Chaohui Zhuang

In Zettel, Wittgenstein considered a modified version of Cantor’s diagonal argument. According to Wittgenstein, Cantor’s number, different with other numbers, is defined based on a countable set. If Cantor’s number belongs to the countable set, the definition of Cantor’s number become incomplete. Therefore, Cantor’s number is not a number at all in this context. We can see some examples in the form of recursive functions. The definition "f(a)=f(a)" can not decide anything about the value of f(a). The definiton is incomplete. The definition of "f(a)=1+f(a)" can not decide anything about the value of f(a) too. The definiton is incomplete.<br><br>According to Wittgenstein, the contradiction, in Cantor's proof, originates from the hidden presumption that the definition of Cantor’s number is complete. The contradiction shows that the definition of Cantor’s number is incomplete. <br><br>According to Wittgenstein’s analysis, Cantor’s diagonal argument is invalid. But different with Intuitionistic analysis, Wittgenstein did not reject other parts of classical mathematics. Wittgenstein did not reject definitions using self-reference, but showed that this kind of definitions is incomplete.<br><br>Based on Thomson’s diagonal lemma, there is a close relation between a majority of paradoxes and Cantor’s diagonal argument. Therefore, Wittgenstein’s analysis on Cantor’s diagonal argument can be applied to provide a unified solution to paradoxes.

The problem of Free Will is an important topic in religion, philosophy and neuroscience. We will introduce a new model of free will: free choice under constraints. Under outer and inner constraints, human still have the ability of free choice. Outer constrains include physical rules, environment and so on. Inner constrains include customs, desires, habits, preferences and so on. Given a specific context, human have the ability of deciding Yes/No on a specific preference. The free choices are caused, but not determined, by the outer and inner constraints. In addition, the free choices will refashion the inner constraints. Preferences are gradually formed by human, thus they may be gradually de-formed by human. Human can keep the positive preferences, while de-form the negative preferences. When we de-form the negative preferences, we have the chance to pursue another new possibility. So are scientists. They can keep the positive aspects of the known theory, while eliminate the negative aspects. When they eliminate the negative aspects of old conceptual schema, they have the chance to pursue another new conceptual schema.