Logic Warning: You are not logged in. Your IP address will be publicly visible if you make any edits. If you log in or create an account, your edits will be attributed to your username, along with other benefits.Anti-spam check. Do not fill this in! ===Mathematical logic=== {{main|Mathematical logic}} [[File:Bertrand Russell 1949.jpg|thumb|left|alt=Photograph of Bertrand Russell|Bertrand Russell made various contributions to mathematical logic.{{sfn |Irvine |2022}}]] The term "mathematical logic" is sometimes used as a synonym of "formal logic". But in a more restricted sense, it refers to the study of logic within mathematics. Major subareas include [[model theory]], [[proof theory]], [[set theory]], and [[computability theory]].{{sfnm|1a1=Li|1y=2010|1p=ix|2a1=Rautenberg|2y=2010|2p=15|3a1=Quine|3y=1981|3p=1|4a1=Stolyar|4y=1984|4p=2}} Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic. However, it can also include attempts to use logic to analyze mathematical reasoning or to establish logic-based [[foundations of mathematics]].{{sfn |Stolyar |1984 |pp=3–6}} 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 [[tautology (logic)|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 theorem]]s.{{sfnm|1a1=Hintikka|1a2=Spade|1loc=[https://www.britannica.com/topic/history-of-logic/Godels-incompleteness-theorems Gödel's incompleteness theorems]|2a1=Linsky|2y=2011|2p=4|3a1=Richardson|3y=1998|3p=15}} 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.{{sfnm|1a1=Bagaria|1y=2021|2a1=Cunningham}} 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.{{sfnm|1a1=Borchert|1y=2006a|1loc=Computability Theory|2a1=Leary|2a2=Kristiansen|2y=2015|2p=195}} Summary: Please note that all contributions to Christianpedia may be edited, altered, or removed by other contributors. If you do not want your writing to be edited mercilessly, then do not submit it here. You are also promising us that you wrote this yourself, or copied it from a public domain or similar free resource (see Christianpedia:Copyrights for details). Do not submit copyrighted work without permission! Cancel Editing help (opens in new window) Discuss this page