Related papers: On construction schemes: building the uncountable …
We develop categorical foundations of discrete dynamical systems, aimed at understanding how the structure of the system affects its dynamics. The key technical innovation is the notion of a cycle set, which provides a formal language in…
Every countable structure has a sentence of the infinitary logic $\mathcal{L}_{\omega_1 \omega}$ which characterizes that structure up to isomorphism among countable structures. Such a sentence is called a Scott sentence, and can be thought…
Usual math sets have special types: countable, compact, open, occasionally Borel, rarely projective, etc. Each such set is described by a single Set Theory formula with parameters unrelated to other formulas. Exotic expressions involving…
Following a line of research initiated in \cite{BBNN}, I describe a general framework for turning reduction concepts of relative computability into diagrams forming an analogy with the Cicho\'n diagram for cardinal characteristics of the…
The classical theory of plane projective geometry is examined constructively, using both synthetic and analytic methods. The topics include Desargues's Theorem, harmonic conjugates, projectivities, involutions, conics, Pascal's Theorem,…
The ordered structures of natural, integer, rational and real numbers are studied in this thesis. The theories of these numbers in the language of order are decidable and finitely axiomatizable. Also, their theories in the language of order…
In combinatorics, the probabilistic method is a very powerful tool to prove the existence of combinatorial objects with interesting and useful properties. Explicit constructions of objects with such properties are often very difficult, or…
Almost two decades ago, Wattenberg published a paper with the title 'Nonstandard Analysis and Constructivism?' in which he speculates on a possible connection between Nonstandard Analysis and constructive mathematics. We study Wattenberg's…
The Shatters relation and the VC dimension have been investigated since the early seventies. These concepts have found numerous applications in statistics, combinatorics, learning theory and computational geometry. Shattering extremal…
In this paper we construct $G$-Hilbert schemes for finite group schemes $G$. We find a construction of $G$-Hilbert schemes as relative $G$-Hilbert schemes over the quotient that does not need the Hilbert scheme of $n$ points, works under…
We give a new definition of the derived category of constructible $\ell$-adic sheaves on a scheme, which is as simple as the geometric intuition behind them. Moreover, we define a refined fundamental group of schemes, which is large enough…
A new construction of naturally reductive spaces is presented. This construction gives a large amount of new families of naturally reductive spaces. First the infinitesimal models of the new naturally reductive spaces are constructed. A…
The constructive approach to mathematics has the advantage that witnesses can be extracted from statements of existence and theorems can be unwound to give algorithms. Even better, constructive theorems can be interpreted in any topos,…
The structures $\langle M,\subseteq^M\rangle$ arising as the inclusion relation of a countable model of sufficient set theory $\langle M,\in^M\rangle$, whether well-founded or not, are all isomorphic. These structures $\langle…
This is my dissertation about digraphs ordered by pp-constructability. We study in particular smooth digraphs, i.e., digraphs without sources or sinks, tournaments and semicomplete digraphs, orientations of paths and cycles, digraphs with…
We analyze the effective content of countable, second countable topological spaces by directly calculating the complexity of several topologically defined index sets. We focus on the separation principles, calibrating an arithmetic…
This paper is a contribution to the study of extensions of arbitrary models of ZF (Zermelo-Fraenkel set theory), with no regard to countability or well-foundedness of the models involved. We present some new constructions of certain types…
I shall describe a general model-theoretic task to construct expansions of pseudofinite structures and discuss several examples of particular relevance to computational complexity. Then I will present one specific situation where finding a…
This essay contains three parts. The first part of essay focuses on the hypothesis of the functional semantic constructions (FSC-Hypothesis). This hypothesis explains that a language, a number, a money are the functional semantic…
We present distributions of countable models and correspondent structural characteristics of complete theories with continuum many types: for prime models over finite sets relative to Rudin-Keisler preorders, for limit models over types and…