Model Theory Of Fields Research Articles

Corrigendum: Model theory of fields with virtually free group actions

AbstractThere is an irreparable error in the proof of Theorem 3.26 in the above‐mentioned paper and we withdraw the claim of having proved that theorem. In fact, that theorem is false in a very strong sense.

MODEL THEORY OF DERIVATIONS OF THE FROBENIUS MAP REVISITED

AbstractWe prove some results about the model theory of fields with a derivation of the Frobenius map, especially that the model companion of this theory is axiomatizable by axioms used by Wood in the case of the theory $\operatorname {DCF}_p$ and that it eliminates quantifiers after adding the inverse of the Frobenius map to the language. This strengthens the results from [4]. As a by-product, we get a new geometric axiomatization of this model companion. Along the way we also prove a quantifier elimination result, which holds in a much more general context and we suggest a way of giving “one-dimensional” axiomatizations for model companions of some theories of fields with operators.

Model theory of fields with free operators in positive characteristic

We give algebraic conditions for a finite commutative algebra B B over a field of positive characteristic, which are equivalent to the companionability of the theory of fields with “ B B -operators” (i.e., the operators coming from homomorphisms into tensor products with B B ). We show that, in the most interesting case of a local B B , these model companions admit quantifier elimination in the “smallest possible” language, and they are strictly stable. We also describe the forking relation there.

Model theory of fields with virtually free group actions

For a group G, we define the notion of a G-kernel and show that the properties of G-kernels are closely related with the existence of a model companion of the theory of Galois actions of G. Using Bass–Serre theory, we show that this model companion exists for virtually free groups generalizing the existing results about free groups and finite groups. We show that the new theories we obtain are not simple and not even NTP 2 .

THE MODEL COMPANION OF DIFFERENTIAL FIELDS WITH FREE OPERATORS

AbstractA model companion is shown to exist for the theory of partial differential fields of characteristic zero equipped with free operators that commute with the derivations. The free operators here are those introduced in [R. Moosa and T. Scanlon, Model theory of fields with free operators in characteristic zero, Journal of Mathematical Logic 14(2), 2014]. The proof relies on a new lifting lemma in differential algebra: a differential version of Hensel’s Lemma for local finite algebras over differentially closed fields.

Model theory of fields with free operators in characteristic zero

Generalizing and unifying the known theorems for difference and differential fields, it is shown that for every finite free algebra scheme 𝒟 over a field A of characteristic zero, the theory of 𝒟-fields has a model companion 𝒟-CF0 which is simple and satisfies the Zilber dichotomy for finite-dimensional minimal types.

Comparing Peano arithmetic, Basic Law V, and Hume’s Principle

This paper presents new constructions of models of Hume’s Principle and Basic Law V with restricted amounts of comprehension. The techniques used in these constructions are drawn from hyperarithmetic theory and the model theory of fields, and formalizing these techniques within various subsystems of second-order Peano arithmetic allows one to put upper and lower bounds on the interpretability strength of these theories and hence to compare these theories to the canonical subsystems of second-order arithmetic. The main results of this paper are: (i) there is a consistent extension of the hypearithmetic fragment of Basic Law V which interprets the hyperarithmetic fragment of second-order Peano arithmetic, and (ii) the hyperarithmetic fragment of Hume’s Principle does not interpret the hyperarithmetic fragment of second-order Peano arithmetic, so that in this specific sense there is no predicative version of Frege’s Theorem.

Diophantine Geometry from Model Theory

Abstract§1. Introduction. With Hrushovski's proof of the function field Mordell-Lang conjecture [16] the relevance of geometric stability theory to diophantine geometry first came to light. A gulf between logicians and number theorists allowed for contradictory reactions. It has been asserted that Hrushovski's proof was simply an algebraic argument masked in the language of model theory. Another camp held that this theorem was merely a clever one-off. Still others regarded the argument as magical and asked whether such sorcery could unlock the secrets of a wide coterie of number theoretic problems.In the intervening years each of these prejudices has been revealed as false though such attitudes are still common. The methods pioneered in [16] have been extended and applied to a number of other problems. At their best, these methods have been integrated into the general methods for solving diophantine problems. Moreover, the newer work suggests limits to the application of model theory to diophantine geometry. For example, all such known applications are connected with commutative algebraic groups. This need not be an intrinsic restriction, but its removal requires serious advances in the model theory of fields.

Does Mathematics Need New Axioms?

(1999). Does Mathematics Need New Axioms? The American Mathematical Monthly: Vol. 106, No. 2, pp. 99-111.

David Marker, Introduction to the model theory of fields. Model theory of fields, Lecture notes in logic, no. 5, Springer, Berlin, Heidelberg, New York, etc., 1996, pp. 1–37. - David Marker. Model theory of differential fields. Model theory of fields, Lecture notes in logic, no. 5, Springer, Berlin, Heidelberg, New York, etc., 1996, pp. 38–113. - Anand Pillay. Differential algebraic

David Marker, Introduction to the model theory of fields. Model theory of fields, Lecture notes in logic, no. 5, Springer, Berlin, Heidelberg, New York, etc., 1996, pp. 1–37. - David Marker. Model theory of differential fields. Model theory of fields, Lecture notes in logic, no. 5, Springer, Berlin, Heidelberg, New York, etc., 1996, pp. 38–113. - Anand Pillay. Differential algebraic groups and the number of countable differentially closed fields. Model theory of fields, Lecture notes in logic, no. 5, Springer, Berlin, Heidelberg, New York, etc., 1996, pp. 114–134. - Margit Messmer. Some model theory of separably closed fields. Model theory of fields, Lecture notes in logic, no. 5, Springer, Berlin, Heidelberg, New York, etc., 1996, pp. 135–152. - Volume 63 Issue 2

Model theory of fields: An application to positive semidefinite polynomials

Model theory of fields: An application to positive semidefinite polynomials