https://www.um.edu.mt/library/oar/handle/123456789/23633 Thu, 09 Apr 2026 12:29:20 GMT 2026-04-09T12:29:20Z https://www.um.edu.mt/library/oar/handle/123456789/24489 Title: The Collection V Editors: Sciriha, Irene; Telenta, Tanya Abstract: Fifth 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/123456789/24489 2002-01-01T00:00:00Z 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/123456789/24472 Title: New necessary and sufficient conditions for the Zarankiewicz conjecture on crossing numbers Abstract: The purpose of this presentation is to give a historical outline about the theory of crossing numbers, and to present the current status of long standing problems on crossing numbers of complete bipartite graphs. A set of new necessary and sufficient conditions are given for the Zarankiewicz conjecture on the crossing number of complete bipartite graphs. These conditions are expressed in terms of the crossing numbers' divisibility properties and their expressibility as polynomials. Tue, 01 Jan 2002 00:00:00 GMT https://www.um.edu.mt/library/oar/handle/123456789/24472 2002-01-01T00:00:00Z https://www.um.edu.mt/library/oar/handle/123456789/24461 Title: Boolean matrices Abstract: In this short note, we will use Boolean matrices to help us find some interesting graph theoretical properties regarding the distance and index of a graph. First we will give some definitions that are necessary for the theorems which follow. Tue, 01 Jan 2002 00:00:00 GMT https://www.um.edu.mt/library/oar/handle/123456789/24461 2002-01-01T00:00:00Z