Is an infinite series of [nonrepeating] random numbers possible? That is, can the term "random" apply to a [nonrepeating] infinite series? It seems to me that Cantor's logic might not allow the operation of [nonrepeating] randomization on a number line approaching infinity.
I'm not sure of an answer, but allow me to put two thought-experiments on the table to discuss. Experiment A: it will produce a rational approximation to a real in the interval (0,1]. Expressed in base 2, the number is constructed as follows: throw a fair coin, and choose the first binary digit (at position 2^-1) accordingly (maybe heads=1, tails=0). Each next coin throw will give you another digit. When you get tired, or consider that your approximation is good enough, stop throwing the coin and use the fraction so far. If you have two of these fractions and are worried that they may be equal, calculate extra digits for each number (with more coin throws) until they differ. Experiment B: produces a random number, not uniformly distributed and possibly repeated (the latter could be solved by throwing away a repeated number and producing a new one), in the interval [0,pi). Proceed as in experiment A, but definitely terminate the experiment at the first 0 digit (tails); then return pi times your fraction.
Thanks, I did mean sequence, and by "infinite" I intend for the count of random numbers in that sequence to approach infinity. By "random" I mean eventually exhausting all relations among numbers. __________ Is the cardinality of random numbers the same as that for reals? Might random numbers by their nature tend toward infinity?
Not entirely ^^ You see, Prime Numbers are NOT random. They are connected by the fact they can only be divided by themselves and one. It's incredibly unlikely if a proof to either of these were found, that it'd be similar to prove the other one.
If it is a sequence, by definition it will be countable. The numbers themselves can be confined to a given interval, since there are (non-countable) infinite numbers available.
Please let me amend my original questions: Is an infinite set of exclusively non repeating random numbers possible? Is "random" analogous to "exhausting all relations among numbers"? __________ Is the cardinality of random numbers the same as that for reals? Might random numbers by their nature tend toward infinity more rapidly than reals?
You have to define random. By the definition of an infinite sequence, there will always be some generating function that represents your sequence (even if it can only be expressed as a mapping). I suppose you could say that it is random if the generating function can't be expressed in terms of certain types of other functions, like elementary functions. But until you specifically define random, by the reasoning that my choice of generating function is random, I can say that {2, 4, 6, ...} is random; because I thought of it randomly.
What do you mean by a random number? Is 6 a random number? Is pi a random number? Do you perhaps mean non-computable or non-definable number?
The set of random numbers exhausts integral order and corresponding numerical magnitude. Random numbers may be generated by exchanging orders with magnitudes.
I know it's sound philosophical, but how would you generate a random sequence of numbers? I mean if you want to 'generate' something, you'll have to use some apparatus (either your mind, computer, cesium atom) and you don't know what makes the apparatus pick its numbers. I don't believe there's something like random numbers.
Perhaps you question is, are there permutations of infinite length? (Just trying to guess your meaning, though.)
Can random numbers be defined but not generated? I'm not sure. How about the permutation between order and number which I mentioned previously?
You're using some words in ways that are unfamiliar to me. * You referred to "the set of random numbers," but I don't know what a random number is. Can you give some examples of random numbers? Is 6 a random number? Pi? * What do you mean "exhausts integral order?" * What do you mean by generating random numbers by "exchanging orders with magnitudes?" Can't imagine what that means. Order is the relation by which 5 < 6, for example. Magnitude is the relation that ignores the difference between -5 and 5. How do you exchange these properties? And how do use these ideas of order and magnitude to generate random numbers? Do you mean particular random numbers or the entire set of random numbers? And what set is that? I know you have something in mind but it's difficult for me to understand what you mean. Are you referring to numbers that are generated unpredictably? Flip a bit if a cosmic ray passes through a particular square millimeter in the next millisecond? Or by "random" do you mean algorithmic randomness? A number is random it's incompressible via a finitely describable algorithm? So any number we can name is not random, but we know that there must be uncountably algorithmically random numbers. Those are the non-constructable or non-definable numbers. [There's a subtle difference between those two concepts? Do any of those questions make sense in terms of what you're trying to do?
I appreciate your patience, SteveL27. Upon consideration, your arguments make a lot of sense. In this regard, allow me to modify my previous speculations to concentrate on a specific random number generator: 1. Start with an exclusive irrational number. 2. Limit its fractional part in digits by its integer value. 3. The resulting string is a random number.
What is an "exclusive" irrational number. What does it mean to limit its part in digits by its integer value? Do you mean just take the part to the right of the decimal point? Can you give an example? Say I start with my favorite irrational, e, the base of the natural log. e = 2.7128... Are you saying that .7128... is a random number? But it isn't, in either sense of the word. * It's not unpredictable. In fact you can can write down the well-known algorithm e = 1 + 1 + 1/2 + 1/6 + 1/24 + 1/120 + ... + 1/n! + ... and crank out as many digits as you like. It's completely deterministic, the opposite of random. * It's not algorithmically random, in fact the finite string (sum as n goes from 0 to infinity of 1/n!) is a small number of symbols that precisely defines e. Is e an "exclusive" irrational by your definition? Or do you mean something else? Do you perhaps mean that the digits of e are normal, in the sense that any block of n digits occurs equally as often as any other block? It's not known if e is normal, but is normality the characteristic you're interested in?
SteveL27, I am beginning to believe that "random number" is an oxymoron. Numbers are defined by value, order and relation overall, while randomness violates those quantities together. Random number sequences manifest near infinite uncertainty and zero probability. There exist tests for non-random numbers, but not for random numbers. No physical measurement has confirmed a random number. The set of random numbers and the set of non-random numbers seem mutually exclusive. Is the set of reals their union? __________ [Strike "exclusive," please.] Some example outputs of my pseudorandom generator (I am assuming normality for irrationals): 3.162277660... becomes .162 2.718281828... becomes .71 A true random number generator is an impossibility, since it requires both non-determinism (randomness) and determinism (predictiveness).
It might help if you think about randomness in different orders of entropy. Entropy is probably the best and most useful way to quantify randomness: maximum entropy on all orders means that there is no advantage you have for having any past information at all since a maximum value on all orders (independent, first order conditional, second order conditional, etc) would mean that the analysis of all past values with respect to each other would not give you an advantage. The best way to describe this is to have all of these distributions be uniform because this distribution is the one that maximizes entropy. If you do this for all possible conditional probabilities, then you will get a distribution that is purely random. From this distribution you will get hints about the kinds of processes that you could construct. If you want something random, but not purely random then you don't have to do anywhere near as much work, but if you want a process that is random the best way possible, you need to construct the above system and from there decide what kind of process would really emulate this distribution.
S=κln|Ω| S=entropy κ=Boltzmann's constant ln=natural logarithm Ω=number of states __________ 1. Does true randomness accompany a transfinite number of states? 2. Is information about states restricted by a finite speed of light? 3. Can multiple states interfere, e.g. achieve minimum entropy? .