Reflecting Upon Godel’s Incompleteness Theorems And Wittgenstein’s Reaction To Them

This paper develops the discussion by centering on Godel’s thought called "generalization process". In short, this is a process that keeps on expanding the domain of knowledge from the least controversial one; and also is a process of constantly incorporating more intellectual content into true range. In this process, the importance of Godel’s incompleteness theorems is reflected in its revelation of the limitation of finite method, thus prompting the necessity of progressing to infinite and abstract concepts. Intuition leads generalization process; It seems to Godel that axiomatic method is the essential means to bring about explicitness and clarify questions, while the original concept of axiom needs to be searched with intuition. Based on clarifying concepts like generalization process and intuition, this paper introduces in the definition of "non-self-sufficiency", thus raising a new demonstration to support generalization process by exploring problems like Isaacson proposition closely related to Godel’s incompleteness theorems.On the other hand, Wittgenstein commented on Godel’s incompleteness theorems that it’s the most controversial part in his philosophy of logic and mathematics. This paper tries to demonstrate that these comments constitute the doubt about generalization process if they are viewed with Wittgenstein’s philosophy of logic and mathematics on a whole. Concretely speaking, Wittgenstein’s emphasis on the form of life questions the necessity of generalization process’ development to infinite and abstract concepts. However, this paper responses to this question, namely demonstrating that the "usage" of mathematics brings certainty to mathematics to some extent based on simply illustrating the current situation of mathematical foundation study, thus pointing out that generalization process can’t proceed unlimitedly.