Let f be a number-theoretic function. A finite set X of natural numbers is called f-large if card(X) ≥ f(min(X)). Let PHf be the Paris Harrington statement where we replace the largeness condition by a corresponding f-largeness condition. We classify those functions f for which the statement PHf is independent of first order (Peano) arithmetic PA.Iffis a fixed iteration of the binary length function, then PHf is independent. On the other hand PHlog ∗ is provable in PA. More precisely let fα(i):=|i | H −1 α (i) where | i |h denotes the h-times iterated binary length of i and H−1 α denotes the inverse function of the α-th member Hα of the Hardy hierarchy. Then PHfα is independent of PA (for α ≤ ε0) iffα = ε0.

The automorphism groups of K0-categorical structures have been studied extensively by both permutation group theorists and model theorists, and this collaboration has turned out to be very fruitful. (See, for example, [10,6,2].) The notion of a recursively saturated structure generalizes that of a (countable)

this paper we will work with a nonstandard model M of PA+ConPA for the language of PA. Since PA is a consistent theory in M, it has a completion C

Abstract. Using the work of several other mathematicians, principally the results of Poonen refining the work of Pop that algebraic independence is definable within the class of finitely generated fields and of Rumely that the ring of rational integers is uniformly interpreted in global fields, and a theorem on the definability of valuations on function fields of curves, we show each infinite finitely generated field considered in the ring language is parametrically biïnterpretable with N. As a consequence, for any finitely generated field there is a first-order sentence in the language of rings which is true in that field but false in every other finitely generated field, and, hence, Pop’s conjecture that elementarily equivalent finitely generated fields are isomorphic is true. 1.

As the first illustration of a potential satisfier for the ‘platitudes for truth ’ in the appendix to his engaging recent discussion of the concept of truth (Wright 1999), Crispin Wright has proposed a notion of ‘truth conceived as coherence ’ for arithmetic. This paper attempts to clarify certain aspects of Wright’s proposal. Take the standard first-order language of arithmetic L. 1 Let B be some axiom system for arithmetic, which Wright calls the ‘coherence base’. With small notational modifications, Wright proposes the following inductive definition of the concept ‘coheres with B ’ (for L-sentences): (CAt) If ϕ is atomic, then ϕ coheres with B iff B � ϕ. (C¬) ¬ϕ coheres with B iff ϕ does not cohere with B. (C∧) ϕ ∧ ψ coheres with B iff ϕ and ψ cohere with B. (C∨) ϕ ∨ ψ coheres with B iff either ϕ or ψ coheres with B. (C→) ϕ → ψ coheres with B iff either ϕ does not or ψ does cohere with B. (C∀) ∀xϕ coheres with B iff, for each number n, ϕ(n) coheres with B. (C∃) ∃xϕ coheres with B iff, for some number n, ϕ(n) coheres with B. First, note that except for the basis clause (CAt) this is the same as the usual Tarskian inductive definition of truth for arithmetic: 2 (TAt) If ϕ has the form t = u, then ϕ is true iff val(t) = val(u). 3 1 The terms of L are defined recursively from a basis of variables, the constant 0 and the operation symbols s, + and ×. The numerals of L are written n, meaning 0 prefixed by n occurrences of the successor symbol s. The atomic formulas of L are equations of the form t = u (with t, u terms) and complex formulas of L are defined by recursion on complexity as usual. Below, Sent(L) is the set of Lsentences of L, AtSent(L) is the set of atomic L-sentences and Form(L) is the set of L-formulas. In the arithmetic formalization of semantics, we use SentL(x) and ClTmL(x) to mean arithmetic formulas expressing respectively that x is (the code of) a sentence of L or that x is (the code of) a closed term of

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.

In this undergraduate thesis the independence of Goodstein's Theorem from Peano arithmetic (PA) is proved, following the format of the rst proof, by Kirby and Paris. All the material necessary for its understanding is developed, beginning with the foundations of set theory, followed by ordinal numbers and a proof of Goodstein's Theorem, and concluded with basic model theory and the independence proof. Other relevant information, such as an outline for an alternative independence proof and an application to dynamical systems is also included. Contents 1.

1 Descriptive use of logic and Intended models 1 1.1 Standard models of arithmetic.......................... 1 1.2 Axiomatics and Formal theories......................... 3 1.3 Hintikka and the two uses of logic in mathematics.............. 5

General and diophantine definability in number rings and their polynomial rings are studied from a model-theoretic point of view. The main tool used is a modern form of the Theorem of Beth. For a ring R we consider the monoid EndR (R ) of all embeddings of a nonstandard enlargement R in itself which fix the standard elements. If R is a number ring or any field, the application of natural restriction ResR from End R[T ] (R[T ] ) to EndR (R ) is a well defined homomorphism of monoids. We give connections between the diophantine definability of the integers Z in a number ring R, a phenomenon of transfer of definability from the polynomial ring R[T ] to the ring R, and properties of the homomorphism ResR . In the case of the ring Z itself we get as a byproduct that the restriction ResZ : End Z[T ] (Z[T ] ) ~ \Gamma! EndZ (Z ) is an isomorphism of monoids, fact which is equivalent with the combination of two well known theorems of Y. Matiyasevich and J. Denef. We pr...

into reduced powers