I never realised that I had a "nearly infinite" amount of money.

Practically, J. D. is right - depending on context it could easily mean simply "more than you could make use of in the course of your life" in much the same way that "it would take longer than the life of the universe" is a usual standard in declaring some time-dependent random event to be "impossible" ( contrary to the saying, a million moneys would take longer than the universe has been in existance to actually produce Hamlet).

However, what kind context is such that 100 is "nearly infinite" on even this level? I'm putting my money of innumeracy for now.

I think your problem, as expressed above by other posters, is trying to reconcile Plato's "realm of mathematics" to the real world. It can't be done, there are no perfect mathematical forms in our universe (that I know of), and you're just going to have to accept that people will misuse words outside of their mathematical meaning.

I do remember reading a book called "1,2,3 ... Infinity" by George Gamow when I was young, as I recall it did a very good job of explaining mathematical and physical concepts in plain english.

What about the "necessary and proper" clause? necessary means indispensable, not convenient, and and just adds a second requirement, it's not an alternative. However...

Keep it up and you'll arrive at "nearly perfect".
(Or "absolutely relative", for that matter.)

Last sentene should say what you mean by 'ALMOST infinite'.

...First ask the listener: "What is Epsilon?" Suppose he says, "Epsilon equals .01″. Then you can say that any number greater than "Infinity minus .01″ is "almost infinite". That includes Infinity, of course. Whether a number such as "infinity minus .009″ is different from "infinity" is a question too hard for me.

for more in this vein, see http://www.rasmusen.org/x/2006/05/02/1158/

Is it really that much worse than "more unique"?

logicnazi, measure theory doesn't do it. There's no topology on the $\sigma$-algebra of measurable sets that doesn't make the ones of measure 1 dense in the whole. Every set is "nearly infinite".

The problem with these mathematical uses of "infinity" is that "infinity" is shorthand for giving up. The closest Greek term to this sense was "apeiron", which literally means "a mess". In order to apply rigorous tools to the infinite, you need to rigorize the notion of the infinite, which generally turns it into what Cantor termed the "transfinite".

To try to inject mathematical rigor into the original context, "nearly infinite" should be replaced as "effectively infinite" (as Eh Nonymous suggests), or by verbiage connoting unboundedness rather than the messy "infinite".

Economists consider 100-year bonds to be "nearly infinite" and calculate their present value the same they would for truly infinite bonds. Considering what making the expiration that large does to the function, it's perfectly reasonable.

Economists consider 100-year bonds to be "nearly infinite" and calculate their present value the same they would for truly infinite bonds. Considering what making the expiration that large does to the function, it's perfectly reasonable.

It's reasonable in the sense of giving an answer that is close to the mathematically correct answer, but I'd say that it's unreasonable relative to the additional effort of calculating their actual present value, which is trivial (the effort, not the actual present value).

For some reason, the discussion of "nearly infinite" brings to mind an old math joke: 1+2=4 for sufficiently large values of 1.

logicnazi,

I think what you're looking for is an "outer measure." To get technical about it, let the underlying set be Z, the positive integers, consider the ring of all possible subsets of Z (set-theoretic definition of ring, not the algebra definition of ring....), and define the outer measure, M, of any set to be n/(n+1) for any finite set, where "n" is the number of elements in the set, and 1 for any infinite set. (Outer measures need only be finitely SUB-additive, not additive, i.e. M(evens U odds) = 1 <= M(evens) + M(odds) = 2, since both sets are infinite.)

Then you could talk about finite sets being within some particular bound of infinity, if 1 - bound < M(set) < 1. Of course, this is a lot of mathematical baggage just to say that finite sets are practically infinite if they have more than, say, 200 elements, or whatever.

Personally, when I'm looking at a stack of 80 beginning algebra assignments to grade, 80 is practically infinite, while when I'm looking at days until the start of fall semester, 80 is practically zero.

Yea, what bobh said: wouldn't "nearly infinite" mean "approaching infinity asymptotically?" Eugene, you did computer stuff, so you probably know more calculus than everyone else in this virtual space combined, surely "nearly infinite" can be given meaning that way?

In fact, doesn't calculus have infinite limits? I know I vaguely recall such a thing from some explorations...

Also when I used to be a physics major there were plenty of situations that would come up that would justify some finite amount being called 'almost infinite'.

For instance consider an electron trapped in some potential well. If the energy barrier provided by the potential well is so huge compared to the energy scales the electron is at (though in human terms still small) then we might call the potential barrier an almost infinite potential barrier since we can approximate this situation very well by an electron in an infinite well.

Alright so perhaps it would be a bit better to call it an effectively infinite potential well.

--
Basically the reason I am objecting to this idea that it is absurd to talk about being 'almost infinity' is that it seems to all be a matter of perspective. I mean if we are comfortable saying (for real numbers) that y is almost 0 if |y|< epsilon then why not just say that z is almost infinity if |1/z|< epsilon (and if you want + infinity z is positive).

It all just depends on the application you are looking at things for. Not that I am disputing whatever particular case you saw which was probably eggregious.

so long as something is finite, it's not nearly infinite, no matter how much it grows

Well, I think bobh and Paul Gowder are on the right track (though I don't think CS folks are famed so much for their calc/real analysis skills as for their discrete-math chops, but as an economist I could be wrong on that).

In any case, most statistical research (both the good and the bad) that is done these days involves appeal to precisely the idea that sample size can be treated as infinite as long as it is large "enough" and/or can be thought of growing unboundedly.

The reason is simple: most estimators (including sample means, for those of you who think I'm delving into measure theory-like esoterica) that most people use involve sample averages of smooth functions of data. Such estimators are generally (except in pathological cases) what's called asymptotically normal. "Normal" refers to the probability distribution of the estimator---some people call the density function associated with the normal distribution a bell curve.

In this context, "asymptotic" means that the sample is large enough that it might as well be treated as being infinite. Asymptotically normal estimators are random variables whose distributions become arbitrarily close to some normal distribution (which one depends on the probability limit of the random variable as well as its limiting variance, which sometimes have to be defined with some care).

If you're familiar with the idea that statistical significance requires a sample mean to be at least two standard deviations away from zero (actually that's just one definition of significance), then you are implicitly familiar with the use of the kind of result I just described. This result is generally referred to as "the" Central Limit Theorem (quotes since there are multiple CLTs, each of whose appropriateness depends on the data generating process).

A CLT lets us act as if estimators---like sample average wages, or fraction of 1Ls in the law school whose LSAT score topped 170---can be treated as normally distributed, even when it's easy to show that the estimators have discrete distributions (as is true with both average wages and fractions with scores above a cutoff level), whereas the normal is continuous. CLTs imply that the more the sample size grows, the closer the sample mean's distribution is to an appropriate normal distribution. "Closer" in this context is a term of mathematical art that I won't pursue further here.

The advantage of this result is that it justifies using rules like

when the mean differs from zero by more than 2 units of (estimated) standard deviation, then we can reject with 95% confidence the null hypothesis that the true population mean equals zero.

Now, you might worry about the fact that the CLT explicitly involves taking the limit of the estimator's distribution as the sample size grows without bound (the mathematical term is "diverges", Paul Gowder). To paraphrase Eugene, no sample size ever actually is infinite, so how can we use the CLT?

Well, suppose that the sample size is so great that any deviations between the true (cumulative) distribution (function) and the normal are so small that they affect only digits far to the right of the decimal point. Well, then using the normal approximation and the true distribution will virtually never lead to different conclusions about the population mean.

So here's a working definition of "almost infinite" in the context of large-sample statistical theory (which, by its very name, might actually be called the study of "almost infinite" sample sizes):

When the sample size is large enough that a given rejection rule would not be affected by replacing the normal approximation with the true distribution function, then the sample size is almost infinite.

This definition, which is analogous to (a special case of?) more general suggestions in the comments above, has one disadvantage: whether or not the sample size is almost infinite will depend on the level of precision chosen for the rejection rule (e.g., do we use 2 standard deviations, 1.96, or even a more precise figure?). In other words, "almost infinite" has a contingent definition. But in practice that won't much matter, especially if there are clear conventions in place. And anyway, lots of mathematical concepts are contingent (for example, a faster convergence rate for an estimator requires that we have a large "enough" sample size to have any bite).

Anyway, the reason I bring up this specific example is that the whole field of large-sample stats is implicitly founded on the (presumed) appropriateness of the concept of "almost infinite". And every one of you who has ever written an empirical paper -- about law, econ, law &econ, physics, sociology, demography, etc -- has most likely appealed to these concepts, knowingly or not.

Silicon Valley Jim: "For some reason, the discussion of "nearly infinite" brings to mind an old math joke: 1+2=4 for sufficiently large values of 1."

Or for sufficiently small values of 4.