Neuro-eliminationists like to trumpet that we only ever retroactively identify with what are just neuronal states. So that when I intend to open this door, my brain has already started doing that. Then I add my intention on top of it.
But isn't this a beautiful observation that Schelling already makes? There is the blind rotation of the drives, and then I "own" them—I become "me" at that point.
(I am of course compressing the process here, for brevity's sake, because strictly "I" don't exist before this decision, this Entscheidung (as Schelling puts it, strangely anticipating Hilbert). Then again, neither does the neuronal process as "just a bunch of neuron firings." That only comes into view because of the gap between it and my weird me-ness.)
What the neuro-eliminationist seems to want to ward off is precisely this gap. They want a world without gaps, where there is smooth continuity between things. This might be very difficult to find in a reality in which there are transfinite sets, which are intrinsically gappy, to the max. For instance.
What is the gap, more precisely? It's the gap between levels of a hierarchy created by a strange level crossing feedback loop. What this means is that the eliminationist is worried about the possibility of downward causality, or, to put it a little more poetically, the possibility of a spectral dimension that exerts causal pressure on a physical one. We're not talking about something purely "mental" influencing something purely "physical." We're talking about a strange third realm that is neither "physical" nor "spiritual" in a strict sense. We're talking about how a specter or spirit, a kind of emergent pattern, can exert downward causality on the system from which it emerges.
We're talking about how causality is aesthetic, which is perfectly standard Kant: sufficient reason is on the side of phenomenon, not thing.
It seems perfectly plausible that a level crossing feedback loop can happen, and indeed, denying that it can happen means you have to take issue with Gödel. Good luck!
No amount of torture—strangely medieval Inquisitorial practice for a supposedly modern neuro-eliminationist—will prove me wrong.