Related papers: Models for Metamath
The construction of first-order logic and set theory gives rise to apparent circularities of mutual dependence, making it unclear which can act as a self-contained starting point in the foundation of mathematics. In this paper, we carry out…
We formally define a "mathematical object" and "set". We then argue that expressions such as "(Ax)F(x)", and "(Ex)F(x)", in an interpretation M of a formal theory P, may be taken to mean "F(x) is true for all x in M", and "F(x) is true for…
Numerous formalisms and dedicated algorithms have been designed in the last decades to model and solve decision making problems. Some formalisms, such as constraint networks, can express "simple" decision problems, while others are designed…
We present a new fragment of axiomatic set theory for pure sets and for the iteration of power sets within given transitive sets. It turns out that this formal system admits an interesting hierarchy of models with true membership relation…
While model checking has often been considered as a practical alternative to building formal proofs, we argue here that the theory of sequent calculus proofs can be used to provide an appealing foundation for model checking. Since the…
To model is to represent. The threshold of decidability defines two epistemological choices: one model (or a finite number of models) suffices for representing the dynamics below the undecidable; above this threshold (defined as…
Recently, in Axioms 10(2): 119 (2021), a nonclassical first-order theory T of sets and functions has been introduced as the collection of axioms we have to accept if we want a foundational theory for (all of) mathematics that is not weaker…
In a previous paper, we showed that profinite $L$-algebras (where $L$ is a variety of modal algebras generated by its finite members) are monadic over $\mathbf{Set}$. This monadicity result suggests that profinite $L$-algebras could be…
To most applied statisticians, a fitting procedure's degrees of freedom is synonymous with its model complexity, or its capacity for overfitting to data. In particular, it is often used to parameterize the bias-variance tradeoff in model…
Unlike computation or the numerical analysis of differential equations, simulation does not have a well established conceptual and mathematical foundation. Simulation is an arguable unique union of modeling and computation. However,…
We present a categorical theory of the composition methods in finite model theory -- a key technique enabling modular reasoning about complex structures by building them out of simpler components. The crucial results required by the…
This study aims to reinforce the theoretical foundation for diverse systems--including the axiomatic definition of intelligence--by introducing a mathematically rigorous and unified formal structure for the concept of 'state,' which has…
G\"odel proved in the 1930s in his famous Incompleteness Theorems that not all statements in mathematics can be proven or disproven from the accepted ZFC axioms. A few years later he showed the celebrated result that Cantor's Continuum…
G\"odel's Incompleteness Theorems suggest that no single formal system can capture the entirety of one's mathematical beliefs, while pointing at a hierarchy of systems of increasing logical strength that make progressively more explicit…
A set $F$ of formulas is complete relative to a given class of logics, if every logic from this class can be axiomatized by formulas from $F$. A set of formulas $F$ is {\L}-complete relative to a given class of logics, if every logic of…
In a Systems Engineering setting, various models are produced using a variety of methods and tools. Focusing on a type of models -- called descriptive models -- which we shall describe, we argue that, while the clarity and precision of…
Despite extensive research both on the theoretical and practical fronts, formalising, reasoning about, and implementing languages with variable binding is still a daunting endeavour - repetitive boilerplate and the overly complicated…
This paper proposes a definition of recognizable transducers over monads and comonads, which bridges two important ongoing efforts in the current research on regularity. The first effort is the study of regular transductions, which extends…
Tree sets are abstract structures that can be used to model various tree-shaped objects in combinatorics. Finite tree sets can be represented by finite graph-theoretical trees. We extend this representation theory to infinite tree sets.…
Let m>2 be an integer. We show that ZF + "For every integer n, Every countable family of non-empty sets of cardinality at most n has an infinite partial choice function" is not strong enough to prove that every countable set of m-element…