送交者: steven 于 2006-2-23, 00:22:13:
回答: why? Let us know why? 由 insight 于 2006-2-22, 23:38:55:
Your idea is called psudomathematics, similar to 蒋春暄. Remember if your idea work, there won't be any undecidable problems, and there are many proven undecidable problems. In fact, there are more undecidable problems than decidable problems. You think all the mathematicians in this world cannot match you?
in mathematical logic, there is only deduction, because induction in philosophical term is just a form of deduction. Deduction in mathematical logic is defined as drawing conclusion from premises and syllogisms. Unlike traditional logic, in this case the premises, say clause X doesn't have to be true, it can be assumed true like "induction." Then by applying syllogism like modus ponens, (which means "imply", step a imply b, imply c and etc till conclusion). This is what we call a sentence obtained from X, the collection of all these kind of sentences which are true is called the consequence of a theory of A, which is a set of all sentence closed under logical operation. Even provided the all the sentences in the consequence of theory of A denoted Cn Th A, are finite strings, and the Cn Th A is recursive enumerable, the Godel theorem says this Cn Th A cannot be complete or consistent at the same time. For example, the Cn Th Number is like that. That means there are properties of Number that cannot be proven to be true nor false. However, for the Cn Th Set, it is different, Cn Th Set can be complete. The problem is Cn Th Set is not recursively enumerable, which means the set of Cn Th Set is like the set of real number, which is continuum.
However, this doesn't limited the advance of knowledge, because even for Cn Th Number, it is infinite. There means there are infinite number of sentences that can be proven to be true or false. That means there are infinite number of problems that we can solve by science. The catch is the set of undecidable is always bigger than the set of decidable.