For some reason I have never seen people state the correct resolution to the Jailor paradox (I mean I'm sure it's in like proper logic papers and stuff¹ but I mean on the internet), so making a note of it here.

A jailor will execute a prisoner on one of the following weekdays (Mon-Fri). He tells the prisoner: "When I will execute you, it will come as a surprise to you." The prisoner reasons: If I am not executed Mon-Thu, I will know for sure I will be executed on Friday; thus I cannot be executed on Friday (as it will be no surprise). Similarly by induction, I cannot be executed on any day. Yet, the jailor comes in on Tuesday and executes the prisoner, taking him by surprise.

The correct answer is: the jailor is just a Godelian program to the prisoner (does the opposite of what the prisoner predicts). The moment the prisoner reasons "I cannot be hung on Friday", this belief becomes wrong. A sound prisoner must therefore remain uncertain (incomplete) about this.

More precisely, suppose we define "prisoner P being surprised by being executed on day X" as: P_X ⊢ ¬X. Then the prisoner's reasoning is (where 1-5 represent the events of being executed on each day) as follows.

Axioms:

P₀ := P; P_X + 1 := P_X + {¬X} 1 ∨ 2 ∨ 3 ∨ 4 ∨ 5 (exclusively) ∀X ∈ [1,5], X ⟹ P_X ⊢ ¬X Then on any day P may reason ¬(1∨2∨3∨4) ⟹ 5 ⟹ P₅ ⊢ ¬5 , and thus that: ¬(1∨2∨3∨4) ⟹ ¬Con(P₅). The only way he can actually establish ¬5 (leading to the paradox) though, is by proving Con(P₅), which he doesn't (after all P₅ is a stronger version of P). Now it is of course true that P₅ itself would reason that ¬5 (and that 5), but that's fine: P₅ is inconsistent.

This is a strong definition of surprise: we may instead define it as ¬P_X ⊢ X and modify (2) accordingly. Then similarly P reasons "5 ⟹ ¬P₅ ⊢ 5, but P₅ ⊢ 5 due to its additional axioms (0), so ¬5". Likewise, it can reason ¬4, …¬1. So P is just actually inconsistent with this modified (2).

These are both illustrations with specific definitions of "surprise", but the general principle ("the jailor is a Godelian program") explains why any prisoner capable of reasoning about the jailor must be incomplete or inconsistent.