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I am currently reading the section on Gentzen's Consistency Proof in this article on Proof Theory from the Stanford Encyclopedia. There it says:

Given a natural ordinal representation system $\langle A, \succ, \dots \rangle$ of order type $\tau$ let $\textbf{PRA} + \mathrm{TI}_{qf}(< \tau)$ be $\textbf{PRA}$ augmented by quantifier-free induction over all initial (externally indexed) segments of $\succ$.

No doubt this is fairly trivial, but I have a hard time understanding what it is supposed to mean. Therefore, I would be happy if someone could spell out the details behind the statement above. To be more precise, my questions are:

  • What exactly is a natural ordinal representation system? Basically, an answer to this question would be very helpful.
  • What is meant by "quantifier-free induction over all initial (externally indexed) segments of $\succ$"?

As I understand it, the rule of quantifier-free induction gives $${\varphi(0) \quad \varphi(x) \rightarrow \varphi(S(x)) \over \varphi(y)}$$
for every predicate $\varphi$. Reading the Wikipedia article on $\textbf{PRA}$ I concluded that this rule is already part of $\textbf{PRA}$. But why should we then augment $\textbf{PRA}$ by quantifier-free induction if it is already part of $\textbf{PRA}$? I assume that I got something wrong here, but I don't know what.

Any help is appreciated!

bs_math
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    Maybe useful: Anna Horská, Where is the Godel-point hiding Gentzen's Consistency Proof of 1936 (Springer, 2014) – Mauro ALLEGRANZA Oct 06 '21 at 11:37
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    The issue is not so "transparent": in order to prove consistency of $\mathsf {PA}$, due to G's Incompleteness Th, we cannot use $\mathsf {PA}$ itself. Thus, Gentzen's strategy was to use as meta-theory a theory that from one side is "weaker" than $\mathsf {PA}$: induction is limited to quantifier-free formulas - but from another side is stronger: induction is not limited to the natural numbers (i.e. up to ordinal $\omega$) but is extender "higher" in the ordinal hierarchy (up to $\epsilon_0$). 1/2 – Mauro ALLEGRANZA Oct 06 '21 at 11:42
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    This is due to the fact that the procedure used by Gentzen "transformed an alleged PA-proof of an inconsistency into another proof such that the latter gets assigned a smaller ordinal than the former. [...] Gentzen’s proof, though elementary, was very intricate. As it turned out, the obstacles to cut elimination, inherent to PA, could be overcome by moving to a richer proof system, albeit in a drastic way by going infinite. This richer system allows for proof rules with infinitely many premises." – Mauro ALLEGRANZA Oct 06 '21 at 11:43
  • See also Contents of Gentzen's consistency proof of PA – Mauro ALLEGRANZA Oct 06 '21 at 13:02

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