About Incompleteness
Gödel's two incompleteness theorems are among the most important results in modern logic. These discoveries revolutionized the understanding of mathematics and logic, and they also had dramatic implications for the philosophy of mathematics. The incompleteness theorems concern the limits of provability in formal axiomatic theories. The first incompleteness theorem states that in any consistent formal system F within which a certain amount of arithmetic can be carried out, there are statements of the language of F which can neither be proved nor disproved in F. According to the second incompleteness theorem, such a formal system F cannot prove that F itself is consistent (assuming that F is indeed consistent). The overall aim of this workshop was to revisit these groundbreaking discoveries, recall the milestones of their derivations, reveal their indispensable background conditions, and consider their impact in several frameworks given by specific formal theories.
Abstracts and Slides
Kentaro Sato
Remarks on Gödel's Incompleteness Theorems
The focus will be on the statements, especially, on the explanation of the meanings, and the needs of the preconditions of Gödel's first and second incompleteness theorems, rather than on the proofs. Some typical misunderstandings of the theorems will be presented, and explained how wrong they are. If time permitted, it will also be explained how these theorems serve the basis for the present-day logic researches, rather than the negative impacts of the theorems.
Graham Priest
Paraconsistent Logic and Some of its Metatheory
In this talk I will provide the details of one of the most basic paraconsistent logics, LP, and some aspects of its metatheory. In particular, I will explain the Collapsing Lemma, and its use to construct models of inconsistent theories, including arithmetic.
Graham Priest
Gödel’s (First) Incompleteness Theorem
Gödel’s First Incompleteness Theorem is usually phrased as: an axiomatic theory of arithmetic (of a certain strength) is incomplete. What it should really say is that such a theory is either incomplete or inconsistent. Of course, using classical logic, an inconsistent theory really isn’t very interesting! But using a paraconsistent logic, matters are quite different. In this talk I will discuss the implications of Gödel’s Theorem from this perspective.
Lorenz Halbeisen
The Concept of Existence in Incomplete Theories
By Gödel's first incompleteness theorem we know that Peano Arithmetic as well as other theories like Set Theory are incomplete, which means that there are sentences which can neither be proved nor disproved within the theory. The situation is particularly interesting in the case when the undecidable sentence states the existence of a certain object. Based on an example, we shall confront the existence of certain sets with Paul Bernay's concept of existence given in his article "Mathematische Existenz und Widerspruchsfreiheit".