I've just received word that I'll be presenting my paper on epistemic value, evidentialism, and the problem of epistemic encroachment at the Epistemic Goodness conference. It's rare that there are conferences are held in easy driving distance from Dallas, so I don't do as many conferences as I'd like these days. I should have a draft to post soon, but not too soon. It's grading and job season.
Here's the question. Must an essentially omniscient being have an infinite number of beliefs? I'd think that the answer is obviously 'Yes'. But, how to justify that answer?
Suppose you say that the essentially omniscient being is omniscient and has the property of being omniscient essentially. If we say that an omniscient being knows every true proposition and believes no false propositions, we'd have to assume that there is an infinite number of true propositions known and believed by this being. I suppose you might say 'Well, there clearly is an infinite number of such propositions. There's an infinite number of mathematical truths.' That seems okay, but I suppose if someone really wanted to defend the idea that the omniscient being had but a finite number of beliefs or items of knowledge, they might say that this being's perfect knowledge of mathematics could be understood in terms of a finite set of beliefs in light of which this being is disposed to answer correctly any question about mathematics. But, this response assumes that the set of all mathematical truths could be derived from this finite stock of starting beliefs. I think, but I'm far from certain, that this has been ruled out by Godel's first incompleteness theorem.
But, what if someone just modifies the definition of omniscience. Suppose they think that no being could have an infinite number of beliefs and say that (a) the omniscient being knows everything that can be known or (b) knows as much as can be known. You couldn't argue that there are no omniscient beings by, say, arguing that such a being would have to have an infinite collection of beliefs and then argue that there are no actual infinities. I think this is sort of a cop out, but is there any contradiction to saying that the essentially omniscient being fails to know some mathematical truths if we work with this definition of omniscience? It seems that there is an infinite stock of propositions that can be known, but while on (a) that means that the omniscient being would have to have an infinite number of beliefs on (b) it's not so clear.