Very weak metatheories

[Timothy Y. Chow]

Pedro Sanchez Terraf 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)?

A similar question came up on MathOverflow a couple of years ago.  You 
might find some relevant information in the answers and comments.

https://mathoverflow.net/questions/378777/did-edward-nelson-accept-the-incompleteness-theorems

Tim