Kurt Gödel 100 jaar
Nieuws |
de redactie
12 mei 2006 | Wat is waarheid en wat kunnen we van haar te weten komen? Dit zijn centrale vragen in de filosofie. De grote mathematicus en denker Kurt Gödel bestudeerde deze vragen waarbij hij zich concentreerde op wiskundige waarheden. Zijn bekendste resultaten zijn de onvolledigheidsstellingen. In zekere zin stellen deze theorema's dat het gat tussen de wiskundige waarheid en wat we van deze waarheid in een formeel systeem kunnen bewijzen heel groot is. Dit jaar is het honderd jaar geleden dat Gödel geboren is en met een Utrechts symposium wordt hem en zijn werk eer betoond.
Naast een algemeen overzicht van Gödels leven, werk en filosofische thema’s wordt er tijdens dit symposium speciaal aandacht besteed aan zijn onvolledigheidstellingen en zijn bijdrages aan de settheorie en Einstein’s relativiteitstheorie. Hier onder leest u vast enkele van de abstracts van de bijdragen die u kunt verwachten op 26 mei van 10:00 tot 17:00 uur in de Jaarbeurs te Utrecht.
Benedikt Löwe ( Amsterdam)
Gödel’s Contribution to Set Theory
In the last thirty years of his life, Kurt Gödel was mostly concerned with questions of the foundations of mathematics. The foundational viewpoint often associated with him, the so- called ‘Gödelian Platonism’ is best understood in the context of the set-theoretic foundations of mathematics and the Gödel’s own contributions to that field. His 1938 work on the continuum hypothesis not only provided half of the solution to Hilbert’s First Problem but also created a new research area in set theory, the area that we nowadays call ‘Inner Model Theory’. In this talk, I shall focus on Gödel’s work of the late 1930s and its consequences for the development of set theory.
Dennis Dieks ( Utrecht)
Gödel on Time and Relativity
In the nineteen-forties Gödel studied Einstein’s general theory of relativity, and discovered a new type of solutions of the field equations. The characteristic feature of these solutions is that they contain closed time-like loops: worldlines that go back into their own past. Gödel famously concluded from the existence of these solutions that “time is ideal” (in the sense of idealistic philosophy); that there is no objective passage of time.
In the talk I will review Gödel’s argument and comment on it from a modern point of view.
Dick de Jongh ( Amsterdam) & Albert Visser (Utrecht)
The Incompleteness Theorems: What They Say and What They do not Say.
Gödel’s Incompleteness Theorems are central to the metamathematical tradition. They effectively showed that Hilbert’s programme was not feasible in any naive interpretation of the programme. The method of their proofs had profound impact on the development of Recursion Theory and Computer Science.
The Theorems have a strong appeal to the popular imagination. This is witnessed by the rich yearly harvest of papers that either claim to refute Gödel’s work by pointing out a subtle, persistently overlooked, mistake or claim, to the contrary, to take Gödel’s work a step further by adapting his work to show that arithmetic is inconsistent. (The Master stupidly overlooked the crucial step! ) Moreover, the appeal of the Theorems is also evident in the extensive and, in intention, serious discussions of the Theorems on internet.
In our talk, we will first discuss the historical context in which Gödel discovered the Incompleteness Theorems. Then, we produce a precise statement of the Theorems. We will give a brief sketch of their proofs. Finally, we will address the issue of the philosophical significance of the theorems. Do they show Fundamental Limitations of the Human Intellect? Or, do they establish the Intrinsic Inferiority of Machines in comparison to Humans? We will argue that all such conclusions are just non- sequiturs.
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