MixedNuts comments on Mixed Reference: The Great Reductionist Project - Less Wrong
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If all levels of the Turing hierarchy are about as real, it's extremely unlikely our universe is at level zero. Yet Church-Turing looks pretty solid.
Combine this with the simulation hypothesis; a universe can only simulate less computationally expensive universes. (Of course this is handwavy and barely an argument, but it's possible something stronger could be constructed along these lines. I do think that much more work needs to be done here.)