User blog comment:Edwin Shade/A Complete Analysis of Taranovsky's Notation/@comment-30118230-20180129200050/@comment-27513631-20180129212609

To clarify my position, I'm not personally convinced that Taranovsky's system is stronger that Pi12-CA.

Now, I'd argue that, yes, each 'n' probably corresponds to an additional level of reflection, and this sequence plausibly has a nonprojectible ordinal as its supremum (KP + nonprojectible universe ~= Pi12-CA).

However, there seems little intrinsic reason to accept that the system is stronger than this. The structure of even much weaker collapsing functions requires strong set-theoretic assumptions and moreover uses the structure provided by the large cardinals as effectively as it reasonably can - it seems unreasonable that a much simpler, less systematically justified notation can reach ground-breaking strengths.

Taranovsky even speaks of Σ1 definability in Lκ+a+1 (and of little more complex) when proving the consistency of the Ci systems. I don't see anythong that obviously can't be recast to replacing cardinals with admissible ordinals, which would place that system firmly at most at Pi12-CA.

(Note that a nonprojectible ordinal α is an ordinal such that Lα is a limit of its Σ1-elementary substructures (equivalently, α is a limit of α-stable ordinals), so as non-Σ1-definability in Lκ+α corresponds to α-stability, the above paragraph's argument follows through.)