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A classic omnipotence paradox asks, "can an omnipotent God create a stone so heavy that He cannot lift it?" The problem here is that we take omnipotence to mean "capable of anything that human languages can express". Since human languages can express logical contradictions, paradoxes ensue.

Recently, I have read about the notion of Turing degrees in computer science. In summary, the Turing degree of a subset of natural numbers measures its computational undecidability. A set with higher Turing degree would need a more powerful oracle machine to decide whether it contains a given natural number.

This got me thinking: can we define the notion of an "omnipotent God" in terms of computational power?

Let me make up the following definition:

A computationally omnipotent God is an "entity" that, given any subset A of natural numbers and any natural number n, can decide whether A contains n in a single computational step.

Note that this is NOT a rigorous mathematical definition. We cannot mathematically define a machine that can decide every subset of natural numbers. The Turing degrees have no upper bound, so there is no oracle machine that is more powerful than every other oracle machine. Nevertheless, we can still say something like "an 'entity' that is more powerful than every oracle machine" in plain English, which is what my definition tried to achieve.

Now, with this definition, the questions are:

  1. Is this definition free of logical omnipotence paradoxes?
  2. What would be the theological consequences, if the Christian God is defined this way?
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  • Presumably, 1) no, and 2) definitive non-existence? – Sofie Selnes Nov 7 '20 at 17:45
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    Mixing talk of the nature of God and computation involves no intersection or conjunction. One does not belong with or follow the other. – user37981 Nov 7 '20 at 19:13
  • Computational complexity is realization invariant up to multiplicative constants, so if doing what you describe is accomplished in one step that step will not be "computational". And while we can define an oracle that can do it in one step, that is far weaker than God's omnipotence (and God can do this in zero steps due to omniscience, so he needn't do it at all). So this definition has little to do with either God or computations, and has no consequences for either. – Conifold Nov 7 '20 at 19:45
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  • Technically speaking, you can define anything however you want. Definitions are an integral part of formal systems. What function your formalism 'computational model of God' serves would, to the best of my knowledge, be original philosophy and likely highly idiosynchratic. You'll note support for Charles M Saunders comment is a common sentiment. Good luck! – J D Nov 9 '20 at 2:29
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I'd like to hear more about what you mean by "computational step." If by "computational" you mean what we normally mean when we speak of computation, then my hunch is that there is no such entity, since we finite humans have a pretty good grasp on what computation is, and there are some sets with Turing degrees >1.

If you mean something by "computational" which goes beyond our sense of the term, though, the question runs the risk of being trivial - i.e., let's say you mean 'computational*'. I could define a single computational* step as a concatenation of any finite number of computational steps; then all kinds of non-God entities would satisfy your definition.

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Your "computationally omnipotent God" is really just an oracle in disguise (as it should be). Call it oracle and be done.

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