Wednesday, January 2, 2013

A common error regarding recurrences in an infinite multiverse

The Life of Pi, and Other Infinities By Natalie Angierm (Published: December 31, 2012) contains a statement by Anthony Aguirre, an associate professor of physics, with which I must take exception. (NYT did not provide for comments to this article on its site).
If you take a finite physical system and a finite set of states, and you have an infinite universe in which to sample them, to randomly explore all the possibilities, you will get duplicates.
This is well and fine but after this I think Professor Aguirre gets a bit sloppy:
If I ask, will there be a planet like Earth with a person in Santa Cruz sitting at this colored desk, with every atom, every wave function exactly the same, if the universe is infinite the answer has to be yes.
No it does not. That there will be duplicates, indeed infinitely many duplicates, does not mean that any particular duplicate must occur.

Consider this infinitely long sequence of digits:

12 122 1222 12222 122222 ... etc.

This can represent a universe of infinite possibilities, (it goes on forever yet there is always further out a sequence you have not seen before) and it is built from finite set of states, (each digit is limited to being either '1' or '2').

Any finite number of '2's in a row will appear infinitely often, as will the single digit '1' - those are the inevitable duplicates. Yet you will never find the sequence '121212' anywhere and '1212' occurs just once at the beginning. To suppose that any particular duplicate should have to occur requires more than just an assumption of infinity. There would have be a particular structure in the infinity to guarantee it.

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