What is the flaw in Cooper's argument?

Question

Lately I have been studying in the subject of degree theory, specifically definability results related to $\mathcal{D}$. A famous conjecture in the field due to Slaman and Woodin is that the only automorphism group of $\mathcal{D}$ is the trivial one. However, it seems in the 1999 paper: "Upper cones as automorphism bases" Cooper gives a counterexample which is pointed out in other sources to be wrong. What is the fault of his argument?

Answer 1

The MathSciNet review of Cooper’s paper by Steffen Lempp points out many key definitions that are unclear/imprecise, to the extent that the proof is unverifiable — essentially, not even wrong. Disclaimer: I’m not an expert in the area, and haven’t managed to find a full text of Cooper’s paper, so I can’t myself vouch for any of this — but the review seems to pretty much directly answer your question, so I’m posting it here as such. Full details are at the link, for MathSciNet subscribers, but here are enough examples to make the point:

I have tried repeatedly in the past, and again now, to follow Cooper's proof, but have been unable to do so. […]

Let me try to point out some of the places where I get stuck:

(1) The absolutely crucial Definition 1.7 on pages 21–23 presents the definitions of “configured”, “preconfigured”, and “inner preconfigured”, upon which all of the later work hinges. Unfortunately, this definition is very unclear to me. σ being “configured” relative to π according to ⟨α⟩ seems to mean that one can compute the string σ from the string π by successively applying in order the functionals mentioned in the tree node α. The other two notions, however, are a mystery to me: Why should, in clauses 3) and 4) on page 22, σ's being configured relative to π according to either ⟨α⟩ or ∅ imply that σ is preconfigured relative to π? This allows using the functionals mentioned in ⟨α⟩ repeatedly, contradicting my limited intuition of this notion being defined inductively. The role of the parameters αˆ, j, and σ′ in clause 4) on page 22 is unexplained.

(2) Page 24 presents the notion of “augmentation”. Given that Cooper does not explicitly define the notion “contains all the axioms relevant to the definition of”, it is unclear to me why the notions of augmentation for ε and ∘ε differ from that for +ε. (3) Page 25 presents the notion of stratified rank. It appears to me, however, that the εs(…,π) increase in each component as π increases, so it is unclear to me in what sense Cooper is taking the “union” of the cross-sections on line 10.

[…four more such notes…]

(7) Definition 2.4 is very vague; in particular “imprinting” is never formally defined.

Since the construction is so vague for me, I cannot comment much on the verification in Section 3 of the paper, which starts on page 37.