I How is it possible to win one-in-infinity odds?

[PeterDonis]

The problem is whether non-constructive methods can apply to an "experiment". Or, whether they rely on precise mathematical axioms that cannot be assumed to hold for physical objects (or even for spacetime itself).

Some mathematicians don't even consider non-constructive methods to be mathematically valid. Since the Axiom of Choice is logically independent of ZF set theory, their objections cannot just be dismissed even on mathematical grounds (apart from the question of what mathematical objects can validly be used to describe physical measurement results).

 

[FactChecker]

$\pi$ is computable. That's why you are able to communicate it as a specific real number. Only a countable subset of the real numbers are computable. That's why I said in a previous post that unless you know this, there's no discussion, as you are simply claiming what's mathematically impossible.

Whether it is computable is not the issue. In an experiment, if you can not measure or determine whether a position on a line is exactly $3.1415926... (=\pi)$ then you would say, by your earlier posts, that the point is not a possible result of the experiment. On the other hand, if you use a different scale on the line and the position previously identified as $3.1415926...$ is now $1.000000...$, then you would say that the point is a possible result of the experiment.

I would like to distinguish between the result of an experiment versus the recorded result of an experiment. I think that I am talking about the first and you are talking about the second.

 

[Dale]

how is it possible to win odds of one-in-infinity in this scenario? Am I thinking of zero the wrong way? One is infinitely far from infinity, which I think would make it infinitely small.

Ok, leaving aside the question about whether an experiment can produce a real number, the key is that probability 0 doesn’t mean impossible.

Suppose I have a real-valued random variable with a uniform probability in the interval from 0 to 1. The probability that the variable is equal to exactly 0.3 is zero, but 0.3 is possible because it is in the interval from 0 to 1. Its probability density is non-zero. On the other hand, 1.3 is not possible because it is outside that interval. So impossible and probability zero are different concepts.

 

[PeterDonis]

Suppose I have a real-valued random variable with a uniform probability

The OP is actually asking about a countably infinite set (an infinite roulette wheel), which raises a different issue: there is no such thing as a "uniform probability" on a countably infinite set, as @PeroK noted in an early post in this thread. You can find valid probability distributions on such sets (he gave one example), but they will not be uniform: the probabilities for each individual member of the set will not all be the same.

In this respect countably infinite sets are actually more problematic for our intuitions than real-valued variables, for which we know how to define a uniform probability density.

 

[Dale]

The OP is actually asking about a countably infinite set (an infinite roulette wheel),

The OP is asking about a countably infinite set with uniform probability. This is a contradiction. Others have already answered about a countably infinite set with non-uniform probability. I have answered about an uncountably infinite set with uniform probability.

 

[PeroK]

Whether it is computable is not the issue. In an experiment, if you can not measure or determine whether a position on a line is exactly $3.1415926... (=\pi)$ then you would say, by your earlier posts, that the point is not a possible result of the experiment.

Not at all. It's not just about infinite decimal expansions. I can have a scale that reads $0, \frac \pi 4, \frac \pi 2, \frac {3\pi} 4,\pi$.

I would like to distinguish between the result of an experiment versus the recorded result of an experiment. I think that I am talking about the first and you are talking about the second.

This is a different matter. Let's take the position of an electron. The problem is that you have the HUP (Heisenberg Uncertainty Principle), which effectively implies that the electron cannot have an infinitely precise position in the first place. Or, to put it another way, the wave-function that would define a single, infinitely precise position is not a physically realisable state.

This is part of the problem. It only appears at the macroscopic scale that an object has a well-defined precise position. This principle, which you are relying on, does not apply at the microscopic scale.

 

[PeroK]

The OP is asking about a countably infinite set with uniform probability. This is a contradiction. Others have already answered about a countably infinite set with non-uniform probability. I have answered about an uncountably infinite set with uniform probability.

Let's take a closer look at this.

I would say that measure theory has an abstract mathematical definition of probability that does not define a physically realisable process in this case. I would put this in the same category as the Banach-Tarski paradox:

https://en.wikipedia.org/wiki/Banach–Tarski_paradox

You could also look up the proof that there exists an unmeasurable subset of $\mathbb R$. That set, I suggest, does not correspond to any physical object.

There are many examples of an uncountable infinite set or an uncountable process that are well-defined mathematically, but do not correspond to any physical set or process. And, in fact, the Gambler's Ruin is an example of a countable mathematical process that is physically impossible: namely, how to dispose of an infinite number of coins in finite time.

The uniform distribution on $[0,1]$ is probably the simplest and the most contentious. It looks innocent and many people claim (without really thinking about it), that you can choose a real number between $0$ and $1$ from this distribution. However, when they conclude that "something impossible can happen" this ought to make them stop and think more deeply about the relationship between mathematics and experiments: i.e. what it is physically possible to do.

If you assume elementary particles are point particles and that classical mechanics (and not QM) applies, then you can assert that a point particle must be precisely at a specific location, one of uncountable many, and may remain at rest at that point. That is clear. It's still not possible to describe an arbitrary point with infinite precision, but you could claim that your classical particle has chosen for you a point out of uncountably many in an interval.

But, elementary particles are quantum mechanical and cannot be pinned down, even theoretically. It's not possible to appeal to the notion that the particle "must be somewhere".

You can, of course, still claim that spacetime has uncountably many points. If we allow that to be in doubt, and say that there are no uncountable sets in nature, then there is no possibility of physically choosing from an uncountable set. For the purpose of this discussion, therefore, we have to assume that spacetime is uncountable.

The problem is then how to choose an arbitrary point in spacetime, without reference to physical particles, with their inherent uncertainties?

The only idea I can see in this thread is to pick a particle (or large set of particles, such as a dart, or ink spot on paper) and ignore QM. That doesn't cut it for me.

The last try, IMO, is to somehow map some portion of spacetime to another and appeal to the fixed-point mapping theorem to pick out one of the uncountably many points. But, it feels like the transition from pure mathematics to a real physical scenario will remain problematic, even in this case.

 

[Dale]

@PeroK I have already told you that I am not going to further pursue the question about the experimental realization of a real-valued measurement. I don’t find your arguments convincing (especially re finite precision which doesn’t seem relevant to me) nor do I find @PeterDonis arguments convincing. But I also have not thought about this carefully myself and don’t have contrary arguments that I find convincing yet.

Until I have convinced myself all I have is doubts which I am done arguing. Please don’t try to draw me in again. I will simply answer wrt the math without focusing on any experimental realization. As I did above.

 

[FactChecker]

Not at all. It's not just about infinite decimal expansions. I can have a scale that reads $0, \frac \pi 4, \frac \pi 2, \frac {3\pi} 4,\pi$.

This is a different matter. Let's take the position of an electron. The problem is that you have the HUP (Heisenberg Uncertainty Principle), which effectively implies that the electron cannot have an infinitely precise position in the first place. Or, to put it another way, the wave-function that would define a single, infinitely precise position is not a physically realisable state.

This is part of the problem. It only appears at the macroscopic scale that an object has a well-defined precise position. This principle, which you are relying on, does not apply at the microscopic scale.

A distribution (usually) has a mean that is often used as the result.

 

[jbergman]

Yet, some event, $\{ X=r \}$ where $r \in [0,1]$ will result from an experiment.

This isn't quite accurate. There is no real physical experiment where this will happen. In any real experiment there will be some measurement error which necessarily discretizes the space.

 

[jbergman]

@Dale can you suggest an experiment that would produce any real number (in an interval, say) and prove (or at least justify) why any real number could result.

I assume you accept that most real numbers are indescribable (uncomputable), and the set of computable numbers is countable with measure zero. From that point of view, your experiment could at best claim to have chosen a real number, but could not specify which one. And, in particular, if two such experiments were carried out there would be no algorithmic way to test whether the numbers are equal.

This is a key paradox of the real numbers. We can test mathematically that $x = y$, where $x, y \in \mathbb R$. But, there is no terminating algorithm to check whether two real numbers are equal. Unless you restrict things to the computable subset. IMO, that is a good example of where a simple piece of mathematics (If $x = y \dots$), is not actually physically/algorithmically possible.

Exactly. I don't think people understand how nasty real numbers and that most are uncomputable.

 

[.Scott]

Are you sure about that? I think you are maybe claiming that we cannot measure an element of a continuum. But I am not sure that is true.

Forget about measuring a point on a continuum - we'll never get to spin the wheel.

As soon as I ask people to place their bets, I will need enough paper (or other material) to record their selections.
If I number the bins in decimal and the players record their selections on pieces of paper, how many digits does each piece of paper need to hold?

 

[Dale]

If there is a such thing as a real-valued measurement then bets can be placed the same way. E.g. if marking a piece of paper is real-valued then bets can be submitted as marked pieces of paper.

Anyway, you are now the third person that I have had to tell that I don’t find their arguments convincing. Again, I also have not thought about this carefully myself and don’t have contrary arguments that I find convincing yet. Until I have convinced myself, all I have is doubts which I am done arguing. Please don’t try to draw me in again. I will open a new thread when I am ready.

 

[PeterDonis]

Thread closed for moderation.

 

[PeterDonis]

After moderator review, the thread will remain closed.