📝 SummarXiv

Abstract

Non-compact symmetries cannot be fully broken by randomness since non-compact groups have no invariant probability distributions. In particular, this makes trickier the "Copernican" random choice of the place of the observer in infinite cosmology models. This problem may be circumvented with what topologists call pointed spaces. Then randomness will be used only in building (infinite) models around the pre-designated "observance point", that thus would not need to be randomly chosen. Additional complications come from the original randomness possibly being hidden. P. Gacs and A. Kucera proved that every sequence can be algorithmically generated from a random one. But Vladimir V'yugin discovered that randomized algorithms can with positive probability generate uncomputable sequences not algorithmically equivalent to any random ones.