What are the disadvantages of mathematical logic

mathematical logic

The mathematics developed in this limited framework is also called constructive or intuitionist mathematics. The most important contribution of intuitionism to basic research consists primarily in the strict delimitation of constructive and non-constructive mathematics, because one can only really calculate (i.e. solve problems algorithmically) within the framework of constructive mathematics.

Anyone who rejects the principle of two-valued comes to a different kind of logic in which there are more than two truth values. In this context, the modal logic, the multi-valued logic and the fuzzy logic emerged.

The endeavor to specify the intuitive concept of calculability or to define it mathematically exactly led to the concept of recursive functions, which, due to their definition, are evidently calculable in the naive sense. Since such fundamental terms as "calculable, decidable, constructable" are closely linked to the term recursive function and many problems required an algorithmic solution, a new theory, the so-called recursion theory, arose from this within the framework of mathematical logic. It deals in the broadest sense with the properties of the recursive functions. Although various efforts have been made to fully characterize the concept of predictability, this has not yet been fully achieved. It is only clear that all recursive functions are calculable, the inversion could only be assumed hypothetically. Church's (hypo-) thesis, which is now generally accepted, states that the computable functions are precisely the recursive ones. Further investigations are often made under this hypothesis, which may be unproven but seems very reasonable.

With his program of a finite justification of classical mathematics, Hilbert tried to take a different path in overcoming the fundamental crisis. He campaigned resolutely for the retention of Cantor's ideas on set theory, but under modified conditions. He had in mind a comprehensive axiomatization of geometry, number theory, analysis, Cantor's set theory and other fundamental areas of mathematics. From this basic idea, which puts certain axioms at the beginning, specifies the evidence and only allows conclusions to be drawn on this basis, a new direction in mathematical logic emerged, the theory of proof. Hilbert's program to completely axiomatize mathematics in essential parts turned out to be unrealistic with the appearance of Gödel's results on the incompleteness of arithmetic (Gödel's incompleteness theorem). As a consequence of the incompleteness theorem, the important insight immediately arises: Is L. the language of arithmetic, & Nopf; = 〈N, +, o, < ,="" 0,="" 1⟩="" das="" standardmodell="" der="" arithmetik="" und="">T is the set of all in & Nopf; valid statements from L, then is T apparently completely. Hence is T cannot be axiomatized recursively. In other words, if Σ is an arbitrary recursive subset of T, then there is always a statement ? in L. so that neither ? still ¬? are provable from £. Every such system is therefore incomplete, and so even such a fundamental theory as arithmetic cannot be axiomatized. Analogous considerations apply even more to every axiomatic system of set theory in which arithmetic can be interpreted.