When does it make sense to say that something is almost infinite?
Sebastian Wright 
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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$\begingroup$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".
$\endgroup$ 2 $\begingroup$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.
$\endgroup$ 0 $\begingroup$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.
$\endgroup$ 4 $\begingroup$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..."
$\endgroup$ $\begingroup$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.
$\endgroup$ $\begingroup$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".
$\endgroup$ $\begingroup$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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