In this talk I will highlight various of Godel's own statements about the significance of his incompleteness theorem, and discuss their relevance to mathematical practice.
At the end of the first meeting the time of the seminar will be discussed. In this quarter we go through the book Melvin Fitting and Richard L. Mendelsohn. First order modal logic.
The book describes present state of first order modal logic with detailed philosophical motivation and (sometimes less detailed) historical information. Standard course in logic like 160a plus elementary information on possible world semantics (Phil 169 is more than enough) provide sufficient background.
When thinking about structures in the physical world, modal logicians have traditionally looked at Time as a main area of interest, because it fits so well with an interest in the flow of information and computation. Spatial logics have mostly been foot- notes to this tradition. But in reality, Space is equally interesting, both for general mathematical reasons, and given the growing importance of work in CS on visual reasoning and image processing. Studying space can be done at many different levels, depending on one's special interest. One must choose some grain level of mathematical structure (topological, affine, metrical, or yet other) providing the right invariances. Next, at least a logician will look for some appropriate language that brings out interesting laws, preferably in a calculus of some reasonable complexity. We will look at a number of languages of this sort, surveying recent results and questions: (a) modal logic in its topological interpretation This was started by Tarski and McKinsey, and recent work includes a nice completeness proof by Mints. We show how to view this in terms of bisimulation games. (b) plausible logical strengthenings of this This brings tools from current 'extended modal logics' to the analysis of spatial patterns. (c) ditto geometrical strengthenings of this. This may be viewed as exploring 'modal fragments' of complex logics like Tarski's elementary geometry. These systems arise from generalizing existing logics. In this, they may be more supply- than demand-driven. As a counterpoint, we also look at (d) 'mathematical morphology' a relatively new theory of shapes in image processing - and show how it is a rather interesting sort of linear and modal logic over vector spaces. References (a) M. Aiello & J. van Benthem, 2000, 'Modal Patterns in Space', to appear in Proceedings LLC 9, CSLI Publications, Stanford, http://www.wins.uva.nl/~aiellom/publications/lops.ps.gz (b) M. Aiello, J. van Benthem & G. Bezhaneshvili, 2001, 'Reasoning about Space, the modal way', to appear. (c) J. van Benthem, 2000, 'Logical Structures in Mathematical Morphology', http://turing.wins.uva.nl/~johan/MM-LL.ps