https://www.um.edu.mt/library/oar/handle/123456789/36159 Mon, 06 Apr 2026 06:30:15 GMT 2026-04-06T06:30:15Z https://www.um.edu.mt/library/oar/handle/123456789/24487 Title: Godel's theorem Abstract: At the beginning of the 20th century the mathematician David Hilbert posed a set of problems to the mathematical community that should have been the so-called road map oftasks to accomplish during the following hundred years. Among them was a problem which he posed in collaboration with Ackermann dealing with the question of whether a formal system of mathematical logic can be considered complete - where completeness implies that every true statement can be expressed within the system, possibly without a paradox. This was probably inspired by the recent discovery ofa series of paradoxes in Russell and Whitehead's Principia Mathematica which is now a de facto standard for defining and proving mathematical statements. The well-known Russell's paradox - formulated in a hundred different ways - has been catered for by denying the possibility of having a set being a member of itself However, other forms of paradoxes are not that easy to eliminate. Epimenides' paradox falls into this category: "I am a liar" or in logic-speak: "This statement is false". Godel's seminal work in 1931 not only managed to show that the PM system was inconsistent, but that any sufficiently powerful formal system is bound to be littered with paradoxes. It is worth stating how series this matter is: practically speaking he stated that there might exist theorems that cannot be proved or disproved - theorems about number theory itself, for instance. The approach to Godel's proofI am going to use is a simplified version based on the work of Douglas R. Hofstadter, "Godel, Escher, Bach: an Eternal Golden Braid". A book which I thoroughly recommend to anyone interested in the question of how animate matter can result out of combinations of inanimate matter. Tue, 01 Jan 2002 00:00:00 GMT https://www.um.edu.mt/library/oar/handle/123456789/24487 2002-01-01T00:00:00Z https://www.um.edu.mt/library/oar/handle/123456789/24457 Title: The Collection VI Editors: Sciriha, Irene; Telenta, Tanya Abstract: Sixth issue of The Collection, a journal by the Department of Mathematics at the University of Malta. Tue, 01 Jan 2002 00:00:00 GMT https://www.um.edu.mt/library/oar/handle/123456789/24457 2002-01-01T00:00:00Z https://www.um.edu.mt/library/oar/handle/123456789/24428 Title: The duplex of a graph Abstract: A very interesting property that the duplex has is that its spectrum contains the spectrum of the original graph G. To prove this, however we first need a lemma from matrix theory, which will be quoted without proof. Tue, 01 Jan 2002 00:00:00 GMT https://www.um.edu.mt/library/oar/handle/123456789/24428 2002-01-01T00:00:00Z https://www.um.edu.mt/library/oar/handle/123456789/24427 Title: Foreword [The Collection, 6] Abstract: The Collection VI workshop proved to be one of the most interesting. During the meeting, the lecturers in the mathematics department were very happy to discover that in their classes they have students who can discern and question established results. We were surprised that we managed to trigger off the interest of the students precisely in areas that they came across first at university. The motivation of some of them was so impelling that they somewhat shyly admitted that they spent the summer engaged in this new research. The department of mathematics is committed to encourage such students and confesses that they are the driving force behind its endeavors. Tue, 01 Jan 2002 00:00:00 GMT https://www.um.edu.mt/library/oar/handle/123456789/24427 2002-01-01T00:00:00Z