This question is about a technical imprecision which is easily avoidable but whose details I'd like to understand better. When we refer to "the consistency strength of $\mathsf{PA}$" (say) we aren't properly referring to $\mathsf{PA}$ as a set; rather, we care about the specific way in which $\mathsf{PA}$ is enumerated. "The set of sentences which $\mathsf{ZFC}$ proves are $\mathsf{PA}$-axioms" is another way to define the same (hopefully!) set, but properly speaking it is equiconsistent with $\mathsf{ZFC}$.


It occurs to me that I don't know whether the usual description of the $\mathsf{PA}$ axioms is optimal with respect to consistency strength. Below, let $(T\_e)\_{e\in\omega}$ be some appropriate enumeration of the c.e. theories. Suppose WLOG that $T\_0$ is the usual c.e. presentation of $\mathsf{PA}$.



> 
> **Question**: Is there an index $e$ such that ("extensionally") $T\_e=\mathsf{PA}$ but $\mathsf{PA}+\mathsf{Con}(T\_e)\not\vdash \mathsf{Con}(T\_0)$?
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> 


One natural way to try to get a positive answer to this question is to enumerate the axioms of $\mathsf{PA}$ very slowly, waiting for contradictions, along the following lines: given a $\mathsf{PA}$-provably-total computable function $f$, let $T\_f$ be the c.e. theory which enumerates the $n$th axiom of $\mathsf{PA}$ in the usual ($T\_0$) sense iff there is no contradiction in $\mathsf{PA}$ using fewer than $f(n)$-many symbols. As long as $f$ is reasonably-fast-growing (say, dominating $n\mapsto 2^n$) it seems "$\mathsf{PA}$-believable" that $T\_f$ detects a contradiction before falling prey to it. On the other hand, I don't see how to show the consistency of $\mathsf{PA}+\mathsf{Con}(T\_f)+\neg\mathsf{Con}(T\_0)$ for any specific $f$.


([This older question](https://mathoverflow.net/questions/417484/proving-short-consistency-can-we-do-better-than-brute-force-search) seems thematically similar, but I don't actually see a concrete connection.)



Is the usual enumeration of $\mathsf{PA}$ "minimal for consistency strength"?
No (to the title), yes (to the question), see


Sy-David Friedman, Michael Rathjen, and Andreas Weiermann: *Slow consistency*, Annals of Pure and Applied Logic 164 (2013), no. 3, pp. 382–393, doi [10.1016/j.apal.2012.11.009](https://doi.org/10.1016/j.apal.2012.11.009).


The idea is that an axiom of usual PA of Gödel number $n$ is padded to length $f(n)$, where $f$ is a recursive function growing faster than all provably total recursive functions of PA (namely, $F\_{\epsilon\_0}$).


