What is the "simple logical truth" that makes omniscience self-contradictory?

Question

Patrick Grim claims in "Problems with Omniscience" that there is a "simple truth well established as a logical theorem", which shows that omniscience is a contradictory concept. I am failing to see the contradiction. What is the "simple truth" and where is the contradiction?

Here is the relevant excerpt (p.3):

"For any such system it is well known that we will be able to encode formulae recoverably as numbers. We will use A̅ to refer to the numbered encoding for a formula A. It is well known that for any such system we will be able to define a derivability relation I such that ⊢I(A̅, B̅) just in case B is derivable from A. Let us introduce a symbol '∇' within such a system, applicable to numerical encodings Ȧ for formulae A. We might introduce '∇' as a way of representing universal knowledge, for example—the knowledge of an omniscient being within at least the realm of this limited formal system. Given any such symbol with any such use we would clearly want to maintain each of the following:

If something is known by such a being, it is so:∇(A̅)→ A.

This fact is itself known by such a being:∇(∇(A) → A).

If B is derivable from A in the system, and A is known by such a being, B is known by such a being as well:I(A̅, B̅) →(∇ (A̅) → ∇ (B̅)).

The simple truth, however, well established as a logical theorem, is that no symbol can consistently mean what we have proposed '∇' to mean, even in a context as limited as formal as arithmetic."

UPDATE: I found the answer to the question, but it is a bit too complex for me, and it is only partially available via preview, so what does this link mean, and does it indeed answer the question?

Answer 1

I am not surprised at the confusion because the theorem in question is neither simple nor entirely logical. It is Tarski's undefinability of truth theorem, which says roughly that one can not define a faithful "truth predicate", which unerringly detects when a sentence is true. More precisely, there is no formula T() in the first order arithmetic such that T(A̅) is true if and only if A is true. Grim's "knowledge of an omniscient being" predicate ∇ is essentially Tarski's truth predicate.

Undefinability of truth is not simple because it is equivalent to Gödel's incompleteness theorem, and while both can be explained to a pedestrian, their actual meanings, not to mention proofs, are quite subtle and technical. It is not quite logical, because both the statement and the proof require the use of some arithmetical concepts and methods, admittedly basic ones, but beyond what is understood by logic traditionally (they would fall under Frege's Logic).

I have to say that I find Grim's argument far short of compelling. Both Tarski's and Gödel's theorems apply to the first order theories with arithmetic, they are no longer true in the second order logic. But even without that, predicates are finite crutches that beings like us use to handle the infinite, an omniscient being has no need for such tools. It can know every true instance separately, all infinitely many of them, without any need for a definable predicate. Such knowledge of instances need not be "definable" (i.e. algorithmic) in any way, let alone in the first order logic. In other words, it produces no truth predicate, and no contradiction with the first order undefinability of truth.

There is an interesting "knowability paradox" that shows existence of unknowable truths if there are any unknown ones. But exactly for an omniscient being its reasoning fails, since no truths are unknown to such being. I am not aware of good arguments that omniscience by itself is contradictory. Omnipotence is another matter, a being either can or can not create a stone it can not lift. Either way, it is not omnipotent without qualifications.