Βιβλίο θεωρίας γραφημάτων και μαθηματικής λογικής από το ανοικτό πανεπιστήμιο,

Model theory is a branch of mathematical logic that has found applications in several areas of algebra and geometry.

It provides a unifying framework for the understanding of old results and more recently has led to significant new results, such as a proof of the Mordell-Lang conjecture for function fields in positive characteristic.

Perhaps surprisingly, it is sometimes the most abstract aspects of model theory that are relevant to these applications.

Model Theory is the part of mathematics which shows how to apply logic to the study of structures in pure mathematics. On the one hand it is the ultimate abstraction; on the other, it has immediate applications to every-day mathematics. The fundamental tenet of Model Theory is that mathematical truth, like all truth, is relative. A statement may be true or false, depending on how and where it is interpreted. This isn’t necessarily due to mathematics itself, but is a consequence of the language that we use to express mathematical ideas.

In these notes and lectures I will discuss some of the differences between classical and constructive logic.

In the first few sections I will try to place the issues in a broader philosophical, mathematical, and historical context.

After that, I will discuss what one can concretely say about the relationship between the two kinds of logic.

Lecture notes for Formal models of computation, unsolvability, and Gödel’s incompleteness theorems.

Lecture notes for a graduate course on model theory.

This is a set of lecture notes for the course Foundations of Mathematics I, offered as Mathematics 558 at the Pennsylvania State University, most recently in Spring 2007.

This a very brief introduction to Graph Theory in 29 slides.

This is the home page for a series of short interactive tutorials introducing the basic concepts of graph theory. There is not a great deal of theory here, we will just teach you enough to wet your appetite for more!