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

[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]

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[PeterDonis]

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