In 1956, the Dutch graphic artist M.C. Escher made an unusual
lithograph with the title `Print Gallery'. It shows a young man
viewing a print in an exhibition gallery. Amongst the buildings
depicted on the print, he sees paradoxically the very same gallery
that he is standing in. A lot is known about the way in which
Escher made his lithograph. It is not nearly as well known that it
contains a hidden `Droste effect', or infinite repetition; but
this is brought to light by a mathematical analysis of the studies
used by Escher. On the basis of this discovery, a team of
mathematicians at Leiden produced a series of hallucinating
computer animations. These show, among others, what happens
inside the mysterious spot in the middle of the lithograph that
Escher left blank.

This talk looks at the relationship between three foundational systems: Goedel's Constructible Universe of Sets, the naive conception of set found in consistent fragments of Frege's Grundgesetze, and the intensional logic of Church's Logic of Sense and Denotation. One basic result shows how to use the constructible sets to build models of fragments of Frege's Grundgesetze from which one can recover these very constructible sets using Frege's definition of membership. This result also allows us to solve the related consistency problem and joint consistency problems for abstraction principles with limited amounts of comprehension. Another basic aim of this paper is to show how to "factor'' this result via a consistent fragment of Church's Logic of Sense and Denotation: so one may use the constructible sets to build models of Church's Logic of Sense and Denotation, from which one may then define models of the consistent fragments of Frege's Grundgesetze.
Preprint: https://www.dropbox.com/s/afhcz8bzy4pdsoc/walsh-sean-CU%2BNC%2BIL-11-19-...
 

 
In this short talk, we will present an alternative approach to computational fluid dynamics，reaction-diffusion and flows in porous media. We outline its physical basis and mathematical derivation along with numerous applications. Some strengths and limitations of the method will be also pointed out.