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:
- Is this definition free of logical omnipotence paradoxes?
- What would be the theological consequences, if the Christian God is defined this way?