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Feferman-Schütte ordinal is sometimes said to be:

....first impredicative ordinal, though this is controversial, partly because there is no generally accepted precise definition of "predicative". Sometimes an ordinal is said to be predicative if it is less than $Γ_0$.

Now Nik Weaver introduces a formal system called Tarski-something-something which I do not even understand.

As I searched it seemed there has been a correspondence between Nik Weaver and Solomon Feferman which is linked here. At this stage, the concept has become highly abstract for me to understand.

Has any research been done on formalizing the concept? And does the vicious circle principle inhibit from "sort of" going beyond the "system", (intuitively speaking of course)? I am cognizant of Takeuti's ordinal diagram, which again is beyond scope of my understanding at this phase.

Also, how accepted is Nik Weaver's formal system of Tarski? Can one formalize predicativity without invoking vague, philosophical notions?

Edit: As I was searching I found that a question of this nature was asked in Discussion in nLab

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"Predicativity" is inherently a vague, philosophical notion. At best one can present a formal system and a philosophical argument that the system is predicatively valid. The formalization itself can be perfectly precise but the argument that the system is acceptable is not a mathematical argument. –  Carl Mummert Jan 23 '12 at 2:48
    
@CarlMummert I suspect when you mentioned the part about acceptibility you were referring to my wording of "Nik Weaver's formal system of Tarski" . Actually what I had in mind by acceptability is whether his theory has been peer-reviwed and for lack of better term _acknowledged_(?). Then again, essentially aren't all of mathematical philosophy notions boil down to being peer-reviwed and published to gain "acceptance"? On a different note (and at the risk of begging), will progress be made to solve problems, by the notion of predicativity being precisely defined à la models of computation? –  Sniper Clown Jan 23 '12 at 3:02
    
@Mahmud, no, Weaver's ideas have pretty much no mainstream acceptance. If you look in the FOM list archives in 2006, you can see Harvey Friedman's discussion with Weaver, and it's clear how marginal Weaver's viewpoint is. –  Keshav Srinivasan Dec 19 '13 at 18:36
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