Very weak metatheories

Richard Kimberly Heck richard_heck at
Wed Jul 27 22:14:38 EDT 2022

A couple comments on this.

> 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).
You can do Gödel numbering in extremely weak theories: Even Q can do 
that. A proper development of syntax needs a bit more, but Buss's theory 
S^1_2 is (as Albert Visser once remarked) almost exactly right for 
syntax. That theory, IIRC, is synonymous with I\Delta_0 + \Omega_1, and 
both of these are interpretable in Q, so they are 'very weak' in a 
well-defined sense.

> 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?
In so far as these are based on interpretability, I believe they will, 
yes. There are some subtleties here, which are discussed in the 
literature on interpretability logic: In very weak theories, we have to 
distinguish between 'axioms interpretability' and 'theorems 
interpretability', and relative consistency depends upon the latter. 
Visser's paper "The Formalization of Interpretability" discusses all of 
this and introduces an intermediate notion that he calls "smooth 
interpretabilty". He also shows that if we help ourselves to just a bit 
more, namely \Sigma_1 (equivalently, \Delta_0) collection, then 
everything works out just fine. That still keeps us below PRA, I believe.


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