Work in set theory showed that almost all ordinary mathematics can be formalized in terms of sets, although there the continuum encyclopedia of symbols pdf some theorems that cannot be proven in common axiom systems for set theory. 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 first half of the 20th century saw an explosion of fundamental results, accompanied by vigorous debate over the foundations of mathematics.

Boole to develop a logical system for relations and quantifiers, which he published in several papers from 1870 to 1885. 1879, a work generally considered as marking a turning point in the history of logic. The two-dimensional notation Frege developed was never widely adopted and is unused in contemporary texts. 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.

Boole and Schröder but adding quantifiers. Peano was unaware of Frege’s work at the time. Euclid were not in fact provable from his axioms. Among these is the theorem that a line contains at least two points, or that circles of the same radius whose centers are separated by that radius must intersect. This would prove to be a major area of research in the first half of the 20th century.