In a number of posts over the past several years, I’ve explored various ways to make a countably infinite fair lottery machine (assuming causal finitism is false), typically using supertasks in some way.
Here’s another, slightly simplified from a construction in Norton. Suppose we toss a countably infinite number of fair coins to make an array with infinitely many infinite rows that could look like this:
HTHTHHHHHHHTTT...
THTHTHTHTHHHHH...
HHHHHTHTHTHTHT...
...Make sure that nobody looks at the coins after they are tossed. Here’s something that could happen: each row of the array contains one and only one tails. This is unlikely (probability zero; Norton originally said it's nonmeasurable, but that was a mistake, and we're coauthoring a correction to his paper) but possible. Have a robot scan the array—a supertask will be needed—to verify whether this unlikely event has happened. If not, we have failed to make the machine. But if yes, our array will look relevantly like:
HHTHHHHHHHHHHH...
HHHHHTHHHHHHHH...
HHTHHHHHHHHHHH...
...Continue making sure nobody looks at the coins. Put a robot at the beginning of the first row. Now, you have an countably infinite fair lottery machine that you can use over and over. To use it, just tell the robot to scan the row it’s at, announce the position of the lone tails, and move to the beginning of the next row. Applied to the above array, you will get the sequence of results 3,6,3,….
Of course, it’s very unlikely that we will succeed in making the machine (the probability is zero). But we might. And once we do, we can run as many paradoxes of infinity as we like. And we might even find ourselves lucky enough to be in a universe where some natural random process has already generated such a lucky array, in which case we don’t even have to flip the coins.
Once we have the machine, we can have lots of fun with it. For instance, it seems antecedently really unlikely that the first hundred times you run the machine, the numbers you get will be in increasing order. But no matter how many numbers you've pulled from the machine, you are all but certain that the next number will be bigger than any of them.
3 comments:
If you have finite mental capacity, the machine may not be so much fun – it’s practically certain that the numbers you get from it will be too big for you to grasp. Then you won’t be able to compare them. In the unlikely event that you get a number you can grasp, you will indeed be practically certain that the next number will be bigger.
But here is what puzzles me. Suppose you have a handy gadget that can take an arbitrarily large number and answer simple questions about it. Is it even? Square? Prime? … Attach the gadget to the infinite lottery machine. Set it to tell you whether the successive lottery outcomes are even or odd. For example, you might get a sequence starting ‘odd’, ‘even’, ‘odd’… What would the sequence be like?
Note that ‘even’ inherits no objective chance from the original coin flips, at least in standard probability theory. Lest you be tempted by indifference, note that the setup has, as Norton says, ‘label independence’ - the even numbered columns can be permuted to the columns at multiples of 3 (0r 4 or 5 etc.). So it seems you have to say that the sequence is ‘non-probabilistically’ random. But what, if anything, could this mean? Could you distinguish the sequence from one produced by an objectively chancy fair coin - not with certainty of course, but inductively, with reasonable confidence, after seeing many outcomes? If not, would ‘non-probabilistic’ randomness make sense? For that matter, would objective chance make sense?
An interesting article on bias in coin tossing:
http://econ.ucsb.edu/~doug/240a/Coin%20Flip.htm
Perhaps coins are not as fair as they seem.
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