FOM: Banach-Tarski and Borel sets

[Kanovei]

>   There is a larger question here: how is TBU meant to be used? One
>can restrict one's attention to Borel sets in the contexts where it is
>appropriate to do so without having a formal theory like TBU around.
>People have been doing it for the last 100 years or so.

>John Steel

Perhaps the full strength of 2nd order PA can be not 
necessary to develop such a theory, a kind of 
"Borel PA-2" or perhaps "Delta^1_2 PA-2" will work,  
however this is perhaps not so nice as Feferman's 
reduction to PA itself.

Vladimir Kanovei