# The riddle of infinite number

Here is a riddle,
‘Can a number be chosen randomly from an infinite set of numbers?’

First, assuming that there are infinite sets of numbers…, how could we present them all to make a random choice?
Second, there are no infinite sets of numbers, a number is an application of a mathematical law. If we stipulate that a number is to be presented without a limit, then the law for making a number cannot be applied. Accordingly, an infinite number is not an infinite number, or an infinitely large number, or even a number.
Third, a number chosen at random is not a number, as such a number breaks the sequential aspect of number. A number chosen at random is not a number, but is a numeral.

JJ

If you can measure something, it is not infinite, a number is a measurement.

1. Select a number, randomly, in the close interval [0,1], lets call this number x.

2. Select another number, randomly, in the same close interval, call this number y.
Transform y as follows:[list]
if y <0.5, y=-1; if y >=0.5, y=1

3. The random number between minus infinty to plus infinty is then:
z=y*((1/x)-1)[/list:u]

Your example demonstrates what is today claimed in the name of mathematics. I can only repeat the metaphysical mistakes that are made by mathematics which I outlined in my first post.
First, by ‘random’ number, you mean instead ‘any’ number. A random number is not random if it is represented in an equation. Why? Because a number is counted. A number selected at random is not counted and is not a number, but a numeral. A numeral per se cannot be represented in an equation.
Second, infinity in mathematics belongs not to mathematics but to what I can only describe as misplaced mysticism. The injunction to repeat a function or to count indefinitely, sets no limit. If no limit is set, then no number is created. Infinity in mathematics by this route, falls.
Third, I cannot select numbers, from above reasoning (1). However, there is a stronger conclusion than this. Let us say that even though I cannot select numbers from the close interval [0,1] I can, at least, substitute numerals for all numbers within the close interval [0,1]. Very well, but how do I present all numerals? If I can represent all numerals, the number of numerals is fixed, for numerals are nothing if not a representation. The problem is similar to ‘can I present all patterns’. The answer is that patterns are only present where they are represented. So numerals, and their mapping to numbers are not represented within intervals, such as the interval [0,1], UNLESS they are presented.

JJ

The are several mangled concepts in your response.

The concept of a number is apart from its representation, which could be a numeral. And a number is a property of a set of things which are countable. And such a set can consists of numbers themselves, eg the set of natural numbers. Specifically countability is the motivation for natural numbers. For real numbers the motivation is continuity rather than countability.

You are right that strictly speaking infinity is not a number. It is a property of a type of countable set, namely a set which has a proper subset that has the same count or number as itself.

But why is it not a mathematical concept? Why should countability or continuity or congruency be mathematical concepts but not infinity? And infinity is just a derivative concept from countability.

Perhaps you are talking about a different infinity, a non-mathematical one.

But a random number can be a variable in an equation. The x’s, y’s and z’s in my equation are random numbers.

Why is randomness not the same as anything? What is the difference, in meaning, when I offer you a sack of candies, and asked you to choose one at random or to pick any one?

What do you mean? As I understand it is not. We can create formulae to generate as many numerals as we want.

I do not have to make representations or actually count all the elements in a set to know that its countability is infinite. All I need to show is that it has a proper subset whose count is the same as itself. So in the set of natural numbers, such a proper subset is the set of even, or odd, numbers.

unfortunately, JJ is winning this arguement, despite his garblings of a few concepts. Your equation for z is a non-linear transformation. Disregard y for now, simply look at the range [0,infinity).

By definition, within a given set, if all elements have an equal probability of being picked, then any “pick” will be random. z is such a function, shoosing fromt eh set of all numbers between 0 and 1.

Look at z=(1/x)-1 (ignoring y for now, it doesn’t change anything except for the sign). If 1>x>.1, then 9>z>0. in other words, 90% of the time, z is between 0 and 9. a number in the range (9,infinity) has only a 10% chance. Thus, all numbers do not have the same chance of being picked, and z is not a true random function.

I’ll post the complete derivation soon, using calculus, if you’re curious. suffice to say, you can only transform the random function linearly to preserve it’s randomness.

here is a linear transformation of x:
z=x*a, where a is any constant, will yield a range of random numbers from 0 to a.

lim (a–>infinity) of x*a = random number from the set [0,a] as a–> infinity.

In that sense JJ is right, the set must be bounded to choose a random number from it. the set can have an infinite number of elements within those boundaries. This is implicit in the random function x.

A better question would be, how can you make X in the first place?

simple…you can’t…not perfectly,anyway

Randomness does not imply equality probability does it? A random variable need not have a uniform pdf. It can be Gaussian, Poisson, etc.

Definition of random number from mathworld:A random number is a number chosen as if by chance from some specified distribution such that selection of a large set of these numbers reproduces the underlying distribution. Almost always, such numbers are also required to be independent, so that there are no correlations between successive numbers.
[Emphasis mine]

in other words, to reproduce the underlying distribution, all of the members of the underlying distribution have to appear close to the same number of times in a large sample.

for instance: roll a die (considered to be truely random) 6000 times. each side should land up about 1000 times each. On average, this corresponds to each side coming up about 1/6th of the time. Now, the die has 6 states. In the original set, each state is 1/6th of the set. so, these results are essentially reproducing the set {1,2,3,4,5,6} 1000 times. Therefore, by your own definition, each state must have the same probability of appearing to reproduce the original distribution. [Emphasis mine, beyotch]

JJ-stop smokin crack

The definition is wrong, and is wrong in a noticeable way. It seems to me that mathematics has just ‘appeared’ and is taken for granted. This may be because it has forgotten its roots, and hence does not, and cannot, account for its present form. The above definition would be a case in point. I will say why: A number correlates to other numbers through succession. Numbers, as they are used in equations, are successive. If they are not, then they are not numbers. Too often, a number is confused with its shape.

The other gentleman in this debate is right (against yourself) to say that we can only select from a bounded set of objects, but then he ignored the point in which I showed that this is not true of numbers. I pointed out (my point 3) that ‘selecting a number’ is selecting a numeral.
We can pick ‘numerals’ out of a bag ‘randomly’, but we do not pick ‘numbers’ out of a bag. We cannot select a number in any circumstance, for a number has its application in a form that recognises succession, such as an equation. He would do well to regard this fact.
I also do not wish to cast any aspersions on the equation that you derived, but I must say that I cannot understand how you arrived at it as I am not a mathematician, nor in this debate need I be one. But it looked as if it could be useful.
JJ

First of all, your nomenclature is just wrong. a number is a concept…the quantity 5. a numeral is the character used to represent that number. everything in math works with numbers. I may have to slap you if you bring that definition up again. never mention numerals to me again, they are totally irrelevant to this discussion.

what the definition is saying is that the choice you make is not influenced by your previous choice.

I’m not surprised you don’t understand the equation. the very concept of an equation is over your head if you don’t even know what a fucking number is.

My equation says that choosing a random number between 0 and 1 and multiplying the result by 2 is the same as choosing a random number between 0 and 2. chanbengchin’s equation does not produce a random number because it makes smaller numbers more likely to be picked.

so, to summarize according to JJ’s “points”

1. you can only choose a random number from a bounded set. Why this is a stunning conclusion is beyond me.

2. Bounded sets can have infinite quantity of elements (the set of allnumbers from 0 to 1). Unbounded sets have infinite # of elements by definition.

3. JJ is just wrong with point 3

The problem for us, and it is quite a problem, is how to count the appearance of the numerals. If you wish to skip this you can read my conclusion below, last paragraph.
We cannot count the numerals by mapping them to the act of throwing for all acts of throwing are the same. Why is this a problem? It is a problem because we have no means of distinguishing and hence counting, two throws that result in the same numeral appearing on the die.
So how do we distinguish between a particular numeral appearing on the die and the same numeral appearing on the die? We cannot say ‘they appear at different times’ because time is extraneous to your example. If this is the case, then how can we say that it is ‘probable’ that there is a one in six chance of a numeral appearing? If it is not probable in the framework of time, in what framework is it probable? It seems that we must invoke a metaphysical concept that is used regularly by mathematics. This concept underpins mathematics yet goes unrecognised. It is the idea that there is an underlying continuum, an abstract form that mathematics appeals to, that does not repeat itself. I will say more. The idea that a continuum does not repeat itself means that appearances can be mapped to it and so counted and distinguished from instances of each other and one from another. An example that uses this mapping to this amorphous continuum is A = A. Another example is found in mathematics of probability.

In order to be able to count we must assume at some point an amorphous non-repeating medium to which we can map our throws and so be able to count them as different instances. The construction of the die and its thrown outcomes requires such a concept, or abstract medium, to work mathematicaly.
JJ

If the underlying distribution is a uniform one. It need not be. You seemingly do not want to acknowledge this. If I want a random number with a Guassian or normal distribution I would not use the dice to generate the numbers.

Randomness is not about probability distribution functions but about independence between choices. Apparently you understood this, for:

And that’s exactly right.

For example if I have a bag of 10 blue balls, 3 red ones and 7 green ones. The underlying distribution of the colour is not uniform. And unless my selection of these balls are random I cannot reproduce the underlying distribution.

goddamn it JJ, shut the hell up.

no, chan, i’m right…i just didn’t explain it correctly . First of all, the definition of randomness does NOT specify what the distribution is. All it must do is reproduce it. If you start with a Gaussian distribution, then a random function will, with sufficient iterations, reproduce that gaussian distribution in the results. A gaussian is a probability function…it is not a type of random distribution.

See you are confused because I never said the elements in the set had to be different. if the set is {2,2,2,3,4,5,5,6,6,7,7,7,7,8} then the probability that each of the 14 elements will be picked is the same with a random function. however, the distribution of NUMBERS will be different. 2 appears 3/14 of the time, 3 1/14 of the time, etc.

your ball example supports my point and contradicts yours. See? what i’m saying is the probability of each of the 10 balls being picked is equal. That way, when you draw 100 balls, each one will be picked 10 times, giving you the original distribution.

If ithe probability of each individual ball being picked was diffferent, say 4 green ones were greased and slipped out of your hand 50% of the time, then the final distribution would have fewer greens and more of everything else. the uniform distribution would not be reproduced.

regardless, you equation is non-linear, so it doesn’t preserve the randomness.

What is a random distribution? a uniform distribution?

Randomness pertains to the choosing, not the outcome of the choice.

My choosing of a number in the interval [0,1] is random. The outcome is not.

I am totally aware that the resulting distribution from my equation is not a uniform distribution. I am not in disagreement with you there at all. But my choosing of ‘any’ number in [0,1] is random, in that successive choice is not dependent on previous ones.

But if JJ question, or the way you interprete it, as generating a uniform probability distribution function from minus to plus infinity, then, admittedly, I have not answered that question, deliberately. I was kinda cheating.

But JJ question actually touched on something very fundamental, although we take it as possible intuitively, ie how can we choose something that has no ‘representation’. And for that mathematicians have no choice but to resort to an axiom, namely an axiom of choice. Look it up.

now we’re getting somewhere. I never argued with you taking a random number between 0 and 1. It’s not possible to make a physical function that will give you a true random number, but that’s a different debate, really.

We’re saying the same thing now…finally.

Don’t you start too…JJ hasn’t touched on a goddamn thing. You want a die that has 2 sides that both have to dots? Make an 8 sided die, and label the sides 1,2,2,3,4,5,6,7. there…now, the number 2 appears more often. I don’t see what’s so had about this.

Of course you can’t take a random number from an unbounded set. if the # of elements in a set is "A, then the chance of it being picked is 1/A. as A approaches infinity, the probability approaches zero. in otherwords, the limit of the probability as A → infinity is 0.

If you want to debate the axiom of choice, fine…i’ll even start a thread for you guys. Seems like a watse of time, because greater minds already debated and accepted it a century ago.

Hey, great question! I think the answer is yes. Here is my method.

Take a stick. Break in in a random place. Measure the length of one of the pieces. Since the Reals are an infinite set (even between bounds) you have just picked random number from an infinite set.

It is not universally accepted that numbers dot not exist. Some people believe that mathematical truths exist independant of our knowing them, and that maths is more a discovery than a construction.

Your third point makes no sense to me.

-T

Here is an easy example that should keep you all happy:

I put my hand into a bag that contains shapes 1,2,3,4,5,6, in the form of carved wooden blocks. I then pick a shape out. Now do I randomly pick out a number or a numeral? I pick out a numeral. Is this not so? It is so. Now you tell me, if you can, at what point they become numbers.

The numbers on a die are not a set of numbers. They are a set of numerals. Now you tell me, if you can, at what point they become numbers.

Probability theorems are not founded in modern mathematics. ‘Why not?’ I here you ask politely. I can show you most easily by asking you this question: How do you distinguish, and hence count, a numeral on the thrown die from multiple instances of the numeral on the thrown die? Mathematically, you can’t unless you set a limit. Probability sets no limit, and so mathematically has no means of distinguishing similar throws.

Infinity sets no limit. A number is a limit. This limit is the act of no longer applying a function. Therefore, where there is no function applied, there is no number. Infinity is not applicable to number.

I would advise you that you are getting too caught up in the particulars of maths and not getting to the root of it.

JJ

JJ

Yes, your idea is that there is a point on a line that corresponds to a random position on it. My point is that there are no random positions and that positions need to be constructed. So, you used the example of a real position, and not a mathematical position, that arises from breaking a stick. This example seems to demand that there is a random point on the whole stick that corresponds to the position of the break.
For your example to work we have to assume that there are positions for all points on a line. But I still can’t construct a position on the line from a point on the line. If you say that the position is measured, I think we have to assume points on the measuring rod for all positions. And this is the same problem as finding positions on the line corresponding to all points on the line. Indeed, we can’t even assume all points. We have to make them.
Briefly, your example defines a point on the line, but not a position, so no random position (number) is created.

JJ