Update: Simple concise version
Thanks to Nick R for pointing in the right direction.
The statement P0: "this statement is false" is undecidable.
The statement P1: "this statement is undecidable" is also undecidable.
P0 can be proved to be undecidable but P1 cannot. How could one categorize these different "levels" of undecidability?
Let's first consider the basic contradictory self-reference:
P0 = "Statement P0 is false"
P0 can't be either true or false without contradiction. This is similar to NaN (Not a Number) in mathematics, only with boolean instead of numeric values. Let's define the truth value of P0 as NaB (Not a Boolean).
The statement in the title:
P1 = "It's not possible to deduce whether P1 is true or false"
is similar in that it's self-referencing but doesn't directly contradict itself. Is the truth value of P1 same as P0, which I defined NaB, or something different? A more complex class of instability?
- If P1 is false, then it should be possible to deduce that. I.e. it must be shown that P1 being true leads to contradiction.
- If P1 is true, then it might very well be, we just don't know it.
Both options are possible - neither one can be ruled out, therefore it's not possible to deduce the truth value of P1.
But that's exactly what P1 says, so we've just proved that it is true! Unlike with P0, even supposing that it's neither true or false makes it back to true. Almost as if it's oscillating between not 2 but 3 values (true, false, neither).
Does this mean that the truth value of P1 is some kind of meta-NaB: "Neither a boolean nor a NaB"?
EDIT: Reworded again for clarification. Question is about P1, not P0. P0 is unrelated only there for the comparison