Distance from Earth to its nearest copy.

To Infity to Beyond is one of those superb Horizon programmes. The UK fascinates me for many reasons but the quality of British television is one of them. ( I recently found a lot of Avengers episodes, the Spi Sci series from the sixties with John Steed, Mrs. -Emma- Peel and later with Tara King and 'Mother', what a pleasure to be able to watch them all again. But that's off-topic. )

To the Horizon programme they show how to calculate the distance between our planet Earth and the nearest exact copy. Yes. IF, the Universe is indeed infinite than there must be a planet Earth somewhere which is an exact copy of ours. This is not only some mental experiment like Hilbert's Hotel but it is possible to calculate the distance to the first copy ( there are many, of course ) in meters.

In meters: \[ 2^{10^118} * 10^26 m\] where \[ 10^118 \] is the number of particles in our universe, and \[ 2^{10^118}\] is thus the number of possible universes and \[ 10^26 m \] is simply the diameter of our own universe.

UPDATE:
I thought about this, watched the programme once more. I think the actual distance is significantly larger than the number above. The example in the program is an extrapolition of a universe consisting of 4 particles of which there are two types. The argument remains valid though. If the universe is infinite then there must be a copy of our Earth somewhere.