I did see a proof some 14 years ago, but it was not about "picking", the question asked was if an unlikely event will happen or not.

Are you seriously suggesting an equivalent alternative to the measure and probability theory with something you haven't even formulated properly? That's like coming to a physics discussion and say: "Guys that's your opinion, however I don't really agree with all those Newton, Einstein & co theories, so I developed my own set of rules. I haven't thought it through yet, and I'm yet to verify my theories can soundly explain 1 cm ball movement, but that doesn't stop me from getting different results by using them, and offer my calculations as an equal alternative".

An alternative for such strongly established theories (that are grounds for many more theories in math and other fields), should at least be a result of some serious PHD work that has been reviewed and tested by experts from the field, and even in such case, I would only humbly and feebly mention it as an alternative.

More specifically, I'm not sure I understood your ideas but it sound to me that they are based on some misconceptions:

<Amadeus: The resulting set will have n/2 elements, for any finite n, and the same holds true as after infinitely N steps. In a 'Cantorian' theory you can try to create a mapping like 1-1, 2-3, 3-5, 4-7, n-(2n-1), that's ok. But here I have to ask: which infinite process am I going to use to regulate this? If it is N, than you'll have a lot of numbers that do not belong to your basic Natural set. So, only if the infinite set that regulates this infinite operation is N/2, you'll have a set with all odd numbers that belong to N.>

Not sure what u mean by "regulate", but I see no problem letting N do the lead here rather then N/2: For every element n from N, I take the element (2n-1) from N/2. The point is that since both sets are infinite, I am never short of elements in N/2. That's the meaning of "infinite". The concept that the odd numbers has only half the number of elements on infinity as well, is inherently wrong. Inf/2 is not half of Inf, it is still Inf. Maybe you mean that [lim (n/2)/n , n->Inf = ½], but that does <not> mean that Inf/2 is smaller then Inf, it only means that the property is kept for every <finite> n.

<Another example, better example: [..] 1- 1^1, 2- 2^2, 3- 3^3, n- n^n etc. But after N steps, we would have N-N^N, and that shouldn't be a natural number even according to the usual theory, and actually most square numbers in this process are not in our basic set either.>

This sounds like a mix up between measuring a set and measuring individual numbers. There is no problem calculating the [lim n^n, n->Inf] within the natural numbers, it safely goes to infinity there without problems. Its true that the set of natural numbers cannot cover a set with ordinal N^N, like the set of real numbers, but that doesn't mean that the real numbers end up with a "bigger infinity" then the natural numbers, they both go to the same infinity, its only that the real numbers are more <dense> as a set.

As an example, we can look at 2 finite sets: A={1,10,20,…, 100}, B={1,2,3,..100}. So A cannot cover B since B is more dense, but they both reach to the same 100, its not like the "100" in B is bigger then the "100" in A..

You need to read my post again slo. I never said we have other number then zero that can do the job, what I said was, that if we use that zero we end up with an <invalid discrete distribution>.

Unlike continuous distributions, with discrete distributions, each element does have weight, meaning that the question "what is the probability to get element X" has a meaning. So if you say that all your elements have zero probability, you actually say there is no chance getting <any> element from the set, and that's impossible. What I am saying is that we can randomly pick an element from every group of size M, no matter how large is M, but not from a <countable infinite> size group.

And the concept of "infinite attempts" is sure not "crappy", it is actually the way we <define> probabilities: The probability of getting element X, is the ratio between the number of success and the number of attempts, when the number of attempts (or pickings), goes to infinity.

<alexmagnus: I don't know what happens in this infinite case, that's why I asked the question. The probability remains 0 for any finite number of attempts.>

Sounds like you are still in the opinion that we <can> define a uniform distribution over the rational numbers. You don't even take <zarg>'s brother word for it alex?

In that specific post I was referring to picking from the set of rational numbers, not from an interval.

The thread became a little messy, but basically we were discussing only 2 cases: picking a number from an interval and picking a number out of the rational numbers.

We already agreed that countable number of pickings is not enough when picking from an interval, but zarg still haven't given up on trying to find a way to actually perform such uncountable number of pickings..

I should have made a more careful presentation. If we take the interval [0,1] and just cut it in a finite number of intervals, then of course no matter how we do it, the sum of lengths of these intervals will equal 1. In order to find the Lebesgue measure of a set A which is not easy to visualize, people like to cover A with the union of a sequence of open intervals.

For example, suppose A is the set of positive integer numbers. Then one can cover A with the intervals

(0.9, 1.1), (1.99, 2.01), (2.999, 3.001), (3.9999, 4.0001), ...

The sum of lengths of these intervals is

0.2 + 0.02 + 0.002 + 0.0002 +... = 0.22222... = 2/9

The same set A can also be covered with the intervals

(0.99, 1.01), (1.999, 2.001), (2.9999, 3.0001), ...

For this second covering the sum of lengths of the intervals is

0.02 + 0.002 + 0.0002 +... = 0.022222... = 2/90

Similarly, we can take a third covering, with the intervals (0.999, 1,001), (1.9999, 2.0001), ..., where the sum of lengths of the intervals is 0.0022222...

The important thing here is how large your infinite set is. If it is as large as your natural set, you have one result, if it is as large as your rational set, then you have other result etc.

<Are you seriously suggesting an equivalent alternative to the measure and probability theory with something you haven't even formulated properly? That's like coming to a physics discussion and say: "Guys that's your opinion, however I don't really agree with all those Newton, Einstein & co theories, so I developed my own set of rules.>

In this theory, we take for granted the existence of an infinity amount, we call it N for simplicity's sake -- I could have used our basic infinity*basic infinity etc --, and, voila, we have created a set of Natural numbers.

There are many ways to define what a rational number is, you can get all philosophical. One way which does not differ much from mainstream would be to define then to be a+b/c; a, b and c are taken from N, and b/c <1. This should work ok, and you have created a fine set of Rational numbers.

What is a real number? Let's say it is a natural number, plus an infinite string of 0&1. How much infinite? N, and this should work fine too. Voila, you have created a set of Reals.

That's basically it. Things won't go to zero in some proofs, but they will be infinitesimal, and I see no reason why we shouldn't recover traditional math from it. One could, of course, try to create a further theory, with higher infinities and such (I guess Russell had one, with classes and such -- I was never able to get past the first pages of principia) --, but I'm a simple man, and I'll keep it simple.

Wow wow you have 2?!

Erdős published with 511 people, and there are only ca. 8,162 people with Erdős number 2, and after a quick scan after Norwegians or physicists, I noticed <only> Atle Selberg and Albert Einstein having that low Erdős number.

This impressive list of physicists have Erdős number 3:

Enrico Fermi
Richard Feynman
Abdus Salam
Erwin Schrödinger
Steven Weinberg
Wolfgang Pauli
Edward Witten

And so on. It is in this sense that the infimum is zero, it is the infimum over all possible sequences of open intervals which cover the given set A. If this infimum is zero, as it is in this case, one then defines the measure of the given set A to be zero.>

Thanks a lot for providing an example, that was to my surprise understandable! Do you also have an example of a Lebesgue measure that is not zero?

The argument was not to "count" a continuum, but to Cover a continuum with your suggested events. You were the one who claimed we can have uncountable pickings using continuum random sources, not me. I just showed what it would have to be, if those sources actually existed, and could actually create uncountable set of random "pickings" (or F(x)=rand() as I represented it in a more formal mathematical way).

<Anyway, I don't see a way for you guys to count unless we are able to box in vacuum somehow>

Actually you <don't> have to box the fluctuations in order to count them: The rational numbers cannot be boxed (since we have infinite rational numbers between any 2 rational numbers), but they can still be counted.

I think its pretty obvious that F(x)=rand() cannot represent any actual events in our world, and so uncountable pickings doesn't exist in our universe.

The continuous range was practically invented to represent the continuous way real events take place in our world: energy that changes in a continuous way, and as you know, Physics is heavily based on calculus, differential equations, etc because of that. And all that would be meaningless if we had behavior like F(x)=rand(): Not only that this function cannot be derived at any point, it is not even continuous at <any> point.

In other words that's the most un-natural function that exist. It describes 100% chaos that nothing can exist in it, not even atoms. So (as YOU said) it can only be compared to the universe before the Big Bang, or in other words: cannot exist in our universe after the beginning of time.

<zarg: So, depending on the location in space-time of your "counting machine", will to another "observer" look less "dense" time-wise [..] This grand emptiness surrounding our universe, which is <not> restricted by speed of light since Big Bang.>

zarg, I think we can safely assume that the probability theory is restricted to:

1. Events that take place after the Big Bang

2. Events that take place inside our universe

3. Use a single frame of reference for all events within the experiment

As difficult as it is, I guess we'll just have to live with those restrictions..

---

Anyways, even if it was possible to perform uncountable random pickings (it isn't possible, but "if"), it still would not have changed the fact that p(x)=0 for every x in the interval.

A Continuous distribution answers the question of: What is the probability to get a number in a Range, and <not> a single number. P(x)=0 is just a by-product when you integrate over a range of zero size (=integrate from x to x), it has no real meaning when talking about a continuous distribution, since each element has zero weight by definition there.

Moreover, the entire probability theory is based on the assumption that we perform a series of experiments (or pickings), taking place one after another. And if we take the number of pickings to a limit, it is always the limit of a series of <countable> pickings, as we can see in the Probability axioms and in its theories, such as the Central limit theorem, Bernoulli process, etc.

I don't know where you got that idea that "math is only limited by your imagination". In fact, math is more restricted then physics, as it is based on axioms and theorems that were established over hundreds of years, when the tiniest change will require 100% clear structure with 100% accurate proof. Math is probably the only field that does <not> leave any room for "opinions", you will have to work very hard proving your opinion if you want anyone to take it seriously.

When talking about measure and probability, math intends to provide measuring tools that are 100% accurate. As all other fields rely on it, you can't afford having any errors there, not in the infrastructure of science.

You can use your imagination to open a new branch in math, provided that it isn't in conflict with the known axioms and theorems. But if it is in conflict with any of them, you will have to prove that your theorems give more accurate measures, and the chances for that are extremely slim, considering all the work done in the field so far, by geniuses and brilliant scientists.

And your suggestion is in a <big> conflict with fundamental things in math:

First, you get different probability results then those we get with the established theories, and this in itself is a big conflict. Those probability/measure calculations have strongly established and deep theories behind them, and were successfully used in many other fields. You can't just come up with different results, supporting only it by the fact that it is your opinion.

Second, using your approach that Inf+1>Inf, will result in many problems. For example we get that: [Sigma(1/n), n->Inf ] < [Sigma(1/n), n->Inf+1], this practically means that a convergent series has no single limit in your math. Among others, it means we cannot have integrals as well (since it relies on Convergent series), and so we cannot have most of our physics theories as well.

Recall that there were 2 different issues:

First we talked about picking a random number out of an <interval>, and that is indeed possible. And you correctly said that even after infinite number of picks, we end up with 0 probability to get one specific real number, since we are in the "wrong infinity" there.

Then alex asked what happens if we have infinite attempts to pick a number out of the <rational numbers>. We all assumed that the probability for a single pick in such case is zero as well, and tried to figure out the probability after infinite number of picks. But then I realized that the whole experiment is not possible. And I said that <zarg>'s intuition, that its impossible to give zero probability to event that can actually happen, was correct in this case, since in case of a countable set like the rational numbers, each element does have a weight, and so it is impossible to have zero weight (=probability) for all the elements that are in the set.

<alexmagnus: I did. But if you cannot define such a distribution then what happens with my hypothetical generator of rational numbers? Is it impossible to give a probability distribution for it? Same question for other countable sets.>

I guess it means that your generator cannot be 100% random, at least not to generate random rational numbers in finite amount of time.

The difference for convergent series should be infinitesimal, so you can recover traditional math with no problem for all practical purposes -- the harmonic series are infinitely large (after N steps, lnN). As for integrals, they could actually be treated as infinite sums when necessary, highly infinite sums. I would be surprised if some tricks were not necessary here and there, in special cases, but, then, that's what math is all about too.

<I don't know where you got that idea that "math is only limited by your imagination". In fact, math is more restricted then physics, as it is based on axioms and theorems that were established over hundreds of years, when the tiniest change will require 100% clear structure with 100% accurate proof. Math is probably the only field that does <not> leave any room for "opinions", you will have to work very hard proving your opinion if you want anyone to take it seriously.>

When it comes to the infinite, one can wonder a lot, and there will be no universe to prove you wrong next month -- maybe an undergraduation teacher... Also, I'd guess that a good name for a theory would be more important than hard work, but it goes without saying that one should expect neither here

As for the hundreds %, over hundreds of years, you are ignoring, eg, Brouwer's thesis concerning the use of the principle of the excluded middle in problems with the infinite. Of course, Brouwer would shot me in the head if he were to read my pet theory, but then Hilbert made sure to exclude him from the Mathematische Annalen, because he didn't like his opinions, so no trouble in there :)

That Brouwer was not just a raving lunatic should be proven by the fact that men like Gödel and von Neumann have used his ideas -- not that they were exactly sane... And since there were mentions to measure, let's remember two Polish mathematicians, Banach and Tarski, who created a paradox to show some of their problems with the axiom of choice.

<You can use your imagination to open a new branch in math, provided that it isn't in conflict with the known axioms and theorems. But if it is in conflict with any of them, you will have to prove that your theorems give more accurate measures, and the chances for that are extremely slim, considering all the work done in the field so far, by geniuses and brilliant scientists.>

If we think about that example, the same argument works if we replace the sequence of positive integer numbers 1, 2, 3, ... by any sequence of real numbers a_1, a_2, a_3, ... One can cover a_1 with an open interval of length 0.1, a_2 with an open interval of length 0.01, a_3 with an open interval of length 0.001 and so on, and the sum of lengths of these intervals will be 0.111111... This would be the first covering. Then one can perform a second covering with intervals ten times smaller, and the sum of lengths will be 0.0111... And so on. The infimum over all such coverings will again be zero. So if we denote by A the set whose elements are the above numbers a_1, a_2, a_3, ..., then A will have Lebesgue measure zero.

In particular by this argument the set of rational numbers has Legesgue measure zero.

<And I presume to point to this, is that the Riemann integral tackle less functions than the Lebesgue integral does, and the prime example here is that function that is 1 for all rational numbers and 0 elsewhere.>

You are absolutely right. This function is not Riemann integrable, but it is integrable with respect to the Lebesgue measure, and its integral over the interval [0,1], or over any other interval, is zero.