Very weak metatheories

[David Auerbach]

Visser has some papers that seem relevant to your interests; among them:
Can We Make the Second Incompleteness Theorem Coordinate Free, Visser 10.1093/logcom/exp048
The arithmetics of a Theory: https://doi.org/10.1215/00294527-2835029 <https://doi.org/10.1215/00294527-2835029>

David Auerbach



> On Jul 27, 2022, at 12:11 PM, Pedro Sánchez Terraf <sterraf at famaf.unc.edu.ar> wrote:
> 
> 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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