text: to use logic to analyze mathematical reasoning or to establish 
  logic-based foundations of mathematics.[165] The latter was a 
  major concern in early 20th-century mathematical logic, which 
  pursued the program of logicism pioneered by philosopher- 
  logicians such as Gottlob Frege, Alfred North Whitehead, and 
  Bertrand Russell. Mathematical theories were supposed to be 
  logical tautologies, and their program was to show this by means 
  of a reduction of mathematics to logic. Many attempts to realize 
  this program failed, from the crippling of Frege's project in his 
  *Grundgesetze*by Russell's paradox, to the defeat of Hilbert's 
  program by Gödel's incompleteness theorems.[166] 
  
  Set theory originated in the study of the infinite by Georg Cantor, 
  and it has been the source of many of the most challenging and 
  important issues in mathematical logic. They include Cantor's 
  theorem, the status of the Axiom of Choice, the question of the 
  independence of the continuum hypothesis, and the modern debate 
  on large cardinal axioms.[167] 
  Bertrand Russell made various 
  contributions to mathematical 
  logic.[163] 
  
  Computability theory is the branch of mathematical logic that 
  studies effective procedures to solve calculation problems. One of 
  its main goals is to understand whether it is possible to solve a given problem using an algorithm. For 
  instance, given a certain claim about the positive integers, it examines whether an algorithm can be found 
  to determine if this claim is true. Computability theory uses various theoretical tools and models, such as 
  Turing machines, to explore this type of issue.[168] 