Hyperations, Veblen progressions and transfinite iterations of ordinal functions

In this paper we introduce hyperations and cohyperations, which are forms of transfinite iteration of ordinal functions. Hyperations are iterations of normal functions. Unlike iteration by pointwise convergence, hyperation preserves normality. The hyperation of a normal function f is a sequence of normal functions so that f^0= id, f^1 = f and for all ordinals \alpha, \beta we have that f^(\alpha + \beta) = f^\alpha f^\beta. These conditions do not determine f^\alpha uniquely; in addition, we require that the functions be minimal in an appropriate sense. We study hyperations systematically and show that they are a natural refinement of Veblen progressions. Next, we define cohyperations, very similar to hyperations except that they are left-additive: given \alpha, \beta, f^(\alpha + \beta)= f^\beta f^\alpha. Cohyperations iterate initial functions which are functions that map initial segments to initial segments. We systematically study cohyperations and see how they can be employed to define left inverses to hyperations. Hyperations provide an alternative presentation of Veblen progressions and can be useful where a more fine-grained analysis of such sequences is called for. They are very amenable to algebraic manipulation and hence are convenient to work with. Cohyperations, meanwhile, give a novel way to describe slowly increasing functions as often appear, for example, in proof theory

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

Laver's results and low-dimensional topology

In connection with his interest in selfdistributive algebra, Richard Laver established two deep results with potential applications in low-dimensional topology, namely the existence of what is now known as the Laver tables and the well-foundedness of the standard ordering of positive braids. Here we present these results and discuss the way they could be used in topological applications

Sharp thresholds for hypergraph regressive Ramsey numbers

The f-regressive Ramsey number R(f)(reg)(d, n) is the minimum N such that every coloring of the d-tuples of an N-element set mapping each x(1),...,x(d) to a color below f(x(1)) (when f(x(1)) is positive) contains a min-homogeneous set of size n, where a set is called min-homogeneous if every two d-tuples from this set that have the same smallest element get the same color. If f is the identity, then we are dealing with the standard regressive Ramsey numbers as defined by Kanamori and McAloon. The existence of such numbers for hypergraphs or arbitrary dimension is unprovable from the axioms of Peano Arithmetic. In this paper we classify the growth-rate of the regressive Ramsey numbers for hypergraphs in dependence of the growth-rate of the parameter function f. We give a sharp classification of the thresholds at which the f-regressive Ramsey numbers undergo a drastical change in growth-rate. The growth-rate has to be measured against a scale of fast-growing recursive functions indexed by finite towers of exponentiation in base omega (the first limit ordinal). The case of graphs has been treated by Lee, Kojman, Omri and Weiermann. We extend their results to hypergraphs of arbitrary dimension. From the point of view of Logic, our results classify the provability of the Regressive Ramsey Theorem for hypergraphs of fixed dimension in subsystems of Peano Arithmetic with restricted induction principles. (C) 2010 Elsevier Inc. All rights reserved

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

Analytic combinatorics, proof-theoretic ordinals, and phase transitions for independence results

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Current research on G\"odel's incompleteness theorems

We give a survey of current research on G\"{o}del's incompleteness theorems from the following three aspects: classifications of different proofs of G\"{o}del's incompleteness theorems, the limit of the applicability of G\"{o}del's first incompleteness theorem, and the limit of the applicability of G\"{o}del's second incompleteness theorem.Comment: 54 pages, final accepted version, to appear in The Bulletin of Symbolic Logi