I Can a truly random number be constructed?

[SSequence]

I can't quite follow your post.

What I was simply saying was that:
IF you have a mapping (total and onto) from N to some subset of real numbers such that given any index you have a program that can calculate*** the corresponding number to arbitrary precision.
THEN it doesn't automatically follow that the corresponding functions "equal" and "greater" are computable (they may or may not be).

Now even though the conclusion doesn't always happen from the premise, for rational numbers both of them happen to be true. However, the rational numbers are certainly not the only subset (of reals) for which both the premise and conclusion above are true. In fact, one can easily describe proper super-sets of rational numbers for which both the premise and conclusion above are true (so at least in that specific sense there is nothing special about rational numbers). This is simply what I was trying to say.*** One can formally create (without much trouble) a further distinction that would disallow somewhat artificial mapping of post#13. I haven't assumed that in the premise here though (so even the artificial mapping of post#13 is allowed).
However, an interesting question would be that if one did disallow artificial mappings(such as mentioned) then can one show that implication described in this post would always hold or it could fail (this possibly might be a more difficult question than it looks)? I haven't thought about that, but it is a bit besides the main point though.

 

[FactChecker]

Selection is meaningless if the difference between one selection and another is opaque.

So many posts are talking about the inability to determine or record an infinite number of digits. That is irrelevant. A physical process can "select" an exact position in space without any human recording or determining that exact position.

 

[Buzz Bloom]

We should distinguish between the concepts of selecting a number versus recording the selected number in some human understandable way. They are not the same.

Hi FactChecker:

Yes, they are conceptually not the same, but so far in this thread these concepts remain undefined. For example, is selection a process? If so, does it require a description of how the process works? Does the process have to complete in a finite time?

Here is a simpler question. How do you define a process for selecting a random positive integer such that each integer has the same probability of being selected as any other integer?

Regards,
Buzz

 

[Buzz Bloom]

Now, I assume people will say, "hey, there is no uniform mapping of the irrationals on [0, 1] to the rationals on [0, 1]." To that, I say see https://www.physicsforums.com/threads/selecting-a-natural-and-a-real-uniformly-at-random.911544/

Hi Aplanis:

If you are using the axiom of choice then perhaps you might want to consider that it is a mathematical axiom rather than a description of a process that is possible to implement in a practical manner in the real world.

Regards,
Buzz

 

[PeroK]

So many posts are talking about the inability to determine or record an infinite number of digits. That is irrelevant. A physical process can "select" an exact position in space without any human recording or determining that exact position.

The original challenge in post #1 was to do something with the selected number.

The implication of your theory about space is that the uncountable infinity of the reals must exist in reality. That may be so, but equally it may not be so.

Not least, because if you believe QM, you cannot point at a particle and say that the position of that particle at a specified instant defines a single real number.

The conclusion of this thread, I believe, is that real numbers can be conjured mathematically, but not by any physical process.

 

[FactChecker]

Not least, because if you believe QM, you cannot point at a particle and say that the position of that particle at a specified instant defines a single real number.

That is not what I was suggesting. I am not talking about the position of a particle, but rather the existence of a position in space. There are many ways to define that position even if it would not be physically possible for a human to determine it with complete accuracy. That begs the question of whether spacetime is quantized. I believe that even that issue is not relevant to whether a numerical position can be "selected" with a continuous uniform distribution.

 

[Stephen Tashi]

That is not what I was suggesting. I am not talking about the position of a particle, but rather the existence of a position in space. There are many ways to define that position even if it would not be physically possible for a human to determine it with complete accuracy.

The thread didn't begin as a question about physics, but I suppose if a physical process could select a number from a known continuous probability distribution then it is theoretically possible to evaluate the CDF of that distribution at the number selected to obtain a number selected from a uniform distribution on [0,1].

The question of whether position in space is a continuum is not precisely the same question as whether Nature can pick a number from a continuous probability distribution. If Nature picks a number from a uniform probability distribution on [0,1] then an event with zero probability happens. To the mathematical theory of probability, this is not disturbing because it doesn't comment on whether events in a probability space "actually" happen. However, most formulations of physics, including QM, have the "common sense" notion that things measured in experiments actually happen.

One can take the attitude that a continuous probability distribution is "just a model" that is only an approximation to Nature's workings. A less convincing approach is to assume Nature can implement continuous probability distributions "for real" and argue from human limitations - i.e say we can only measure the results of Nature to a finite precision. Human limitations in measuring a result don't explain how a not-precisely-known zero probability result can actually happen. Arguing that limitations in measurement explain zero probability results is just another way of saying that the concept of a continuous probability distribution is "just a model".

 

[chiro]

Hey PeroK.

If you assume any number can be chosen uniformly then the answer will be zero because of how probability works and the details are found in stochastic processes that are studied in graduate probability courses in university.

One intuitive way of explaining this is that there are infinitely many values to sample from and you will have to state how you can decide a value given infinitely many possibilities.

If you can describe how to make a decision in a finite number of time of what is chosen given an infinite set of possibilities then you can describe how to sample the space.

Being able to sample the space and knowing why you can't do it [or whether you can] will give the answer as to why it is possible [or impossible] to decide what the value is and ultimately what the probability.

Think of taking a set of values with a simple ordering function and depending on the decision you taker the left half of the set or the right half.

Keep doing so and it means that for n decisions you can uniquely decide 2^n values [think of it like branches of a tree].

What happens when n -> infinity?

 

[PeroK]

Hey PeroK.

If you assume any number can be chosen uniformly then the answer will be zero because of how probability works and the details are found in stochastic processes that are studied in graduate probability courses in university.

One intuitive way of explaining this is that there are infinitely many values to sample from and you will have to state how you can decide a value given infinitely many possibilities.

If you can describe how to make a decision in a finite number of time of what is chosen given an infinite set of possibilities then you can describe how to sample the space.

Being able to sample the space and knowing why you can't do it [or whether you can] will give the answer as to why it is possible [or impossible] to decide what the value is and ultimately what the probability.

Think of taking a set of values with a simple ordering function and depending on the decision you taker the left half of the set or the right half.

Keep doing so and it means that for n decisions you can uniquely decide 2^n values [think of it like branches of a tree].

What happens when n -> infinity?

You obviously didn't read post #1, which was a challenge.

Can you solve the challenge problem or not?

 

[chiro]

You asked whether something can be done and I'm giving some insight into whether it can.

I'm not solving the problem for you - but like many others on the forum, I'm giving my input or 2c as it were.

You need to tell me how you sample the information and then I can tell you more about whether you can do it.

You see - rational numbers have constraints - i.e. you have a/b where a and b are integers and b != 0.

The sampling method here is key - if you sample one way you will miss certain numbers and if you sample another way you will miss others.

If you can describe your sampling technique then I can build on that. How you measure something will dictate whether you can do something in statistics and if the sampling technique allows one to get more information about whether a number is rational then the probability will increase.

This is why I bring this up - you want an answer and I'm trying to help you get it.

Answer the sampling problem and maximize the probability given that sampling technique and you should be able to solve the problem.

 

[Zafa Pi]

Here's a challenge of sorts, inspired by some previous discussions.

You must choose a random number uniformly on the interval $[0, 1]$. If the number is rational, someone wins a £1 million prize. If the number is irrational, no prize is won.

It is your task to devise the method by which the random number is chosen and checked for rationality. All numbers between $0$ and $1$ must have an equal probability (density) of being chosen.

What we know from probability theory is that the probability of a rational number being chosen is 0, but that it is still "possible", in some sense.

My belief is that the challenge is impossible, although I am happy to be proved wrong.

Go to an optics lab and use a beam splitter to generate a sequence of 0s and 1s. After n bits you will be within 2^-n of the "random number." This is no worse than what transpires with computable reals which are countable and a mere chimera of [0,1] with no uniform distribution.

There is clearly no way a finite string will reveal if the ultimate number is rational, but don't worry because it isn't.
Only someone irrational would doubt the previous sentence. :-)

 

[bahamagreen]

Here's a challenge of sorts, inspired by some previous discussions.

You must choose a random number uniformly on the interval $[0, 1]$. If the number is rational, someone wins a £1 million prize. If the number is irrational, no prize is won.

It is your task to devise the method by which the random number is chosen and checked for rationality. All numbers between $0$ and $1$ must have an equal probability (density) of being chosen.

What we know from probability theory is that the probability of a rational number being chosen is 0, but that it is still "possible", in some sense.

My belief is that the challenge is impossible, although I am happy to be proved wrong.

Choosing or selecting a random number uniformly suggests a degree of resolution (a specification that uniformity is satisfied after some number of digits behind the decimal, maybe like a variance bound). Or is infinite resolution required for "choosing uniformly"?

Constructing a random number looks interesting (I'm thinking of a fair Roulette wheel with each digit equally represented, subsequent spins constructing the series of digits). If you stop, your number is rational, but until you stop your number is irrational if you keep going... is the requirement to keep going "choosing uniformly"?

It seems like the construction process offers two paths; a rational number path if you stop, and an irrational number path if you proceed - as long as the process is ongoing its steps don't seem to eliminate either a potential rational or irrational result...? :)