129 research outputs found
Binary Relations as a Foundation of Mathematics
We describe a theory for binary relations in the Zermelo-Fraenkel style. We choose for ZFCU, a variant of ZFC Set theory in which the Axiom of Foundation is replaced by an axiom allowing for non-wellfounded sets. The theory of binary relations is shown to be equi-consistent ZFCU by constructing a model for the theory of binary relations in ZFU and vice versa. Thus, binary relations are a foundation for mathematics in the same sense as sets are
From coinductive proofs to exact real arithmetic: theory and applications
Based on a new coinductive characterization of continuous functions we
extract certified programs for exact real number computation from constructive
proofs. The extracted programs construct and combine exact real number
algorithms with respect to the binary signed digit representation of real
numbers. The data type corresponding to the coinductive definition of
continuous functions consists of finitely branching non-wellfounded trees
describing when the algorithm writes and reads digits. We discuss several
examples including the extraction of programs for polynomials up to degree two
and the definite integral of continuous maps
Mereology then and now
This paper offers a critical reconstruction of the motivations that led to the development of mereology as we know it today, along with a brief description of some questions that define current research in the field
Presheaf models for constructive set theories
This chapter introduces new kinds of models for constructive set theories based on categories of presheaves. It concentrates on categories of classes rather than sets, following the lines of algebraic set theory. It defines a general notion of what is a categorical model for CST, and shows that categories of presheaves provide examples of such models. To do so, it considers presheaves as functors with values in a category of classes. The models introduced are a counterpart of the presheaf models for intuitionistic set theories defined by Dana Scott in the 1980s. In this work, the author has to overcome the challenges intrinsic to dealing with generalized predicative formal systems rather than impredicative ones. An application to an independence result is discussed
The World Is Either Digital or Analogue
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