Related papers: Finiteness and Computation in Toposes
This paper defines a new notion of bounded computable randomness for certain classes of sub-computable functions which lack a universal machine. In particular, we define such versions of randomness for primitive recursive functions and for…
Lattice discretizations of continuous manifolds are common tools used in a variety of physical contexts. Conventional discrete approximations, however, cannot capture all aspects of the original manifold, notably its topology. In this paper…
Locales have been studied as "topologies without points", mainly by tools of category theory. While traditional topology presents a space as a set of points with specified neighborhoods, localic topology presents a space as a lattice of…
We propose an axiomatic foundation of mathematics based on the finite sequence as the foundational concept, rather than based on logic and set, as in set theory, or based on type as in dependent type theories. Finite sequences lead to a…
There are different definitions of ends in non-locally-finite graphs which are all equivalent in the locally finite case. We prove the compactness of the end-topology that is based on the principle of removing finite sets of vertices and…
This expository paper advocates an approach to physics in which ``typicality" is identified with a suitable form of algorithmic randomness. To this end various theorems from mathematics and physics are reviewed. Their original versions…
Rough set theory is a new mathematical approach to imperfect knowledge. The notion of rough sets is generalized by using an arbitrary binary relation on attribute values in information systems, instead of the trivial equality relation. The…
We investigate the connections between computability theory and Nonstandard Analysis. In particular, we investigate the two following topics and show that they are intimately related. (T.1) A basic property of Cantor space $2^{\mathbb{N}}$…
Most ideas about what an algorithm is are very similar. Basic operations are used for transforming objects. The evaluation of internal and external states by relations has impact on the further process. A more precise definition can lead to…
The topological properties of a set have a strong impact on its computability properties. A striking illustration of this idea is given by spheres and closed manifolds: if a set $X$ is homeomorphic to a sphere or a closed manifold, then any…
Existing models of computation, such as a Turing machine (hereafter, TM), do not consider the agent involved in interpreting the outcome of the computation. We argue that a TM, or any other computation model, has no significance if its…
We study first-order expansions of the reals which do not define the set of natural numbers. We also show that several stronger notions of tameness are equivalent to each others.
Many recent writers in the philosophy of mathematics have put great weight on the relative categoricity of the traditional axiomatizations of our foundational theories of arithmetic and set theory (\cite{Parsons1990a}, \cite{Parsons2008}…
The notion of probability plays an important role in almost all areas of science and technology. In modern mathematics, however, probability theory means nothing other than measure theory, and the operational characterization of the notion…
In this survey, my aim has been to discuss the use of sequences and countable sets in general topology. In this way I have been led to consider five different classes of topological spaces: first countable spaces, sequential spaces, Frechet…
We introduce a notion of computable randomness for infinite sequences that generalises the classical version in two important ways. First, our definition of computable randomness is associated with imprecise probability models, in the sense…
To any finite ordered subset and any finite partition of a group a set of tuples of positive integers, named as configurations, is associated that describes the group's behavior. The present paper provides an exposition of this notion and…
The goals of this paper are to obtain theoretical models of what happens when a computer calculates the rotation set of a homeomorphism, and to find a good algorithm to perform simulations of this rotation set. To do that we introduce the…
Computational pseudorandomness studies the extent to which a random variable $\bf{Z}$ looks like the uniform distribution according to a class of tests $\cal{F}$. Computational entropy generalizes computational pseudorandomness by studying…
By nature, transmissible human knowledge is enumerable: every sentence, movie, audio record can be encoded in a sufficiently long string of 0's and 1's. The works of G\"odel, Turing and others showed that there are inherent limits and…