For Quine’s theory sometimes called „Mathematical Logic“, grundlagen der Geometrie (Classic Reprint) PDF New Foundations. Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics. Mathematical logic is often divided into the fields of set theory, model theory, recursion theory, and proof theory.
Författare: Friedrich Schur.
Since its inception, mathematical logic has both contributed to, and has been motivated by, the study of foundations of mathematics. Each area has a distinct focus, although many techniques and results are shared among multiple areas. The borderlines amongst these fields, and the lines separating mathematical logic and other fields of mathematics, are not always sharp. The mathematical field of category theory uses many formal axiomatic methods, and includes the study of categorical logic, but category theory is not ordinarily considered a subfield of mathematical logic. Theories of logic were developed in many cultures in history, including China, India, Greece and the Islamic world. In the middle of the nineteenth century, George Boole and then Augustus De Morgan presented systematic mathematical treatments of logic. Charles Sanders Peirce built upon the work of Boole to develop a logical system for relations and quantifiers, which he published in several papers from 1870 to 1885.
From 1890 to 1905, Ernst Schröder published Vorlesungen über die Algebra der Logik in three volumes. This work summarized and extended the work of Boole, De Morgan, and Peirce, and was a comprehensive reference to symbolic logic as it was understood at the end of the 19th century. Concerns that mathematics had not been built on a proper foundation led to the development of axiomatic systems for fundamental areas of mathematics such as arithmetic, analysis, and geometry. In logic, the term arithmetic refers to the theory of the natural numbers. Boole and Schröder but adding quantifiers. The 19th century saw great advances in the theory of real analysis, including theories of convergence of functions and Fourier series. Georg Cantor developed the fundamental concepts of infinite set theory.
In the early decades of the 20th century, the main areas of study were set theory and formal logic. The discovery of paradoxes in informal set theory caused some to wonder whether mathematics itself is inconsistent, and to look for proofs of consistency. In 1900, Hilbert posed a famous list of 23 problems for the next century. Georg Cantor had been unable to obtain. To achieve the proof, Zermelo introduced the axiom of choice, which drew heated debate and research among mathematicians and the pioneers of set theory.