Final Report Summary - DEPENDENTCLASSES (Model theory and its applications: dependent classes)
Humans like to classify various objects: A biologist will divide living things into eukaryotes and prokaryotes; an astronomer will divide the heavenly bodies into stars, planets, etc. As researchers in mathematical logic we are interested in classifying mathematical theories (not just various theories for the same mathematical object, but more generally theories that describe all kinds of objects). Here we do not just mean theories that have already been proposed and used in the math. literature, but "all possible theories". This is stated too vaguely, but there is a large body of work on how to formalize the meaning of "mathematical theory".
The classical case deals with the so-called "first order, complete theories" and their classes of so-called "models" (the mathematical structures satisfying these theories). To give a trivial example: If the theory T consists just of the sentence "for every x there is a y such that x+y=0", then the natural numbers {0,1,2...} are not a model of T, while the integers {... -2,-1, 0, 1,2,...} are.
Such classes (that correspond to first order theories) are called elementary classes; their investigation started with Frege in the late 19th century and found their first apex in Godel's completeness theorem (1930). There has been much interest in the classification of these elementary classes, underpinned by the belief that there are interesting divides; for example a division into "understandable, analyzable" cases and "complicated" ones. Such a division is between the so-called stable ones and the unstable ones. A prominent example of the former is the (theory of the) field of complex numbers, and for the second the field of real numbers. The real field includes all positive and negative numbers with possibly infinite decimal expansion, with the algebraic operations addition and multiplication.
When adding an "imaginary number" i, a square root of -1, we get the complex field. While this field looks more complicated at first sight, it turns out that its theory is much easier to understand (in particular, it is "stable", while the theory of real numbers is "unstable".) This phenomenon might give an additional explanation why mathematicians are interested in such imaginary numbers; their structure is easier to understand, and can help us understand the real numbers better.
Such abstract research has eventually helped also in the investigation of other specific mathematical classes. While the real number field is more complicated than the complex one it is still easier to analyze than the (apparently simpler) ring of integers {...,-2 -1, 0, 1, 2, ...}. The important difference between these two theories is formalized by the notion of "dependence": The (theory of the) reals is an example of a dependent class, while the integers are independent. Investigating the difference between dependent and independent classes is a major focus of our project.
Elementary classes (including all of the examples above) cover many of the actual classes that arise in mathematical research, but by no means all of them - there are important non-elementary classes. The classification of such non-elementary classes is a central theme of our project. A major achievement so far has been a generalization of this divide to a large natural family of non-elementary classes, namely those defined by the so-called "good homogeneous diagrams"; and even the proof of the so-called "generic pair conjecture" for many such cases.
The classical case deals with the so-called "first order, complete theories" and their classes of so-called "models" (the mathematical structures satisfying these theories). To give a trivial example: If the theory T consists just of the sentence "for every x there is a y such that x+y=0", then the natural numbers {0,1,2...} are not a model of T, while the integers {... -2,-1, 0, 1,2,...} are.
Such classes (that correspond to first order theories) are called elementary classes; their investigation started with Frege in the late 19th century and found their first apex in Godel's completeness theorem (1930). There has been much interest in the classification of these elementary classes, underpinned by the belief that there are interesting divides; for example a division into "understandable, analyzable" cases and "complicated" ones. Such a division is between the so-called stable ones and the unstable ones. A prominent example of the former is the (theory of the) field of complex numbers, and for the second the field of real numbers. The real field includes all positive and negative numbers with possibly infinite decimal expansion, with the algebraic operations addition and multiplication.
When adding an "imaginary number" i, a square root of -1, we get the complex field. While this field looks more complicated at first sight, it turns out that its theory is much easier to understand (in particular, it is "stable", while the theory of real numbers is "unstable".) This phenomenon might give an additional explanation why mathematicians are interested in such imaginary numbers; their structure is easier to understand, and can help us understand the real numbers better.
Such abstract research has eventually helped also in the investigation of other specific mathematical classes. While the real number field is more complicated than the complex one it is still easier to analyze than the (apparently simpler) ring of integers {...,-2 -1, 0, 1, 2, ...}. The important difference between these two theories is formalized by the notion of "dependence": The (theory of the) reals is an example of a dependent class, while the integers are independent. Investigating the difference between dependent and independent classes is a major focus of our project.
Elementary classes (including all of the examples above) cover many of the actual classes that arise in mathematical research, but by no means all of them - there are important non-elementary classes. The classification of such non-elementary classes is a central theme of our project. A major achievement so far has been a generalization of this divide to a large natural family of non-elementary classes, namely those defined by the so-called "good homogeneous diagrams"; and even the proof of the so-called "generic pair conjecture" for many such cases.