A theory of hyperfinite sets

We develop an axiomatic set theory -- the Theory of Hyperfinite Sets THS, which is based on the idea of existence of proper subclasses of big finite sets. We demonstrate how theorems of classical continuous mathematics can be transfered to THS, prove consistency of THS and present some applications.Comment: 28 page

Computability and analysis: the legacy of Alan Turing

We discuss the legacy of Alan Turing and his impact on computability and analysis.Comment: 49 page

Process as a world transaction

Transaction is process closure: for a transaction is the limiting process of process itself. In the process world view the universe is the ultimate (intensional) transaction of all its extensional limiting processes that we call reality. ANPA’s PROGRAM UNIVERSE is a computational model which can be explored empirically in commercial database transactions where there has been a wealth of activity over the real world for the last 40 years. Process category theory demonstrates formally the fundamental distinctions between the classical model of a transaction as in PROGRAM UNIVERSE and physical reality. The paper concludes with a short technical summary for those who do not wish to read all the detail

Interactive and common knowledge in the state-space model

This paper deals with the prevailing formal model for knowledge in contemporary economics, namely the state-space model introduced by Robert Aumann in 1976. In particular, the paper addresses the following question arising in this formalism: in order to state that an event is interactively or commonly known among a group of agents, do we need to assume that each of them knows how the information is imparted to the others? Aumann answered in the negative, but his arguments apply only to canonical, i.e., completely specified state spaces, while in most applications the state space is not canonical. This paper addresses the same question along original lines, demonstrating that the answer is negative for both canonical and not-canonical state spaces. Further, it shows that this result ensues from two counterintuitive properties held by knowledge in the state-space model, namely Substitutivity and Monotonicity.

Benchmarking in cluster analysis: A white paper

To achieve scientific progress in terms of building a cumulative body of knowledge, careful attention to benchmarking is of the utmost importance. This means that proposals of new methods of data pre-processing, new data-analytic techniques, and new methods of output post-processing, should be extensively and carefully compared with existing alternatives, and that existing methods should be subjected to neutral comparison studies. To date, benchmarking and recommendations for benchmarking have been frequently seen in the context of supervised learning. Unfortunately, there has been a dearth of guidelines for benchmarking in an unsupervised setting, with the area of clustering as an important subdomain. To address this problem, discussion is given to the theoretical conceptual underpinnings of benchmarking in the field of cluster analysis by means of simulated as well as empirical data. Subsequently, the practicalities of how to address benchmarking questions in clustering are dealt with, and foundational recommendations are made

A model of adaptive decision making from representation of information environment by quantum fields

We present the mathematical model of decision making (DM) of agents acting in a complex and uncertain environment (combining huge variety of economical, financial, behavioral, and geo-political factors). To describe interaction of agents with it, we apply the formalism of quantum field theory (QTF). Quantum fields are of the purely informational nature. The QFT-model can be treated as a far relative of the expected utility theory, where the role of utility is played by adaptivity to an environment (bath). However, this sort of utility-adaptivity cannot be represented simply as a numerical function. The operator representation in Hilbert space is used and adaptivity is described as in quantum dynamics. We are especially interested in stabilization of solutions for sufficiently large time. The outputs of this stabilization process, probabilities for possible choices, are treated in the framework of classical DM. To connect classical and quantum DM, we appeal to Quantum Bayesianism (QBism). We demonstrate the quantum-like interference effect in DM which is exhibited as a violation of the formula of total probability and hence the classical Bayesian inference scheme.Comment: in press in Philosophical Transactions