Introduction:
This book adds new mathematics to the research note at the university and describes it. I am happy that I was born and brought to a Japanese. Languages as wide as Japanese do not see examples. In many cases it is difficult to translate into a foreign language. I apologize for the part that can not be translated. I think that mathematics is a worldwide common language.
A mathematical foundation problem and occurrence of fundamental theory of mathematics
Set theory introduced by G.Cantre at the end of the 19th century not only as a means of researching new mathematical objects such as point set theory and order numbers but also redefines various objects and theories of mathematics from more basic concepts It is a research which constitutes. For example, in R. Dedekind 's natural number theory (for number theory we will discuss it later), the system of natural numbers is made up of sets and correspondences, and Dedekind' s irrational number theory also makes real numbers a special set of rational numbers As follows. In this way, the mathematical conceptual structure becomes remarkably smooth and unified by using the collective concept. Despite the fact that the contradiction of the early set theory became clear as an inverse of the set theory, it is gradually recognized that the concept concept is fundamental and useful as a place to develop mathematics It was.
However, the fact that the effective usage of the collective concept is quite similar to the usage leading to the argument, and that the inversion appears almost within the scope of the formal logic is based on mathematical reflections on the conceptual composition method and the logic in mathematics It encouraged, mathematics foundation problem occurred. Depending on the idea of what mathematics is, the standpoint of this reversal comes from the beginning of the occurrence, such as LEJ Brewer's intuitionism, G. Frege and B. Russell's logicalism, D. Hilbert's formalism It split. (My position is axiomatic (described later))
Logicism is a theory that regards it as a field of mathematical logic. The solution of the reversal from this position was chiefly cast by Russell. Russell argued that the argument arises because it ignores the types of various concepts. Logicism is nothing but the basis or background for this claim.
