Reverse mathematics and well-ordering principles

The paper is concerned with generally Pi^1_2 sentences of the form 'if X is well ordered then f(X) is well ordered', where f is a standard proof theoretic function from ordinals to ordinals. It has turned out that a statement of this form is often equivalent to the existence of countable coded omega-models for a particular theory T_f whose consistency can be proved by means of a cut elimination theorem in infinitary logic which crucially involves the function f. To illustrate this theme, we shall focus on the well-known psi-function which figures prominently in so-called predicative proof theory. However, the approach taken here lends itself to generalization in that the techniques we employ can be applied to many other proof-theoretic functions associated with cut elimination theorems. In this paper we show that the statement 'if X is well ordered then 'X0 is well ordered' is equivalent to ATR0. This was first proved by Friedman, Montalban and Weiermann [7] using recursion-theoretic and combinatorial methods. The proof given here is proof-theoretic, the main techniques being Schuette's method of proof search (deduction chains) [13], generalized to omega logic, and cut elimination for infinitary ramified analysis

Unbounded arithmetic

When comparing the different axiomatizations of bounded arithmetic and Peano arithmetic, it becomes clear that there are similarities between the fragments of these theories. In particular, it is tempting to draw an analogy between the hierarchies of bounded arithmetic and Peano arithmetic. However, one cannot deny that there are essential and deeply rooted differences and the most one can claim is a weak analogy between these hierarchies. The following quote by Kaye expresses this argument in an elegant way: "Many authors have emphasized the analogies between the fragments Sigma(b)(n)-IND of IDelta0+(Vx)(xlog x) exists) and the fragments ISigma(n) of Peano arithmetic. Sometimes this is helpful, but often one feels that the bounded hierarchy of theories is of a rather different nature and new techniques must be developed to answer the key questions concerning them." In this paper, we propose a (conjectured) hierarchy for Peano arithmetic which is much closer to that of bounded arithmetic than the existing one. In light of this close relation, techniques developed to establish properties of the new hierarchy should carry over naturally to the (conjectured) hierarchy of bounded arithmetic. As the famous P vs. NP problem is related to the collapse of the hierarchy of bounded arithmetic, the new hierarchy may prove particularly useful in solving this famous problem

Analytic combinatorics for a certain well-ordered class of iterated exponential terms

International audienceThe aim of this paper is threefold: firstly, to explain a certain segment of ordinals in terms which are familiar to the analytic combinatorics community, secondly to state a great many of associated problems on resulting count functions and thirdly, to provide some weak asymptotic for the resulting count functions. We employ for simplicity Tauberian methods. The analytic combinatorics community is encouraged to provide (maybe in joint work) sharper results in future investigations

Phase Transitions for Gödel Incompleteness

Gödel's first incompleteness result from 1931 states that there are true assertions about the natural numbers which do not follow from the Peano axioms. Since 1931 many researchers have been looking for natural examples of such assertions and breakthroughs have been obtained in the seventies by Jeff Paris (in part jointly with Leo Harrington and Laurie Kirby) and Harvey Friedman who produced first mathematically interesting independence results in Ramsey theory (Paris) and well-order and well-quasi-order theory (Friedman). In this article we investigate Friedman style principles of combinatorial well-foundedness for the ordinals below epsilon_0. These principles state that there is a uniform bound on the length of decreasing sequences of ordinals which satisfy an elementary recursive growth rate condition with respect to their Gödel numbers. For these independence principles we classify (as a part of a general research program) their phase transitions, i.e. we classify exactly the bounding conditions which lead from provability to unprovability in the induced combinatorial well-foundedness principles. As Gödel numbering for ordinals we choose the one which is induced naturally from Gödel's coding of finite sequences from his classical 1931 paper on his incompleteness results. This choice makes the investigation highly non trivial but rewarding and we succeed in our objectives by using an intricate and surprising interplay between analytic combinatorics and the theory of descent recursive functions. For obtaining the required bounds on count functions for ordinals we use a classical 1961 Tauberian theorem by Parameswaran which apparently is far remote from Gödel's theorem

Unprovability results involving braids

We construct long sequences of braids that are descending with respect to the standard order of braids (``Dehornoy order''), and we deduce that, contrary to all usual algebraic properties of braids, certain simple combinatorial statements involving the braid order are true, but not provable in the subsystems ISigma1 or ISigma2 of the standard Peano system.Comment: 32 page

Derivation Lengths Classification of G\"odel's T Extending Howard's Assignment

Let T be Goedel's system of primitive recursive functionals of finite type in the lambda formulation. We define by constructive means using recursion on nested multisets a multivalued function I from the set of terms of T into the set of natural numbers such that if a term a reduces to a term b and if a natural number I(a) is assigned to a then a natural number I(b) can be assigned to b such that I(a) is greater than I(b). The construction of I is based on Howard's 1970 ordinal assignment for T and Weiermann's 1996 treatment of T in the combinatory logic version. As a corollary we obtain an optimal derivation length classification for the lambda formulation of T and its fragments. Compared with Weiermann's 1996 exposition this article yields solutions to several non-trivial problems arising from dealing with lambda terms instead of combinatory logic terms. It is expected that the methods developed here can be applied to other higher order rewrite systems resulting in new powerful termination orderings since T is a paradigm for such systems