Abstract not found

We prove that the maximal order type of the wqo of linear orders of finite Hausdorff rank under embeddability is ϕ2(0), the first fixed point of the ε-function. We then show that Fraïssé’s conjecture restricted to linear orders of finite Hausdorff rank is provable in ACA + 0 + “ϕ2(0) is well-ordered ” and, over RCA0, implies ACA ′ 0 + “ϕ2(0) is well-ordered”. 1

Abstract The article starts with a brief survey of Unprovability Theory as of autumn 2006. Then, as an illustration of the subject's model-theoretic methods, we re-prove exact versions of unprovability results for the Paris-Harrington Principle and the KanamoriMcAloon Principle using indiscernibles. In addition, we obtain a short accessible proof of unprovability of the Paris-Harrington Principle. The proof employs old ideas but uses only one colouring and directly extracts the set of indiscernibles from its homogeneous set. We also present modified, abridged statements whose unprovability proofs are especially simple. These proofs were tailored for teaching purposes. The article is intended to be accessible to the widest possible audience of mathematicians, philosophers and computer scientists as a brief survey of the subject, a guide through the literature in the field, an introduction to its model-theoretic techniques and, finally, a model-theoretic proof of a modern theorem in the subject. However, some understanding of logic is assumed on the part of the readers. The intended audience of this paper consists of logicians, logic-aware mathematicians andthinkers of other backgrounds who are interested in unprovable mathematical statements.

We study the computability-theoretic complexity and proof-theoretic strength of the following statements: (1) “If X is a well-ordering, then so is εX”, and (2) “If X is a well-ordering, then so is ϕ(α, X)”, where α is a fixed computable ordinal and ϕ represents the two-placed Veblen function. For the former statement, we show that ω iterations of the Turing jump are necessary in the proof and that the statement is equivalent to ACA + 0 over RCA0. To prove the latter statement we need to use ωα iterations of the Turing jump, and we show that the statement is equivalent to Π0 ωα-CA0. Our proofs are purely computability-theoretic. We also give a new proof of a result of Friedman: the statement “if X is a well-ordering, then so is ϕ(X, 0)” is equivalent to ATR0 over RCA0.