation from the mean (on the order of $\sqrt{n}$, if I remember my statistics).

But whether or not the population is a multiple of $4$ is clearly an irrelevant detail (not to mention totally impossible to determine experimentally). So it is standard practice to do some calculations like this as though there were an infinite number of people. Populations are often modeled as distributions, rather than individual persons, and we can get away with this when the population is, well, close enough to infinity.

The real numbers would be almost useless in practice without the ability to pretend that some things are infinite. A metal beam is not a continuous object, but a finite collection of molecules. An economy is not a distribution of wages and trade preferences, but a finite list of governments, firms, and consumers. But when these lists are, one might say, almost infinite, we can understand them more readily as their continuous, infinite counterparts.

To take an extreme example, here is a memorable quote that I heard at camp many years back. Someone asserted: "Population is a continuous function..." but then he stopped briefly, correcting himself: "...I mean a continuous integer-valued function..."

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This is a common misuse of the word infinite. Anything that is infinite or approaching infinity, is not quantifiable, regardless of the context.

There exists no such quantity that can ever get close to infinity. Therefore, it will never make sense to say that a quantity is "almost infinite".

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Transfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite. The term transfinite was coined by Georg Cantor, who wished to avoid some of the implications of the word infinite in connection with these objects, which were nevertheless not finite. Few contemporary writers share these qualms; it is now accepted usage to refer to transfinite cardinals and ordinals as "infinite". However, the term "transfinite" also remains in use. Lucian from Romania

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