TRIAL AND ERROR MATHEMATICS II: DIALECTICAL SETS AND QUASIDIALECTICAL SETS, THEIR DEGREES, AND THEIR DISTRIBUTION WITHIN THE CLASS OF LIMIT SETS

This paper is a continuation of Amidei, Pianigiani, San Mauro, Simi, & Sorbi (2016), where we have introduced the quasidialectical systems, which are abstract deductive systems designed to provide, in line with Lakatos’ views, a formalization of trial and error mathematics more adherent to the real mathematical practice of revision than Magari’s original dialectical systems. In this paper we prove that the two models of deductive systems (dialectical systems and quasidialectical systems) have in some sense the same information content, in that they represent two classes of sets (the dialectical sets and the quasidialectical sets, respectively), which have the same Turing degrees (namely, the computably enumerable Turing degrees), and the same enumeration degrees (namely, the ${\rm{\Pi }}_1^0$ enumeration degrees). Nonetheless, dialectical sets and quasidialectical sets do not coincide. Even restricting our attention to the so-called loopless quasidialectical sets, we show that the quasidialectical sets properly extend the dialectical sets. As both classes consist of ${\rm{\Delta }}_2^0$ sets, the extent to which the two classes differ is conveniently measured using the Ershov hierarchy: indeed, the dialectical sets are ω-computably enumerable (close inspection also shows that there are dialectical sets which do not lie in any finite level; and in every finite level n ≥ 2 of the Ershov hierarchy there is a dialectical set which does not lie in the previous level); on the other hand, the quasidialectical sets spread out throughout all classes of the hierarchy (close inspection shows that for every ordinal notation a of a nonzero computable ordinal, there is a quasidialectical set lying in ${\rm{\Sigma }}_a^{ - 1}$, but in none of the preceding levels).

Assignment of ordinals to patterns of resemblance

In [2] T. J. Carlson introduces an approach to ordinal notation systems which is based on the notion of Σ1-elementary substructure. We gave a detailed ordinal arithmetical analysis (see [7]) of the ordinal structure based on Σ1-elementarily as defined in [2]. This involved the development of an appropriate ordinal arithmetic that is based on a system of classical ordinal notations derived from Skolem hull operators, see [6]. In the present paper we establish an effective order isomorphism between the classical and the new system of ordinal notations using the results from [6] and [7]. Moreover, on the basis of a concept of relativization we develop mutual (relatively) elementary recursive assignments which are uniform with respect to the underlying relativization.

A Buchholz derivation system for the ordinal analysis of KP + Π3-reflection

In this paper we introduce a notation system for the infinitary derivations occurring in the ordinal analysis of KP + Π3-Reflection due to Michael Rathjen. This allows a finitary ordinal analysis of KP + Π3-Reflection. The method used is an extension of techniques developed by Wilfried Buchholz, namely operator controlled notation systems for RS∞-derivations. Similarly to Buchholz we obtain a characterisation of the provably recursive functions of KP + Π3-Reflection as <-recursive functions where < is the ordering on Rathjen's ordinal notation system . Further we show a conservation result for -sentences.

Parsimony hierarchies for inductive inference

Freivalds defined an acceptable programming system independent criterion for learning programs for functions in which the final programs were required to be both correct and “nearly” minimal size. i.e.. within a computable function of being purely minimal size. Kinber showed that this parsimony requirement on final programs limits learning power. However, in scientific inference, parsimony is considered highly desirable. Alim-computable functionis (by definition) one calculable by a total procedure allowed to change its mind finitely many times about its output. Investigated is the possibility of assuaging somewhat the limitation on learning power resulting from requiring parsimonious final programs by use of criteria which require the final, correct programs to be “not-so-nearly” minimal size, e.g., to be within a lim-computable function of actual minimal size. It is shown that some parsimony in the final program is thereby retained, yet learning power strictly increases. Considered, then, are lim-computable functions as above but for whichnotations forconstructive ordinals are used to bound the number of mind changes allowed regarding the output. This is a variant of an idea introduced by Freivalds and Smith. For this ordinal notation complexity bounded version of lim-computability, the power of the resultant learning criteria form finely graded, infinitely ramifying, infinite hierarchies intermediate between the computable and the lim-computable cases. Some of these hierarchies, for the natural notations determining them, are shown to be optimally tight.