Is there a good name for a mathematical theorem being true "only because of an astronomically large computation"? I kind of want to say "approximately independent", since the limit as the computation goes to infinity would be actually independent.
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Hmm, to me that hides the fact that it actually is true and provable with a finite proof.
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Limits involve a large number of calculations, but calculus can be used to generate the same result with a small number, so proof not dependent on many calculations. A dynamic non-stationary system--eg, climate, economy--needs many calculations, but I know of no proofs here.
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This may help understand what I mean: https://johncarlosbaez.wordpress.com/2012/04/24/enormous-integers …. Some statements are true but the minimal proofs are spectacularly long. More interestingly, some number theory proofs depend on huge calculations to rule out small counterexamples before the analytic bounds kick in.
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I don't know any word for "true with only a vast finite incompressible proof". I imagine we want to exclude mere compressibly vast proofs that would be short in theories with higher proof-theoretic ordinals, e.g., trying to prove Buchholz hydra termination in PA instead of ZF.
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I also can't instantly prove that (hard to say in 280) among true finitely provable sentences of length N, their shortest proof across all variants of ZF adjoined to axioms asserting arbitrarily large ordinals to be well-founded, has proof length growing 'rapidly'.
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...That didn't come out well. I mean that I don't know of a proof that there exist briefly statable mathematical facts with vast but finite proofs, whose proof doesn't compress in the face of axioms bearing the power of huge ordinals (or large cardinals or Busy Beaver numbers).
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Yeah, it's easy to construct short statements that have short proofs only assuming large cardinals, say, and astronomical proofs otherwise. But I'm comfortable with those being inside the definition, excluded only as necessary by saying "natural" or some such.
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Motivation: I made a bet on Facebook yesterday that 6/7 of the Millennium problems would be solved by 2040. Someone asked why I didn't say 7, and the answer I wanted to give was "hedging against one being independent or approximately independent".
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And therefore out of reach even for a superintelligence.
End of conversation
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