Related papers: A Note on the Possibility of Self-Reference in Mat…
We present the model theoretic concepts that allow mathematics to be developed with the notion of the potential infinite instead of the actual infinite. The potential infinite is understood as a dynamic notion, being an indefinitely…
We give a precise definition of a formal mathematical object as any symbol for an individual constant, predicate letter, or a function letter that can be introduced through definition into a formal mathematical language without inviting…
One of the most basic functions of language is to refer to objects in a shared scene. Modeling reference with continuous representations is challenging because it requires individuation, i.e., tracking and distinguishing an arbitrary number…
Godel's First Incompleteness Theorem is generalized to definable theories, which are not necessarily recursively enumerable, by using a couple of syntactic-semantic notions, one is the consistency of a theory with the set of all true…
Deficiencies in Kauffman's proposal regarding a new way for building scientific theories are pointed out. A suggestion to overcome them, and in fact, independently construct mathematical theories which are beyond the reach of Goedel's…
In his paper on the incompleteness theorems, G\"odel seemed to say that a direct way of constructing a formula that says of itself that it is unprovable might involve a faulty circularity. In this note, it is proved that 'direct'…
This article was motivated by the discovery of a potential new foundation for mainstream mathematics. The goals are to clarify the relationships between primitives, foundations, and deductive practice; to understand how to determine what…
In an earlier paper, "Omega-inconsistency in Goedel's formal system: a constructive proof of the Entscheidungsproblem" (math/0206302), I argued that a constructive interpretation of Goedel's reasoning establishes any formal system of…
We recently presented the so-called allagmatic method, which includes a system metamodel providing a framework for describing, modelling, simulating, and interpreting complex systems. Its development and programming was guided by…
According to the math tea argument, there must be real numbers that we cannot describe or define, because there are uncountably many real numbers, but only countably many definitions. And yet, the existence of pointwise-definable models of…
There is an increasing interest in applying recent advances in AI to automated reasoning, as it may provide useful heuristics in reasoning over formalisms in first-order, second-order, or even meta-logics. To facilitate this research, we…
We introduce a model-complete theory which completely axiomatizes the structure $Z_{\alpha}=(Z, +, 0, 1, f)$ where $f : x \to \lfloor{\alpha} x \rfloor $ is a unary function with $\alpha$ a fixed transcendental number. When $\alpha$ is…
A formalisation of G\"odel's incompleteness theorems using the Isabelle proof assistant is described. This is apparently the first mechanical verification of the second incompleteness theorem. The work closely follows {\'S}wierczkowski…
We present the first class of mathematically rigorous, general, fully self-referential, self-improving, optimally efficient problem solvers. Inspired by Kurt Goedel's celebrated self-referential formulas (1931), such a problem solver…
In this note we observe that automated theorem provers (ATPs) that recursively enumerate theorems in a particular way will fail to identify some valid theorems for a reason that is analogous to how G\"odel proved the existence of what are…
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…
This paper proposes a modal typing system that enables us to handle self-referential formulae, including ones with negative self-references, which on one hand, would introduce a logical contradiction, namely Russell's paradox, in the…
Independence of premise principles play an important role in characterizing the modified realizability and the Dialectica interpretations. In this paper we show that a great many intuitionistic set theories are closed under the…
A semantic analysis of formal systems is undertaken, wherein the duality of their symbolic definition based on the "State of Doing" and "State of Being" is brought out. We demonstrate that when these states are defined in a way that opposes…
Abstract algebraic logic is a theory that provides general tools for the algebraic study of arbitrary propositional logics. According to this theory, every logic L is associated with a matrix semantics Mod*(L). This paper is a contribution…