Unprovability and phase transitions in Ramsey theory

The first mathematically interesting, first-order arithmetical example of incompleteness was given in the late seventies and is know as the Paris-Harrington principle. It is a strengthened form of the finite Ramsey theorem which can not be proved, nor refuted in Peano Arithmetic. In this dissertation we investigate several other unprovable statements of Ramseyan nature and determine the threshold functions for the related phase transitions. Chapter 1 sketches out the historical development of unprovability and phase transitions, and offers a little information on Ramsey theory. In addition, it introduces the necessary mathematical background by giving definitions and some useful lemmas. Chapter 2 deals with the pigeonhole principle, presumably the most well-known, finite instance of the Ramsey theorem. Although straightforward in itself, the principle gives rise to unprovable statements. We investigate the related phase transitions and determine the threshold functions. Chapter 3 explores a phase transition related to the so-called infinite subsequence principle, which is another instance of Ramsey’s theorem. Chapter 4 considers the Ramsey theorem without restrictions on the dimensions and colours. First, generalisations of results on partitioning α-large sets are proved, as they are needed later. Second, we show that an iteration of a finite version of the Ramsey theorem leads to unprovability. Chapter 5 investigates the template “thin implies Ramsey”, of which one of the theorems of Nash-Williams is an example. After proving a more universal instance, we study the strength of the original Nash-Williams theorem. We conclude this chapter by presenting an unprovable statement related to Schreier families. Chapter 6 is intended as a vast introduction to the Atlas of prefixed polynomial equations. We begin with the necessary definitions, present some specific members of the Atlas, discuss several issues and give technical details

On the Hierarchy of Natural Theories

It is a well-known empirical phenomenon that natural axiomatic theories are pre-well-ordered by consistency strength. Without a precise mathematical definition of "natural," it is unclear how to study this phenomenon mathematically. We will discuss the significance of this problem and survey some strategies that have recently been developed for addressing it. These strategies emphasize the role of reflection principles and ordinal analysis and draw on analogies with research in recursion theory. We will conclude with a discussion of open problems and directions for future research

Slow Reflection

We describe a “slow” version of the hierarchy of uniform reflection principles over Peano Arithmetic (PA). These principles are unprovable in Peano Arithmetic (even when extended by usual reflection principles of lower complexity) and introduce a new provably total function. At the same time the consistency of PA plus slow reflection is provable in PA + Con ( PA ) . We deduce a conjecture of S.-D. Friedman, Rathjen and Weiermann: Transfinite iterations of slow consistency generate a hierarchy of precisely ε 0 stages between PA and PA + Con ( PA ) (where Con ( PA ) refers to the usual consistency statement)

Type-Two Well-Ordering Principles, Admissible Sets, and Pi^1_1-Comprehension

This thesis introduces a well-ordering principle of type two, which we call the Bachmann-Howard principle. The main result states that the Bachmann-Howard principle is equivalent to the existence of admissible sets and thus to Pi^1_1-comprehension. This solves a conjecture of Rathjen and Montalbán. The equivalence is interesting because it relates "concrete" notions from ordinal analysis to "abstract" notions from reverse mathematics and set theory. A type-one well-ordering principle is a map T which transforms each well-order X into another well-order T[X]. If T is particularly uniform then it is called a dilator (due to Girard). Our Bachmann-Howard principle transforms each dilator T into a well-order BH(T). The latter is a certain kind of fixed-point: It comes with an "almost" monotone collapse theta:T[BH(T)]->BH(T) (we cannot expect full monotonicity, since the order-type of T[X] may always exceed the order-type of X). The Bachmann-Howard principle asserts that such a collapsing structure exists. In fact we define three variants of this principle: They are equivalent but differ in the sense in which the order BH(T) is "computed". On a technical level, our investigation involves the following achievements: a detailed discussion of primitive recursive set theory as a basis for set-theoretic reverse mathematics; a formalization of dilators in weak set theories and second-order arithmetic; a functorial version of the constructible hierarchy; an approach to deduction chains (Schütte) and beta-completeness (Girard) in a set-theoretic context; and a beta-consistency proof for Kripke-Platek set theory. Independently of the Bachmann-Howard principle, the thesis contains a series of results connected to slow consistency (introduced by S.-D. Friedman, Rathjen and Weiermann): We present a slow reflection statement and investigate its consistency strength, as well as its computational properties. Exploiting the latter, we show that instances of the Paris-Harrington principle can only have extremely long proofs in certain fragments of arithmetic

Proof lengths for instances of the Paris–Harrington principle

As Paris and Harrington have famously shown, Peano Arithmetic does not prove that for all numbers k , m , n there is an N which satisfies the statement PH ( k , m , n , N ) : For any k-colouring of its n-element subsets the set 0 , … , N − 1 has a large homogeneous subset of size ≥m. At the same time very weak theories can establish the Σ 1 -statement ∃ N PH ( k ‾ , m ‾ , n ‾ , N ) for any fixed parameters k , m , n . Which theory, then, does it take to formalize natural proofs of these instances? It is known that ∀ m ∃ N PH ( k ‾ , m , n ‾ , N ) has a natural and short proof (relative to n and k) by Σ n − 1 -induction. In contrast, we show that there is an elementary function e such that any proof of ∃ N PH ( e ( n ) ‾ , n + 1 ‾ , n ‾ , N ) by Σ n − 2 -induction is ridiculously long. In order to establish this result on proof lengths we give a computational analysis of slow provability, a notion introduced by Sy-David Friedman, Rathjen and Weiermann. We will see that slow uniform Σ 1 -reflection is related to a function that has a considerably lower growth rate than F ε 0 but dominates all functions F α with α < ε 0 in the fast-growing hierarchy

Mathematical Logic: Proof Theory, Constructive Mathematics

The workshop “Mathematical Logic: Proof Theory, Constructive Mathematics” was centered around proof-theoretic aspects of core mathematics and theoretical computer science as well as homotopy type theory and logical aspects of computational complexity

μ計算と算術の階層構造の探索---ゲール・スチュワートゲームの視点より

Tohoku University横山啓太課