The 'countable infinity', symbolized as 'Aleph zero', is the smallest of all infinities. In the examples that you have cited, the number of elements in the population is countable, not necessarily 'countably infinite'. However, in practice, that is what is assumed, and there is nothing wrong in assuming so. When you draw a sample from such a population, standard statistical techniques that would be used are based on mathematics. Where is the question of doubting the results thus obtained?

In case of fishes in a large lake the usual procedure is to catch a sample, mark and count every fish caught and then let them return to the lake. After some time we catch the fish again and count them all, paying attention to their marked fraction. A very simple calculation tells us how big is the total population. Nowhere the population is treated as "countably infinite". Quite contrary it is only assumed big and unknown, but always finite, at every step of the procedure.

So, why there are any doubts here?

First you need to apply some statistical methods to estimate the uncertainty of the result obtained this way. The main assumption, of course, is that the marked fish was dispersed uniformly in the entire population before the second sample was taken. This has the chance to be satisfied only after long enough time between both samplings, longer than doubled lake's diameter divided by the average speed of fish.

This estimate, in turn, may strongly depend of the season of the year. Anyway, you will never be sure that all marked fish did survive during this time ...

Kangaroo counting has a long literature, and highly contested, well worth looking up. It is important in resource management (economic and biological). This is a large visible population spread over a wide area, and not capable of being serially counted (as we count sheep, or dollar notes); nor weighed (as we count dollar coins); nor by colour intensity in a satellite image (as we count the activity of jelly fish). This is a finite version of alepho 0, which is different to the large number finiteness we deal with in other istuations. This is usually best handled using survey techniques, which rely on some homogeneity assumptions (in the sample frame) and robustness in the method of sampling - fish population, or capture re-capture estimation methods are typical

There are two key methods for estimating animal abundance, and these are distance sampling and capture recapture approaches. There are many additional flavors of these, the literature is extensive and I could not begin to describe them here. These are essentially methods which allow you to convert a sample of detected/captured animals into some density/abundance estimate, by providing a rigorous way of estimating the probability of detection of an animal. Basically, and intuitively, if you detect n animals and the probability of an animal being detected is p, then there are n/p animals. As an example, if in a pond you capture 10 fish, and the probability to detect a fish is 0.25, there should be 40 fish there. The trick is to estimate p and account for all the potential sources of variability, such that you get not only an estimate, but its precision. Provided the assumptions of the methods used are respected (and yes, they often are not, leading to issues and criticism) the estimates and their precision should be valid and reliable. Take a peak at this reference for a review on these methods: Schwarz, C. J. & Seber, G. A. F. Estimating animal abundance: review III Statistical Science, 1999, 14, 427-56

It is very simple: just establish point-to-point (one-one) correspondence with the set of natural numbers.

It is very discouraging to see this thread reset for the *third* time.

What is the problem??

Valeriy,

It is interesting to speculate on a new number where a 1-1 correspondence cannot be established (because this takes time and the population moves in that time); that is finite but which cannever be counted because the correspondence at countable level (say a freeze frame image) cannot be cumulated or extrapolated. Arguably this applies to fish counting and capture-recapture.

When a population is not fixed in time we refer to it as an open population. Again, when sampling animal populations, there's a dichotomy between open population and closed population methods. If one can safely assume that the population is closed (for the duration of he sampling period, at least) life is considerably simpler - closed population methods are much simpler. If that is not the case then one must (potentially) account for immigration, emigration, birth and death processes. That makes the problem much harder, with many more parameters to estimate. So if you can concentrate on a closed population that is advisable.

Thanks, Balazs, Tiago, Marek & Stephen

for the valuable contributions.

One important caveat of an approach like the one suggested by Balazs, which is a mark recapture approach, is what is know as the plague of unmodelled heterogeneity. There are a number of underlying assumptions for that simple capture recapture approach to work, and one of the key ones is that the probability of capturing a fish is independent of it being marked or not. However, it is quite possible that fish which were caught in the first instance are more likely to be caught in the second time (say they are the big fish, or those which like to live near the margin of the lake). This will lead to underestimation of abundance. On the contrary, if by being caught once they became less likely caught next time (a trap shy behaviour, say because the marking process is painful) the population will be overestimated. The size of the bias can render such estimates useless, and so care should be taken. Take home message: all such methods have a assumption, if they do not hold, problems (bias) arise.

Survey sampling uses the concept of clustering, usually to reduce the cost in travel to reach a scattered population. In the animal case we can also use clustering, but in a more biologically astute way: fish go about in schools; we can count schools and we can count school size. Both are related to resources and to predators and population dynamics (the reproduction/replenishment rate of the population). A Sensible sampling approach would be to sample schools (mobs, flights, herds) or territories marked by schools etc (or individuals for solitary animals), both of which can be considered more stable and better measured possibly using 'ecosystem grids', then a following stage estimate the (trend in) size of the socialised populations using captive sampling, such as extended observation; marking, transmitters etc; longitudinal techniques. I dont know how fish associate in lakes; it does seem that it would be a mistake to assume they occupy the space uniformly, so that all would pass one point eventually. If we knew a little more about how they did associate we may be able to cluster our sample design a little better...