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What is Incompleteness

Handbook of Research on Innovations in Database Technologies and Applications: Current and Future Trends
A database is incomplete if there is a sentence whose truth value cannot be ascertained.
Published in Chapter:
A Paraconsistent Relational Data Model
Navin Viswanath (Georgia State University, USA) and Rajshekhar Sunderraman (Georgia State University, USA)
DOI: 10.4018/978-1-60566-242-8.ch003
Typically, relational databases operate under the Closed World Assumption (CWA) of Reiter (Reiter, 1987). The CWA is a meta-rule that says that given a knowledge base KB and a sentence P, if P is not a logical consequence of KB, assume ~P (the negation of P). Thus, we explicitly represent only positive facts in a knowledge base. A negative fact is implicit if its positive counterpart is not present. Under the CWA we presume that our knowledge about the world is complete i.e. there are no “gaps” in our knowledge of the real world. The Open World Assumption (OWA) is the opposite point of view. Here, we “admit” that our knowledge of the real world is incomplete. Thus we store everything we know about the world – positive and negative. Consider a database which simply contains the information “Tweety is a bird”. Assume that we want to ask this database the query “Does Tweety fly?”. Under the CWA, since we assume that there are no gaps in our knowledge, every query returns a yes/no answer. In this case we get the answer “no” because there is no information in the database stating that Tweety can fly. However, under the OWA, the answer to the query is “unknown”. i.e. the database does not know whether Tweety flies or not. We would obtain a “no” answer to this query under the OWA only if it was explicitly stated in the database that Tweety does not fly.
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The Role of Science, Technology, and the Individual on the Way of Software Systems Since 1968
Kurt Gödel wrote the incompleteness theorems in the 1930s. They are still valid as the subject of several of today’s discussions and will continue to be effective in the future. These theorems show that any logical system consists of either contradiction or statements that cannot be proven. Therefore, they help to understand that the formal systems used are not complete. On the other hand, the trend of mathematicians in the 1900s was the unification of all theories for the solution of the most difficult problems in all disciplines. Before Gödel defined any system as consistent without any contradictions, and as incomplete with all or some disproved statements, Hilbert formulated all mathematics in an axiomatic form with Set Theory. Gödel’s theorem showed the limitations that exist within all logical systems and laid the foundation of modern computer science. These theorems caused several results about the limits of computational procedures. A famous example is the inability to solve the halting problem. A halting problem finds out whether a program with a given input will halt at some time or continue to run into an infinite loop. This decision problem demonstrates the limitations of programming. In a modern sense, this means that it is impossible to build an excellent compiler or a perfect antivirus.
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The Relevance of Science-Religion Courses in the Educational Field: Science and Scientism – Theology and Science
In 1902, the logician Bertrand Russel disqualified the certitude given by Cantor’s proofs, presenting his paradox sets. Russel and Alfred Whitehead have developed a strong axiomatic system for framing types of mathematical reasoning in their system. These logical rules were trying to avoid the paradox. The system was, on the other side, very complicated. The mathematician, David Hilbert realized an easier system, trying to include all types of right judgment. He wanted to prove that this system is not contradictious, resulting that mathematics offers certainly knowledge. The essence of Hilbert’s schedule was shattered by the brilliant logician Kurt Gödel. This man, through his theorems of incompleteness, shows that any system with formal rules, large enough that it can contain arithmetical elements, has also unprovable sentences. Gödel’s incompleteness theorem shows that a theory that implies mathematical-axiomatic thinking, cannot be complete, meaning that it can have different internal limits or that cannot completely describe mathematically shaped physical reality. Gödel’s theorem is an expression on what structurally exists in the created reality of the world – meaning the limit. There is a correspondence between Gödel’s logical theorems and Heisenberg’s indetermination principle from quantum physics, by the fact that both express the inherent limitations of each axiomatical thinking or theory. Thereby, a scientific theory cannot pretend anymore that it’s the exclusive result of axiomatical thinking, in which intuition and spiritual experience have no role (Lemeni, 2009 AU14: The in-text citation "Lemeni, 2009" is not in the reference list. Please correct the citation, add the reference to the list, or delete the citation. , p. 141).
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