Section 1 is devoted to the study of countable recursively saturated models with an automorphism moving every non-algebraic point. We show that every countable theory has such a model and exhibit necessary and sufficient conditions for the existence of automorphisms moving all non-algebraic points. Furthermore we show that there are many complete theories with the property that every countable recursively saturated model has such an automorphism. In Section 2 we apply our main theorem from Section 1 to models of Quine's (...) set theory New Foundations (NF) to answer an old consistency question. If NF is consistent, then it has a model in which the standard natural numbers are a definable subclass N of the model's set of internal natural numbers Nn. In addition, in this model the class of wellfounded sets is exactly $\bigcup_{n\in \mathbb{N}}\mathscr{P}^n(\varnothing)$. (shrink)

This paper generalizes results of F. Körner from [4] where she established the existence of maximal automorphisms (i.e. automorphisms moving all non-algebraic elements). An ω-maximal automorphism is an automorphism whose powers are maximal automorphisms. We prove that any structure has an elementary extension with an ω-maximal automorphism. We also show the existence of ω-maximal automorphisms in all countable arithmetically saturated structures. Further we describe the pairs of tuples (¯a,¯b) for which there is an ω-maximal automorphism mapping ¯a to ¯b.

Which groups are isomorphic to automorphism groups of models of Peano Arithmetic? It will be shown here that any group that has half a chance of being isomorphic to the automorphism group of some model of Peano Arithmetic actually is.For any structure, let Aut() be its automorphism group. There are groups which are not isomorphic to any model= (N, +, ·, 0, 1, ≤) of PA. For example, it is clear that Aut(N), being a subgroup of Aut((, <)), must be (...) torsion-free. However, as will be proved in this paper,if(A, <)is a linearly ordered set and G is a subgroup of Aut((A, <)),then there are modelsofPAsuch that Aut() ≅G.Ifis a structure, then its automorphism group can be considered as a topological group by letting the stabilizers of finite subsets ofAbe the basic open subgroups. If′ is an expansion of, then Aut(′) is a closed subgroup of Aut(). Conversely, for any closed subgroupG≤ Aut() there is an expansion′ ofsuch that Aut(′) =G. Thus, ifis a model of PA, then Aut() is not only a subgroup of Aut((N, <)), but it is even aclosedsubgroup of Aut((N, ′)).There is a characterization, due to Cohn [2] and to Conrad [3], of those groupsGwhich are isomorphic to closed subgroups of automorphism groups of linearly ordered sets. (shrink)

We give an examination of the automorphism group Aut of a countable recursively saturated model M of PA. The main result is a characterisation of strong elementary initial segments of M as the initial segments consisting of fixed points of automorphisms of M. As a corollary we prove that, for any consistent completion T of PA, there are recursively saturated countable models M1, M2 of T, such that Aut[ncong]Aut, as topological groups with a natural topology. Other results include a classification (...) of the normal subgroups of Aut of the form [lcub]g: g [uharr] A = idA[rcub], for sets A M, and a highly homogeneous representation of Aut as a subgroup of Aut. (shrink)

The paper presents an outline of the general theory of countable arithmetically saturated models of PA and some of its applications. We consider questions concerning the automorphism group of a countable recursively saturated model of PA. We prove new results concerning fixed point sets, open subgroups, and the cofinality of the automorphism group. We also prove that the standard system of a countable arithmetically saturated model of PA is determined by the lattice of its elementary substructures.

The connections between mathematical logic and combinatorics have a rich history. This paper focuses on one aspect of this relationship: understanding the strength, measured using the tools of computability theory and reverse mathematics, of various partition theorems. To set the stage, recall two of the most fundamental combinatorial principles, König's Lemma and Ramsey's Theorem. We denote the set of natural numbers by ω and the set of finite sequences of natural numbers by ω<ω. We also identify each n ∈ ω (...) with its set of predecessors, so n = {0, 1, 2, …, n − 1}. (shrink)

We consider the question: If M is a countable recursively saturated model of PA and K is an elementary submodel of M, is there an automorphism α of M such that K is the fixed point set of α? We give a survey of the known results and we prove that, if M is arithmetically saturated, then M has continuum many pairwise nonisomorphic elementary submodels which are fixed point sets.

Section 1 is devoted to the study of countable recursively saturated models with an automorphism moving every non-algebraic point. We show that every countable theory has such a model and exhibit necessary and sufficient conditions for the existence of automorphisms moving all non-algebraic points. Furthermore we show that there are many complete theories with the property that every countable recursively saturated model has such an automorphism.In Section 2 we apply our main theorem from Section 1 to models of Quine's set (...) theory New Foundations (NF) to answer an old consistency question. If NF is consistent, then it has a model in which the standard natural numbers are a definable subclass ℕ of the model's set of internal natural numbers Nn. In addition, in this model the class of wellfounded sets is exactly. (shrink)

We give a survey of automorphisms of countable recursively saturated models of Peano Arithmetic.