A uniform approach to fundamental sequences and hierarchies

In this article we give a unifying approach to the theory of fundamental sequences and their related Hardy hierarchies of number-theoretic functions and we show the equivalence of the new approach with the classical one

A Computation of the Maximal Order Type of the Term Ordering on Finite Multisets

We give a sharpening of a recent result of Aschenbrenner and Pong about the maximal order type of the term ordering on the finite multisets over a wpo. Moreover we discuss an approach to compute maximal order types of well-partial orders which are related to tree embeddings

Phase transitions related to the pigeonhole principle

Since Paris introduced them in the late seventies (Paris1978), densities turned out to be useful for studying independence results. Motivated by their simplicity and surprising strength we investigate the combinatorial complexity of two such densities which are strongly related to the pigeonhole principle. The aim is to miniaturise Ramsey's Theorem for $1$-tuples. The first principle uses an unlimited amount of colours, whereas the second has a fixed number of two colours. We show that these principles give rise to Ackermannian growth. After parameterising these statements with respect to a function f:N->N, we investigate for which functions f Ackermannian growth is still preserved

The proof-theoretic strength of Ramsey's theorem for pairs and two colors

Ramsey's theorem for $n$-tuples and $k$-colors ($\mathsf{RT}^n_k$) asserts that every k-coloring of $[\mathbb{N}]^n$ admits an infinite monochromatic subset. We study the proof-theoretic strength of Ramsey's theorem for pairs and two colors, namely, the set of its $\Pi^0_1$ consequences, and show that $\mathsf{RT}^2_2$ is $\Pi^0_3$ conservative over $\mathsf{I}\Sigma^0_1$. This strengthens the proof of Chong, Slaman and Yang that $\mathsf{RT}^2_2$ does not imply $\mathsf{I}\Sigma^0_2$, and shows that $\mathsf{RT}^2_2$ is finitistically reducible, in the sense of Simpson's partial realization of Hilbert's Program. Moreover, we develop general tools to simplify the proofs of $\Pi^0_3$-conservation theorems.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

A note on the consistency operator

It is a well known empirical observation that natural axiomatic theories are pre-well-ordered by consistency strength. For any natural theory $T$, the next strongest natural theory is $T+\mathsf{Con}_T$. We formulate and prove a statement to the effect that the consistency operator is the weakest natural way to uniformly extend axiomatic theories

Honest elementary degrees and degrees of relative provability without the cupping property

An element a of a lattice cups to an element b>ab>a if there is a c<bc<b such that a∪c=ba∪c=b. An element of a lattice has the cupping property if it cups to every element above it. We prove that there are non-zero honest elementary degrees that do not have the cupping property, which answers a question of Kristiansen, Schlage-Puchta, and Weiermann. In fact, we show that if b is a sufficiently large honest elementary degree, then b has the anti-cupping property, which means that there is an a with 0<Ea<Eb0<Ea<Eb that does not cup to b. For comparison, we also modify a result of Cai to show, in several versions of the degrees of relative provability that are closely related to the honest elementary degrees, that in fact all non-zero degrees have the anti-cupping property, not just sufficiently large degrees

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