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Witness for the Prosecution

December 23, 2013

At the end of the 1957 classic Witness for the Prosecution there’s a brief message to the audience:

The management of this theatre suggests that for the greater entertainment of your friends who have not yet seen the picture, you will not divulge to anyone the secret of the ending of ‘Witness for the Prosecution’.

It was truly a more innocent age…

For reasons which need not concern you dear reader, I’ve been thinking about numbers as of late. There’s a common trope out there which states that since π is infinite and never repeats, every finite string of numbers you care to come up with occurs somewhere in π, simply because it is infinite and never repeats itself. I should point out right at the start that this is believed to be true by the majority of mathematicians, but it hasn’t been proven. It is certainly possible to create an irrational number which lacks this property. A favourite example is:

0.0110001111000001111110000000111111110000000001111111111...

which is written in base-10, yet only uses zeroes and ones. Every subsequent group of zeroes and ones is one digit longer than the previous group, hence it never repeats and it is infinitely long. But no matter how far you search, you’ll never be able to find the string ‘492’ in this number. Similarly, we could take π and replace all the nines with zeroes:

Actual π:   3.1415926535897932384626433832795028841971693993751058209...
Modified π: 3.1415026535807032384626433832705028841071603003751058200...

No matter how far you search this new irrational, the string ‘492’ will never occur, yet the number is still just as infinite now as it was before we started dicking around with it.

I don’t think this logic is a shocking revelation, this stuff has been dealt with and is reasonably well understood in a post-Cantor world. The reason I bring it up is because —to me— it sounds like this argument is often trotted out by the multi-verse or infinite-universe crowd as proof of our non-uniqueness. If the universe goes on forever, then there must be infinitely many exact duplicates of Earth (where I am right now writing this exact same blog post) out there because you can only arrange a bunch of particles in so many ways. Is that really true, or is it only as inevitable as being able to find any finite string of symbols in an irrational number?