At 1/18/04, Hal Finney wrote:
Now consider all possible program tapes being run at the same time,
perhaps on an infinite ensemble of (virtual? abstract?) machines.
Of those, a fraction of 1 in 2^100 of those tapes will start with that
100 bit sequence for the program in question.
Now consider another program that is larger, 120 bits.  By the same
reasoning, 1 in 2^120 of all possible program tapes will start with that
particular 120-bit sequence.  And so 1/2^120 of all the executions will
be of that program.

Yes, but if we're really talking about all possible finite bit strings, then the number of bit strings that begin with that 100 bit program is exactly the same as the number that begin with the 120 bit program - countably infinite. You can put them into a 1 to 1 correspondence with each other, just like you can put the integers into a 1 to 1 correspondence with the squares. The intuition that there must be more integers than squares is simply incorrect, as Galileo pointed out long ago. So shouldn't your two programs have the exact same measure?

I don't mean to sound so critical - I'm genuinely asking for information. I know virtually nothing about measure theory. Is there some well-defined way of getting different measures for countably infinite sub-sets of a countably infinite ensemble?

-- Kory

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