Gdel's Work in Set Theory Case Study Example | Topics and Well Written Essays - 2000 words

Gdel's Work in Set Theory - Case Study Example In his Logical Journey, Wings publications indicate that GÃ ¶del’s works began in 1930 when he started studying the consistency problems of classical analysis (Wang, 1996). At the time, there had been no rigorous justifications and explanations on the rigorous mathematics (Feferman, et al., 2003, p. 339). This study got its motivation from Hilbert’s works. Hilbert had been working towards the provision of a directly consistent analysis of the finitary methods. The problems that this work had formed the driving force to his study. Through this, GÃ ¶del’s wanted to prove the constancy of number theory by a finitary numeral theory (Barbara, et al., 1990). He also wanted to prove the dependability of analysis by number theory. He represented real numbers by the predicates in number theory. In so doing, he found out that he had to use the truth concept in order to verify the axioms of the analysis. He came with an enumeration of symbols, sentences, and verifications within the specified order. In so doing, he discovered that the impression of arithmetic truth cannot be given a defined form in arithmetic. He observed that if a way to define the truth within a system existed, it would lead to a liar paradox (Rahman, et al., 2008). This would show that the system is inconsistent with what is being studied. These arguments were later formalized so that they bring meaning to the existence of undecidable propositions without quoting any individual occurrences. It is observable that GÃ ¶del tried to reduce the c onsistency problem to that of arithmetic for ease of solving. At this point, he temporarily changed the direction with the aim of intruding another element. The element would prove an illumination solution to Liar Paradox (Winterburn, 2012, p. 47). This appeared to require the truth definition for the arithmetic. This, in turn, resulted to paradoxes, like the Liar paradox to mean that the sentence is a false one. GÃ ¶del then discerned paradoxes of this form would not necessarily come in existence if the truth were to be replaced with probability.