The notion of a cell and that of a cell decomposition has been a central one in the study of certain first-order theories. A cell is a particular kind of definable set. The notion of a cell was first explicitly considered in [8], in the context of the theory of real closed fields. Collins defined a class of cells in this context, and showed that every definable subset of a real closed field is ...

In the first part of this paper we discuss some aspects of Detlefsen's attempt to save Hilbert's Program from the consequences of GodePs Second Incompleteness Theorem. His arguments are based on his interpretation of the long standing and well known controversy on what, exactly, finitistic means are. In his paper [1] Detlefsen takes the position that there is a form of the cj-rule which is a fi...

The purpose of this paper is to clarify the relationship between various conditions implying essential undecidability: our main result is that there exists a theory $T$ in which all partially recursive functions are representable, yet $T$ does not interpret Robinson's theory $R$. To this end, we borrow tools from model theory---specifically, we investigate model-theoretic properties of the mode...

Porous materials are lightweight, flexible and resistant to hairline cracks, so today with the development of technology porous structure produced for use in various industries. This structure widely use in beams, plates and shells. The purpose of this paper is to investigate the effect of porosity in axial symmetry in bending and buckling load sheet for analysis. For this purpose, a circular p...

As is well-known a language of finite words, considered as labeled linear orders, is definable in monadic second-order logic (MSO) iff it is definable in the existential fragment of MSO, that is the quantifier alternation hierarchy collapses. Even more, it does not make a difference if we consider existential MSO over a linear order or a successor relation only. In this note we show that somewh...

Assume that the problem P0 is not solvable in polynomial time. Let T be a first-order theory containing a sufficiently rich part of true arithmetic. We characterize T∪{ConT } as the minimal extension of T proving for some algorithm that it decides P0 as fast as any algorithm B with the property that T proves that B decides P0. Here, ConT claims the consistency of T . As a byproduct, we obtain a...

We explore how computational ontologies can be impactful vis-à-vis the developing discipline of “data science.” We posit an approach wherein management theories are represented as formal axioms, and then applied to draw inferences about data that reside in corporate databases. That is, management theories would be implemented as rules within a data analytics engine. We demonstrate a case study ...

This document is a tutorial for FoCaLize, describing how to develop proofs of properties using Zenon. Differently from other tutorials, we won’t focalize on mathematical developments, preferring to show the language in action on programs closer to what “usual programers” develop in the “everyday life”. To get in touch with basic Zenon capabilities, we will first address very simple first order ...

We propose a general method for overwriting theories with model conservative extensions in mechanized mathematics systems. Model conservative extensions, which include the definition of new constants and the introduction of new abstract datatypes, are “safe” because they preserve models as well as consistency. The method employs the notions of theory interpretation and theory instantiation. It ...

The key tool in proving inexpressibility results in finite-model theory is Ehrenfeucht-F'rai'ss6 games. This paper surveys various game-theoretic techniques and tools that lead to simpler proofs of inexpressibility results. The focus is on first-order logic and