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The Boltzmann Brain is an idea that pushes the limits of infinity and probability.
On time scales far beyond human comprehension, long into the heat death of the universe where all stars have run out of fuel and all of the remaining black holes have evaporated via Hawking Radiation, there can be nothing but the dark void and random quantum fluctuations within.
Because of the endless scale of infinite time, the random fluctuations will occur in such a way as to occasionally produce real objects.
An infinite series of random letters will eventually contain every book ever written, or that could ever be written.
Random quantum fluctuations will eventually happen to be in the exact state of objects by sheer chance. Infinite time enables the improbable to become certainties.
There is no limit to the complexity of the objects that could randomly pop into existence. An apple, a boat, stars, and even galaxies would eventually be produced by the aligning of these random quantum fluctuations, and would vanish just as fast.
This means that eventually an object as complex as a brain, complete with memories and perceptions would randomly pop into existence, even though it exists completely alone in the void out of complete chance.
Eventually a brain that is structured exactly the same way as yours would also exist, full of your exact memories and sensory input as what you are experiencing right now.
Paradox 1
Given infinite time, this brain wouldn’t form only once, but infinite times. The chances that you are a physical human brain on earth and not one of the infinite Boltzmann brains in an infinite void is, well, pretty low.
Paradox 2
Furthermore, it is much more likely that quantum fluctuations would create a single brain than to create an entire universe for you to be living in. It is much more likely that you are a single Boltzmann brain with false memories and perceptions, than to be living in an external consistent universe.