Related papers: On foundations for deductive mathematics
We review the philosophical framework of mathematical conceptualism as an alternative to set-theoretic foundations and show how mainstream mathematics can be developed on this basis. The paper includes an explicit axiomatization of the…
Choice functions constitute a simple, direct and very general mathematical framework for modelling choice under uncertainty. In particular, they are able to represent the set-valued choices that appear in imprecise-probabilistic decision…
This paper presents mathematics as a general science of computation in a way different from the tradition. It is based on the radical philosophical standpoint according to which the content, meaning and justification of experience lies in…
Inference is the process of using facts we know to learn about facts we do not know. A theory of inference gives assumptions necessary to get from the former to the latter, along with a definition for and summary of the resulting…
ASPIC-style structured argumentation frameworks provide a formal basis for reasoning in artificial intelligence by combining internal argument structure with abstract argumentation semantics. A key challenge in these frameworks is ensuring…
Bayesian inference has theoretical attractions as a principled framework for reasoning about beliefs. However, the motivations of Bayesian inference which claim it to be the only 'rational' kind of reasoning do not apply in practice. They…
The analysis of theory-confirmation generally takes the deductive form: show that a theory in conjunction with physical data and auxiliary hypotheses yield a prediction about phenomena; verify the prediction; provide a quantitative measure…
Argumentation is a non-monotonic process. This reflects the fact that argumentation involves uncertain information, and so new information can cause a change in the conclusions drawn. However, the base logic does not need to be…
The concept of definability of physical fields within a set-theoretical foundation is introduced. We propose an axiomatic set theory and show the Schroedinger equation and, more generally, a nonlinear sigma model come naturally out of the…
This paper introduces an alternative approach to proving the existence of choice functions for specific families of sets within Zermelo-Fraenkel set theory (ZF) without assuming any form on the Axiom of Choice (AC). Traditional methods of…
It is quite well-known from Kurt Godel's (1931) ground-breaking result on the Incompleteness Theorem that rudimentary relations (i.e., those definable by bounded formulae) are primitive recursive, and that primitive recursive functions are…
The reflection principle is the statement that if a sentence is provable then it is true. Reflection principles have been studied for first-order theories, but they also play an important role in propositional proof complexity. In this…
Inference systems are a widespread framework used to define possibly recursive predicates by means of inference rules. They allow both inductive and coinductive interpretations that are fairly well-studied. In this paper, we consider a…
This article discusses what can be proved about the foundations of mathematics using the notions of algorithm and information. The first part is retrospective, and presents a beautiful antique, Godel's proof, the first modern incompleteness…
This article explores the model-dependent nature of set cardinality, emphasizing that cardinality is not absolute but varies across different axiomatic frameworks. Although Cantor's diagonal argument shows the real numbers are…
The foundations of forcing theory are reworked to streamline the presentation and to show how the most basic results are applicable in very general contexts.
We provide a formal introduction into the classic theorems of general topology and its axiomatic foundations in set theory. In this second part we introduce the fundamental concepts of topological spaces, convergence, and continuity, as…
In many real-life settings, agents must navigate dynamic environments while reasoning under incomplete information and acting on a corpus of unstable, context-dependent, and often conflicting norms. We introduce a general, non-modal,…
We introduce a framework uniting algorithmic randomness with exchangeable credences to address foundational questions in philosophy of probability and philosophy of science. To demonstrate its power, we show how one might use the framework…
Using an elementary argument, we prove new fixed point theorems for classical elliptic complexes. We obtain new results for conformal relations and coisotropic intersections. We obtain theorems for the average intersections of families of…