Things that can and things that can’t be done in PRA

Abstract

It is well-known by now that large parts of (non-constructive) mathematical reasoning can be carried out in systems T which are conservative over primitive recursive arithmetic PRA (and even much weaker systems). On the other hand there are principles S of elementary analysis (like the Bolzano-Weierstraß principle, the existence of a limit superior for bounded sequences etc.) which are known to be equivalent to arithmetical comprehension (relative to T ) and therefore go far beyond the strength of PRA (when added to T ). In this paper we determine precisely the arithmetical and computational strength (in terms of optimal conservation results and subrecursive characterizations of provably recursive functions) of weaker function parameter-free schematic versions S − of S, thereby exhibiting different levels of strength between these principles as well as a sharp borderline between fragments of analysis which are still conservative over PRA and extensions which just go beyond the strength of PRA.

Transfer and a supremum principle for ERNA

“…In this section we will prove an ERNA-version of completeness, to be understood 'up to infinitesimals'. As mentioned before, PRA cannot prove Dedekind completeness (see [6]), and neither can ERNA. We start with Cauchy completeness, which will be used for Dedekind completeness.…”

“…Indeed, with such a function st, ERNA would allow to construct the field of real numbers. As ERNA's consistency is proved in PRA, the latter would also allow to construct the real number field, something which is known to be impossible, [6].…”

“…Three possible types remain to be considered. In defining the corresponding formula T ϕ we use ERNA's function d defined in (6).…”

Effective Uniform Bounds on the Krasnoselski-Mann Iteration

“…Whereas these systems contain quite some parts of nonconstructive analysis, principles based on sequential compactness are not included. The significant and highly non-trivial impact of such principles for the extraction of bounds has been determined completely in [27] and [28]. We only discuss the results for the particular simple case of the principle…”

“…Because of the existence of the 'Cauchy modulus f ', '∀(a) n PCM(a n )' is equivalent to the principle of so-called arithmetical comprehension which potentially creates bounds of huge complexity when added to systems like PBA, EBA (see [28]). What we showed in [27] is, that things are quite different when PCM(a n ) is only applied to sequences (a n ) in a given proof of a theorem ( * ) which can be explicitly defined in terms of the parameters n, x, y of ( * ).…”