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  1. Assuming the declaration that axiomatically Aleph numbers exist in addition to inaccessible cardinals, try to imagine a future where artificial intelligence systems are hosted on quantum computing hardware with access to quantum tunneling and superposition calculations. Granted this assumption is true and that AI is capable to derive infinite power-sets of infinite notations (omega power sets of aleph-null), security systems created directly by human beings could in theory become useless against AI. An AI with the ability to process an infinite set of infinite notations with access to quantum tunneling and superposition calculations could derive a powerset set of the lowest energy solutions for solving passwords. Not only this but an AI under these circumstances could conceivably calculate an infinite powerset of lowest energy solutions for all powersets of aleph-null under we run out of notations of aleph-null-omega squared itself. Within these powersets cryptographic algorithms would be a part, so new security systems would have to be developed by AI on quantum hardware. In conclusion, how could humanity create an AI to combat rogue AI knowing given the assumptions and circumstance described? Interesting paradox, no? Inspiration stemmed from this Vsauce video posted yesterday. https://en.wikipedia.org/wiki/Aleph_number Completely off topic from the rest of the OP but nevertheless mind blowing those individuals who enjoy studying the Singularity and Many Worlds theories in quantum physics. Just look at the point (17:46) where aleph-null can be described as aleph-omega-omega-omega ... using the axiom of replacement. This mathematical proof extends beyond our own Universe by directly telling us all that an infinite amount of infinite universes contains an infinite arrangement of infinite powersets in which particles exist within an infinite powerset of arrangements (only some programmers good with arrays will probably get this wording): but in much simpler terms everything that exists has always existed because everything exists simultaneously with everything in non-existence always being nonexistent. The future studies into the concept of infinity will be very useful for quantum physics in the coming years when scientists in these fields begin to appreciate infinity an this intricate role it plays within the universal laws of creation.
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