# Introduction

• H.-D. Ebbinghaus
• J. Flum
• W. Thomas
Part of the Undergraduate Texts in Mathematics book series (UTM)

## Abstract

Towards the end of the nineteenth century mathematical logic evolved into a subject of its own. It was the works of Boole, Frege, Russell, and Hilbert, among others1, that contributed to its rapid development. Various elements of the subject can already be found in traditional logic, for example, in the works of Aristotle or Leibniz. However, while traditional logic can be considered as part of philosophy, mathematical logic is more closely related to mathematics. Some aspects of this relation are:
1. (1)

Motivation and Goals. Investigations in mathematical logic arose mainly from questions concerning the foundations of mathematics. For example, Frege intended to base mathematics on logical and set-theoretical principles. Russell tried to eliminate contradictions that arose in Frege’s system. Hilbert’s goal was to show that “the generally accepted methods of mathematics taken as a whole do not lead to a contradiction” (this is known as Hilbert’s program).

2. (2)

Methods. In mathematical logic the methods used are primarily mathematical. This is exemplified by the way in which new concepts are formed, definitions are given, and arguments are conducted.

3. (3)

Applications in Mathematics. The methods and results obtained in mathematical logic are not only useful for treating foundational problems; they also increase the stock of tools available in mathematics itself. There are applications in many areas of mathematics, such as algebra and topology, but also in various parts of theoretical computer science.

## Keywords

Equivalence Relation Mathematical Logic Mathematical Proof Theoretical Computer Science Left Inverse
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