On The Significance Of G(?)del's Theorems

Godel's Theorems have been quite influential in the 20th century, butthere are misinterpretations about the theorems themselves, the concernedlogic and the formal method. Therefore it's necessary to make some revisions.Firstly, the paper distinguishes two kinds of "completeness" in G?del'sTheorems. One is a property of logic, and inquires the relation of syntacticand semantic reasoning. The other is a theoretical quality, for it depends onthe set of primes and models of a fixed theory. Completeness Theorem showsthat syntactic and semantic consequences coincide in the case of the predicatelogic. Incompleteness Theorem indicates that any formal system rich enoughis an incomplete theory, in which proof can be verified mechanically.Secondly, the paper analyses the significance of Godel's Theorems. It'snecessary to correct some misinterpretations. The Godel's Sentence is not aparadox. Godel's Theorems don't indicate that mathematics is inconsistent,nor do they mean that mind is superior to machine, nor do they completelydeny the Hilbert Program, nor do they undermine logic. The paper mentionsthat Godel's theorems discover the limitations of the formal method andadvance it. The key idea of Incompleteness Theorem lies in the proof that aclass of strong subsystem of mathematics is incomplete. CompletenessTheorem indicates the limitations of formal language. The SecondIncompleteness Theorem demonstrates that the formal method has strongcapabilities to express. The paper points out that Godel distinguishes truth andprovability for the first time. It is noted that no truth predicate exists in theformal arithmetic system, which Godel discovered earlier than Tarski.Lastly, The paper shows that Godel's Theorems have great impact onmodern logic, philosophy, computer science and physics. Therefore thesignificance of Godel's Theorems is worth discussing.