Very weak metatheories

[Pedro Sánchez Terraf]

It seems to be a standard practice to use Primitive Recursive Arithmetic 
as a “finitistic metatheory”; at least that is the take on Kunen's 2011 
Set Theory book.

My question, whose greatest sin is to have emerged out of sheer 
curiosity, is about what happens if one replaces PRA by a weaker, even 
ultrafinitist, metatheory.

  * Do relative consistency results (CH, ¬CH) go through as usual?
  * What about the Reflection Principle?
  * (Your metatheorem of choice)?

After perusing a monograph by Buss, it seems to me (though I'm no expert 
on the matter!), that with a big enough fragment of Bounded Arithmetic 
one might develop Gödel numbering, but I'm not sure if also the 
requisite machinery for the previous results can be set up. I'm curious 
of what happens if one goes even lower (strictly bounded, or even less).

Apologies in advance for any nonsense above.

Best regards,

--
Pedro Sánchez Terraf
CIEM-FAMAF — Universidad Nacional de Córdoba
cs.famaf.unc.edu.ar/~pedro <https://cs.famaf.unc.edu.ar/~pedro>
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