Wikipedia:Reference desk/Archives/Mathematics/2014 October 13

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October 13

Is there a name for something that is finite but not countable?

Is there a name for something that is finite but not countable? The countability of this thing is not in the usual mathematical way, because a Mathematician would argue any finite set is usually countable. What i am thinking about is a finite set where the collection of the amount of something is nearly impossible for some reason like current technology or the laws of physics or etc.

Examples:

• Number of grains of sand in all the world at one instant (no time elapsing) - It is impossible to know how many grains of sand are on Earth at one instant. Surely there are above trillions, but even if we could count every grain, perhaps by the time we were done, some of the first grains we counted would be blown into space somehow, or rocks would have eroded to create new sand that wasn't there before.

• The EXACT length of the coastline of the island containing the United Kingdom. We can approximate it using smaller and smaller lines to lay against the coast, but this measurement of coast length is always changing. The tide comes and goes and screws with the number all the time, and we cant get lines to be infinitely small so as to get an exact measure. Also, on a beach, where is the coastline? Is it at the part of land where water never touches with any wave? Is it where the sand is only covered in water 50% of the time? Is it where the sand is not smoothed at all by wave action? Is it where the sand is dry when the tide is out?

Im afraid i can't come up with more examples right now, but I'm sure there are plenty! What is this sort of number or set called to refer to the property that it is finite but uncountable? Is there anything more formal than "Finite but uncountable" ? I found Uncountable_set here on wikipedia, but it expressly refers to infinite sets.

ALSO, is there anywhere an exploration of the types of reasons these sets exist? The main one i have shown in my examples is where there are so many elements in the set that it is rather inconvenient to try to count them all at a given moment. I wonder what other reasons might exist.

Thanks for any help or resources that can be given!

216.173.144.188 (talk) 14:39, 13 October 2014 (UTC)

You have to distinguish between fundamentally impossible to count and impractical to count. It's the former that mathematicians confine themselves to. For your first example, if you had some sort of omniscient being, they would be able to count all the grains of sand on earth. It's not that it's fundamentally impossible to count them all, it's just completely impractical for us to do so given our human limits. That's a rather fuzzy line, though. Go to some aboriginal tribesman with only a rudimentary number sense, and two dozen eggs is an uncountable number of eggs. If we limit a single person to physical counting, ten billion is an uncountable number (more than three per second across a 100 year lifetime), though if we permit collaboration or mechanization, things become more and more feasible (if you got everyone in New York City on board, you could count out 10 billion objects in an afternoon). Given that what is practical depends on how you do the counting, it's not really a well-defined mathematical concept. For your second example, "counting" doesn't really apply, as what you're looking at isn't discrete. There is the concept of a non-measurable set, though, which looks a non-discrete metrics on sets, though. (Sets are really set up around discrete objects - so for a "set" representing the coastline, we're really talking about a (non-countable) infinite number of discrete points which make up the coastline, which then has a non-measurable but not infinite measure related to its length.). Uncountable set deals only with infinite sets, because if you have a finite, discrete set of objects, some level of omniscience could theoretically be able to count them, no matter how many there are. (Though I should also reference countably infinite for completeness.) -- 160.129.138.186 (talk) 16:46, 13 October 2014 (UTC)

The length of the coastline of Britain is ill defined. That's the appropriate technical term for that case. The number of grains of sand in the world is also ill-defined since it's not precisely clear what the world is, and not at all clear what a grain of sand is. But additionally, even armed with a precise definition, it would be impossible to count the grains, as you said. I think there's no mathematical term for that problem since mathematicians normally ignore such practical issues. Maybe ultrafinitism is related. -- BenRG (talk) 06:48, 14 October 2014 (UTC)

"if you got everyone in New York City on board, you could count out 10 billion objects in an afternoon" - and then you will have to count all errors they've made, etc. --AboutFace 22 (talk) 18:39, 17 October 2014 (UTC)

You might be interested in ultrafinitism. Some ultrafinitists (for example, the recently deceased Edward Nelson) doubt that the exponential function on the natural numbers is total. They might consider it possible, for example, that there is no such number as 10¹⁰⁰⁰, and therefore no genuine way to "count" all thousand-digit decimal numbers. (Oh, I guess I mean all thousand-digit decimal representations — to the hypothesized ultrafinitist, they might not actually be "numbers".) --Trovatore (talk) 18:43, 17 October 2014 (UTC)

Random numbers

It occurred to me that there is no true random number. Think of a number, called "Fred", so large it can be written only on a sheet of paper stretching from the Earth to Rigel. If the sequence of natural numbers is infinite, then the vast majority of numbers in the sequence are larger than Fred. No matter what set of numbers one selects for any practical task requiring random numbers, they will always be at the short end of all the possible numbers and are therefore pseudorandom. I can't believe this thought is new. There are too many smart people out there. However, I have not heard this argument put before. Comments? Captainbeefart (talk) 14:58, 13 October 2014 (UTC)

Typically when you talk about random numbers, you talk about drawing from a particular probability distribution. The uniform distribution is one possibility, but certainly not the only one. For example, even though a random number drawn from a Gaussian distribution could theoretically have all possible real values, most of the probability density is in the region around the mean, so the contribution of extremely large numbers are negligible. In addition, even with the uniform distribution people tend not to talk about random numbers drawn uniformly across *all* integers or all reals. Instead, you normally talk about uniform draws from a given range: a random number drawn uniformly from between 0 and 100, for example. Your observation about how even very large numbers are very small when compared to all possible numbers that can be is something that comes up quite frequently when one talks about the concept of infinity or of large numbers in general, but the connection with random numbers, not so much - usually because applications of random numbers don't typically involve a uniform distribution across all integers/reals. -- 160.129.138.186 (talk) 16:14, 13 October 2014 (UTC)

You are right about this: there is no such thing as a uniformly random natural number. This is because we cannot assign probability density function that gives a (constant) positive probability of each number being chosen that also sums to one. For any finite list of natural numbers, we can assign a uniform probability function. Another thing to point out is that virtually all "random numbers" that are sampled via a computational algorithm are pseudorandom. There a few devices for sampling truly random numbers, based on radioactive decay and other physical properties, but these are rarely seen outside of research labs. However, there is nothing wrong with saying, "let $x\thicksim \mathrm {N} (0,1)$" to indicate a truly random variable following the normal distribution. This is a truly random variable by fiat [1], and there is no upper limit governing how large a given sampled value might be. SemanticMantis (talk) 16:30, 13 October 2014 (UTC)

Benford's law might be relevant, but that depends on how the numbers are generated.

—Wavelength (talk) 18:37, 13 October 2014 (UTC) and 02:04, 14 October 2014 (UTC)

"Numbers" can't be random. 9 isn't random, and $\pi$ isn't random either. A process to choose numbers can be random. Saying "X is a random integer between 1 and 10" isn't as correct as saying "X is a randomly chosen integer between 1 and 10" or "X is a random variable following the discrete uniform distribution on [1, 10]."

Anyway, you seem to be saying there is no uniform distribution on all integers. That is of course correct and can be easily proven. However, such a distribution can be used as an improper prior. There are also many non-uniform distributions on the positive integers, such as $\mathrm {Pr} (n)\propto 1/n^{2}$. -- Meni Rosenfeld (talk) 09:10, 14 October 2014 (UTC)

This comment probably doesn't belong in this thread, but it is somehow related. Probabilistic reasoning can be quite mind-boggling, and can lead astray. Think about throwing darts at the real numbers. More concretely, check out Axiom of symmetry. It was used to argue (strongly) against the continuum hypothesis. It is, initially (to me) quite convincing. Then think about throwing darts at the natural numbers, and modify the axiom of symmetry accordingly. (Map a number to a finite set of numbers.) This "modified axiom of symmetry" is equally convincingly true at first glance, but is (easily) provably false. I certainly don't trust my own probabilistic intuition for infinite sets. YohanN7 (talk) 18:56, 14 October 2014 (UTC)

Yep there's loads of paradoxes when one can choose 'any amount'. I quite like the Two envelopes problem. One fairly reasonable property of a random natural number for instance is that it has a one in seven chance of being divisible by seven, but sticking down the notion so it is watertight is tricky. Dmcq (talk) 10:32, 15 October 2014 (UTC)

See German_tank_problem#Bayesian_analysis for an example where an unknown number N is estimated. Initially it is only assumed that 0≤N<∞, and so no uniform distribution exists. To overcome this obstacle assume that 0≤N<Ω where Ω is some big but finite number, and after the computation is finished, let Ω→∞. Bo Jacoby (talk) 12:48, 15 October 2014 (UTC).

Captainbeefart, while it looks like your question is directly about what it might mean for a natural number to be "random", you might still be interested that there is a whole field of study (or arguably more than one) dealing with the corresponding question on the real numbers.
Maybe the easiest introduction to the topic is to think of a real number in terms of its decimal expansion. The decimal expansion of a real number (every real number, even the real number 0) goes on forever (though in some cases it's a bit repetitive).
Now we can consider the question of whether that decimal expansion can be distinguished from an infinite sequence of decimal digits generated randomly (say, by throwing a 10-side die infinitely many times).
Well, if we're allowed an oracle, then of course the decimal expansion of any given real number can be perfectly predicted, by taking an oracle defined by that real number itself. So in some tautological sense no individual real number can be completely random.
However, it can still be pretty random. For example, it might be impossible for any Turing machine to predict the sequence any better than would be expected by chance. This and similar definitions are the field of study of algorithmic randomness, which is a current hot topic in mathematical logic (or at least it was a few years ago — I'm not entirely up-to-date on these things anymore).
Maybe a little more esoterically, a "random" real number should not be contained in any set of reals having Lebesgue measure zero. Again, this is impossible, because for any real number x, x is always in the singleton set {x}, which has measure zero. But again, we can define larger and larger classes of sets having measure zero, and demand that x not be in any set in the class. This is roughly the idea behind forcing, specifically random real forcing, which is probably a redlink, but probably ought to have an article. --Trovatore (talk) 19:56, 17 October 2014 (UTC)