A1⊢P means that A1 can prove P, not that a proof has been found.
What you're proposing is essentially replacing the axiom (using your notation [[t]]P for "t is a proof of P")
∀P. (∃t. [[t]]P) → P
with an axiom schema
∀P. ([[t]]P) → P
where each value of t gives a different axiom. I'll have to think about whether this works - my initial feeling is no, but I'll look into it more when I have a chance.
Edit: The ∀P should also be interpreted as a schema; it's rather nonstandard to be able to quantify over statements.
In On Explicit Reflection in Theorem Proving and Formal Verification S. Artemov claims that for any t and P, it is derivable that [[t]]P→P, provided that the system has a few properties: the underlying logic is classical or intuitionistic, proofhood in the system is decidable / theoremhood is computably enumerable, the system can represent any computable function and decidable relation, and there is a function rep that maps syntactic terms to ground terms. PatrickRobotham tells me the paper is very messy, so maybe that doesn't matter. I'm clearly over my head.
I tried to write down my idea a few times, but it was badly wrong each time. Now, Instead of solving the problem, I'm just going to give a more conservative summary of what the problem is.
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Eliezer's talk at the 2011 Singularity Summit focused largely on the AI reflection problem (how to build AI that can prove things about its own proofs, and execute self modifications on the basis of those proofs, without thereby reducing its self modification mojo). To that end, it would be nice to have a "reflection principle" by which an AI (or its theorem prover) can know in a self-referential way that its theorem proving activities are working as they should.
The naive way to do this is to use the standard provability predicate, ◻, which can be thought of as asking whether a proof of a given formula exists. Using this we can try to formalize our intuition that a fully reflective AI, one that can reason about itself in order to improve itself, should understand that its proof deriving behavior does in fact produce sentences derivable from its axioms:
AI ⊢ ◻P → P,
which is intended to be read as "The formal system "AI" understand in general that "If a sentence is provable then it is true" ", though literally it means something a bit more like "It is derivable from the formal system "AI" that "if there exists a proof of sentence P, then P", in general for P".
Surprisingly, attempting to add this to a formal system, like Peano Arithmetic, doesn't work so well. In particular, it was shown by Löb that adding this reflection principle in general lets us derive any statement, including contradictions.
So our nice reflection principle is broken. We don't understand reflective reasoning as well as we'd like. At this point, we can brainstorm some new reflection principles: Maybe our reflection principles should be derivable within our formal system, instead of being tacked on. Also, we can try to figure out in a deeper way why "AI ⊢ ◻P → P" doesn't work: If we can derive all sentences, does that mean that proofs of contradictions actually do exist? If so, why weren't those proofs breaking our theorem provers before we added the reflection principle? Or do the proofs for all sentences only exist for the augmented theorem prover, not for the initial formal system? That would suggest our reflection principle is allowing the AI to trust itself too much, allowing it to derive things because deriving them allows them to be derived. Though it doesn't really look like that if you just stare at the old reflection principle. We are confused. Let's unconfuse ourselves.