<Dd> Mathematicians have always worked with logic and symbols, but for centuries the underlying laws of logic were taken for granted, and never expressed symbolically . Mathematical logic, also known as symbolic logic, was developed when people finally realized that the tools of mathematics can be used to study the structure of logic itself . Areas of research in this field have expanded rapidly, and are usually subdivided into several distinct departments . <Dl> <Dt> Model theory </Dt> <Dd> Model theory studies mathematical structures in a general framework . Its main tool is first - order logic . </Dd> <Dt> Set theory </Dt> <Dd> A set can be thought of as a collection of distinct things united by some common feature . Set theory is subdivided into three main areas . Naive set theory is the original set theory developed by mathematicians at the end of the 19th century . Axiomatic set theory is a rigorous axiomatic theory developed in response to the discovery of serious flaws (such as Russell's paradox) in naive set theory . It treats sets as "whatever satisfies the axioms", and the notion of collections of things serves only as motivation for the axioms . Internal set theory is an axiomatic extension of set theory that supports a logically consistent identification of illimited (enormously large) and infinitesimal (unimaginably small) elements within the real numbers . See also List of set theory topics . </Dd> <Dt> Proof theory and constructive mathematics </Dt> <Dd> Proof theory grew out of David Hilbert's ambitious program to formalize all the proofs in mathematics . The most famous result in the field is encapsulated in Gödel's incompleteness theorems . A closely related and now quite popular concept is the idea of Turing machines . Constructivism is the outgrowth of Brouwer's unorthodox view of the nature of logic itself; constructively speaking, mathematicians cannot assert "Either a circle is round, or it is not" until they have actually exhibited a circle and measured its roundness . </Dd> </Dl> </Dd> <Dl> <Dt> Model theory </Dt> <Dd> Model theory studies mathematical structures in a general framework . Its main tool is first - order logic . </Dd> <Dt> Set theory </Dt> <Dd> A set can be thought of as a collection of distinct things united by some common feature . Set theory is subdivided into three main areas . Naive set theory is the original set theory developed by mathematicians at the end of the 19th century . Axiomatic set theory is a rigorous axiomatic theory developed in response to the discovery of serious flaws (such as Russell's paradox) in naive set theory . It treats sets as "whatever satisfies the axioms", and the notion of collections of things serves only as motivation for the axioms . Internal set theory is an axiomatic extension of set theory that supports a logically consistent identification of illimited (enormously large) and infinitesimal (unimaginably small) elements within the real numbers . See also List of set theory topics . </Dd> <Dt> Proof theory and constructive mathematics </Dt> <Dd> Proof theory grew out of David Hilbert's ambitious program to formalize all the proofs in mathematics . The most famous result in the field is encapsulated in Gödel's incompleteness theorems . A closely related and now quite popular concept is the idea of Turing machines . Constructivism is the outgrowth of Brouwer's unorthodox view of the nature of logic itself; constructively speaking, mathematicians cannot assert "Either a circle is round, or it is not" until they have actually exhibited a circle and measured its roundness . </Dd> </Dl> <Dd> Model theory studies mathematical structures in a general framework . Its main tool is first - order logic . </Dd> <Dd> A set can be thought of as a collection of distinct things united by some common feature . Set theory is subdivided into three main areas . Naive set theory is the original set theory developed by mathematicians at the end of the 19th century . Axiomatic set theory is a rigorous axiomatic theory developed in response to the discovery of serious flaws (such as Russell's paradox) in naive set theory . It treats sets as "whatever satisfies the axioms", and the notion of collections of things serves only as motivation for the axioms . Internal set theory is an axiomatic extension of set theory that supports a logically consistent identification of illimited (enormously large) and infinitesimal (unimaginably small) elements within the real numbers . See also List of set theory topics . </Dd>

What are the branches of mathematics and their meaning