Mathematics is Inconsistent

1998

Abstract In [N] the contraction-free and cut-free sequent calculus G3ip for intuitionistic p) ropositiona. l logic was extended by rules for theories of apartness an (l order. The logical content. of the axioms of these theories is expressed by the geometry of sequent calculus rules, which have only atomic formulas as active and principal. In this way also such extensions are contraction-free and cutfree. Cut. elimination permits structural proof analysis, and syntactic proofs of conservativity results.

2007

As the overcrowded bus huddles into the hustle and bustle of Chinatown, after what feels more like a small lifetime than a four-hour journey, you could be almost forgiven for thinking you are in Shanghai, but the signs assure me it is indeed New York that I am finding myself in. Ten minutes later, sitting in a Japanese café and sipping bubble Tapioca tea, I am acting just like one of the locals, well, almost... aside from my wide-eyed, mesmerized expression.

Preliminary proceedings of Advances in Modal Logic (AIML-2004), 2004

We provide a strongly complete infinitary proof system for hybrid logic. This proof system can be extended with countably many sequents. Thus completeness proofs are provided for infinitary hybrid versions of non-compact logics like ancestral logic and Segerberg's modal logic with the bounded chain condition. This extends the completeness result for hybrid logics by Blackburn and Tzakova. Keywords: hybrid logic, strong completeness, non-compact logics, infinitary proof rules

… and complexity: from Leibniz to Chaitin, 2007

Human beings have a future if they deserve to have a future! Gregory J. Chaitin

Randomness and Complexity, From Leibniz to Chaitin, 2007

Human beings have a future if they deserve to have a future! Gregory J. Chaitin

Logic and Theory of Algorithms, Fourth Conference on Computability in Europe (CiE 2008), Local Proceedings

is an informal network of European scientists working on computability theory, including its foundations, technical development, and applications. Among the aims of the network is to advance our theoretical understanding of what can and cannot be computed, by any means of computation. Its scientific vision is broad: computations may be performed with discrete or continuous data by all kinds of algorithms, programs, and machines. Computations may be made by experimenting with any sort of physical system obeying the laws of a physical theory such as Newtonian mechanics, quantum theory, or relativity. Computations may be very general, depending on the foundations of set theory; or very specific, using the combinatorics of finite structures. CiE also works on subjects intimately related to computation, especially theories of data and information, and methods for formal reasoning about computations. The sources of new ideas and methods include practical developments in areas such as neural networks, quantum computation, natural computation, molecular computation, computational learning. Applications are everywhere, especially, in algebra, analysis and geometry, or data types and programming. Within CiE there is general recognition of the underlying relevance of computability to physics and a broad range of other sciences, providing as it does a basic analysis of the causal structure of dynamical systems.