Related papers: Teaching G{\"o}del's incompleteness theorems
We construct a De Morgan algebra-valued logic with quantifiers, where the truth values are in a finite De Morgan algebra, We show that there is a representation theorem of the cylindric algebra of this logic from which a completeness…
Formal logic has often been seen as uniquely placed to analyze mathematical argumentation. While formal logic is certainly necessary for a complete understanding of mathematical practice, it is not sufficient. Important aspects of…
Basic arithmetic is the cornerstone of mathematics and computer sciences. In arithmetic, 'division by zero' is an undefined operation and any attempt at extending logic for algebraic division to incorporate division by zero has resulted in…
Using recent results in topos theory, two systems of higher-order logic are shown to be complete with respect to sheaf models over topological spaces---so-called ``topological semantics''. The first is classical higher-order logic, with…
Modal logics are widely used in computer science. The complexity of modal satisfiability problems has been investigated since the 1970s, usually proving results on a case-by-case basis. We prove a very general classification for a wide…
Stalnaker and Thomason famously proved that the conditional logic \textsf{C2} with first-order quantifiers is complete with respect to a selection function semantics. However, the selection functions used in this completeness result take…
Computational and complexity theory are core components of the computer science curriculum, and in the vast majority of cases are taught using decision problems as the main paradigm. For experienced practitioners, decision problems are the…
This paper exposes a contradiction in the Zermelo-Fraenkel set theory with the axiom of choice (ZFC). While Godel's incompleteness theorems state that a consistent system cannot prove its consistency, they do not eliminate proofs using a…
An alternative proof of the completeness of relational algebra with respect to allowed formulas of first-order logic is presented. The proof relies on the well-known embedding of relational algebra into cylindric algebra, which makes it…
The paper examines the construction of a course in mathematical analysis at a pedagogical university, aimed at developing the ability of future mathematics teachers to detect and solve problems related to finding proofs. Key words: teaching…
I describe some deep-seated problems in higher mathematical education, and give some ideas for their solution -- I advocate a move away from the traditional introduction of mathematics through calculus, and towards computation and discrete…
Philosophy of science attempts to describe all parts of the scientific process in a general way in order to facilitate the description, execution and improvements of this process. So far, all proposed philosophies have only covered existing…
We study relative precompleteness in the context of the theory of numberings, and relate this to a notion of lowness. We introduce a notion of divisibility for numberings, and use it to show that for the class of divisible numberings,…
Goedel Incompleteness Theorem leaves open a way around it, vaguely perceived for a long time but not clearly identified. (Thus, Goedel believed informal arguments can answer any math question.) Closing this loophole does not seem obvious…
As an approach to a Theory of Everything a framework for developing a coherent theory of mathematics and physics together is described. The main characteristic of such a theory is discussed: the theory must be valid and and sufficiently…
We employ the notions of `sequential function' and `interrogation' (dialogue) in order to define new partial combinatory algebra structures on sets of functions. These structures are analyzed using J. Longley's preorder-enriched category of…
We commonly think of mathematics as bringing precision to application domains, but its relationship with computer science is more complex. This experience report on the use of Racket and Haskell to teach a required first university CS…
Differential Linear Logic enriches Linear Logic with additional logical rules for the exponential connectives, dual to the usual rules of dereliction, weakening and contraction. We present a proof-net syntax for Differential Linear Logic…
Recent investigations have been carried out on critical analyses of beginning physics teachers confronted with questionable explanations. These studies raise the question of the choices made by teachers for their teaching once they have…
I have developed a pedagogy and textbook for teaching logic centered on what I call "logical worldviews". A logical worldview examines the close connection between philosophical commitments and the logical principles and method for a…