David Hilbert was a German mathematician and one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas including invariant theory, the calculus of variations, commutative algebra, algebraic number theory, the foundations of geometry, spectral theory of operators and its application to integral equations, mathematical physics, and the foundations of mathematics.
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics.
In mathematics, a theorem is a statement that has been proved, or can be proved. The proof of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems.
Wilhelm Friedrich Ackermann was a German mathematician and logician best known for his work in mathematical logic and the Ackermann function, an important example in the theory of computation.
Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathematics. In this latter sense, the distinction between foundations of mathematics and philosophy of mathematics turns out to be vague. Foundations of mathematics can be conceived as the study of the basic mathematical concepts and how they form hierarchies of more complex structures and concepts, especially the fundamentally important structures that form the language of mathematics also called metamathematical concepts, with an eye to the philosophical aspects and the unity of mathematics. The search for foundations of mathematics is a central question of the philosophy of mathematics; the abstract nature of mathematical objects presents special philosophical challenges.
Ernst Friedrich Ferdinand Zermelo was a German logician and mathematician, whose work has major implications for the foundations of mathematics. He is known for his role in developing Zermelo–Fraenkel axiomatic set theory and his proof of the well-ordering theorem. Furthermore, his 1929 work on ranking chess players is the first description of a model for pairwise comparison that continues to have a profound impact on various applied fields utilizing this method.
Proof theory is a major branch of mathematical logic and theoretical computer science within which proofs are treated as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of a given logical system. Consequently, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature.
In mathematics and logic, an axiomatic system is any set of primitive notions and axioms to logically derive theorems. A theory is a consistent, relatively-self-contained body of knowledge which usually contains an axiomatic system and all its derived theorems. An axiomatic system that is completely described is a special kind of formal system. A formal theory is an axiomatic system that describes a set of sentences that is closed under logical implication. A formal proof is a complete rendition of a mathematical proof within a formal system.
Crispin James Garth Wright is a British philosopher, who has written on neo-Fregean (neo-logicist) philosophy of mathematics, Wittgenstein's later philosophy, and on issues related to truth, realism, cognitivism, skepticism, knowledge, and objectivity. He is Professor of Philosophical Research at the University of Stirling, and taught previously at the University of St Andrews, University of Aberdeen, New York University, Princeton University and University of Michigan.
Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. Its defining method can briefly be described as "going backwards from the theorems to the axioms", in contrast to the ordinary mathematical practice of deriving theorems from axioms. It can be conceptualized as sculpting out necessary conditions from sufficient ones.
Thoralf Albert Skolem was a Norwegian mathematician who worked in mathematical logic and set theory.
In mathematical logic, the theory of infinite sets was first developed by Georg Cantor. Although this work has become a thoroughly standard fixture of classical set theory, it has been criticized in several areas by mathematicians and philosophers.
In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding.
Martin Hugo Löb was a German mathematician. He settled in the United Kingdom after the Second World War and specialised in mathematical logic. He moved to the Netherlands in the 1970s, where he remained in retirement. He is perhaps best known for having formulated Löb's theorem in 1955.
A timeline of mathematical logic; see also history of logic.
In proof theory, a branch of mathematical logic, elementary function arithmetic (EFA), also called elementary arithmetic and exponential function arithmetic, is the system of arithmetic with the usual elementary properties of 0, 1, +, ×, , together with induction for formulas with bounded quantifiers.
In mathematical set theory, the axiom of adjunction states that for any two sets x, y there is a set w = x ∪ {y} given by "adjoining" the set y to the set x. It is stated as
Stephen George Simpson is an American mathematician whose research concerns the foundations of mathematics, including work in mathematical logic, recursion theory, and Ramsey theory. He is known for his extensive development of the field of reverse mathematics founded by Harvey Friedman, in which the goal is to determine which axioms are needed to prove certain mathematical theorems. He has also argued for the benefits of finitistic mathematical systems, such as primitive recursive arithmetic, which do not include actual infinity.
Juliette Kennedy is an associate professor in the Department of Mathematics and Statistics at the University of Helsinki. Her main research interests are mathematical logic and the foundations of mathematics. In the course of her work she has published extensively on the works of Kurt Gödel.
