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Message in a bottle: is the Fermi Paradox just a matter of time?

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The Fermi Paradox is the conflict between the lack of clear, obvious evidence for extraterrestrial life and various high estimates for their existence. Numerous hypotheses have been offered to explain this paradox. I’m left wondering whether we aren’t just missing the obvious. What if we and other sentient, sufficiently advanced, life forms simply exist in time spans that never overlap? Consider: how long has humanity had the capacity to generate any significant radio signals? 100 years? And how long have we been capable of finding radio signals of a similar kind in the cosmos? Half of that? Our species has existed 1-3 million years and civilisation has only formed in the last 6-10 thousand. How long does a species last once it develops technology of the spacefaring kind?  Every indication suggests that our civilisation does not have all that much longer to run in relative terms. It could well end in the next 50-100 years - be it from nuclear war, uncontrollable pandemics or (more cert