Mathematical logic theory and math

Research in mathematical logic at the University of Notre Dame largely resides in two areas: computability theory and model theory. Computability theory concerns computability and complexity, often measured by Turing reducibility. Model theory at Notre Dame deals particularly with classification theory and o-minimality.

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ND Mathematical Logic Calendar in a window by its self. In addition to faculty from the math department, the Logic Research Group includes faculty from the philosophy department. Sergei Starchenko. Gabriel Conant. Daniel Max Hoffmann. Charles McCoy. Anibal Medina Mardones.

mathematical logic theory and math

Priyanko Rajan. Alessio Sammartano. Amanda Katharine Serenevy. Charles Stanton. Daniel Studenmund. Michael Wibmer. Peter Cholak Professor Computability theory. Julia Knight Professor, Charles L. Huisking Logic and Computable Structure Theory.

Anand Pillay Professor, William J. Hank Family Model theory. Sergei Starchenko Professor Model theory. Nicolas Chavarria Gomez. Gabriel Conant Lumpkins Postdoctoral Fellow Model theory and its interactions with combinatorics, group theory, and number theory. Bruce Driver Affiliate Professor. Xiaobo Liu Affiliate Professor. Charles McCoy Affiliate Professor.Mathematical logic.

These areas share basic results on logic, particularly first-order logic, and definability. In computer science particularly in the ACM Classification mathematical logic encompasses additional topics not detailed in this article; see Logic in computer science for those.

Since its inception, mathematical logic has both contributed to, and has been motivated by, the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and analysis.

Mathematical Logic

In the early 20th century it was shaped by David Hilbert's program to prove the consistency of foundational theories. Work in set theory showed that almost all ordinary mathematics can be formalized in terms of sets, although there are some theorems that cannot be proven in common axiom systems for set theory.

Contemporary work in the foundations of mathematics often focuses on establishing which parts of mathematics can be formalized in particular formal systems as in reverse mathematics rather than trying to find theories in which all of mathematics can be developed.

First-order logic. First-order logic — also known as first-order predicate calculus and predicate logic — is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. Model theory.

In mathematics, model theory is the study of classes of mathematical structures example: groups, fields, graphs, universes of set theory from the perspective of mathematical logic.

The objects of study are models of theories in a formal language. A set of sentences in a formal language is called a theory; a model of a theory is a structure example: an interpretation that satisfies the sentences of that theory. Model theory recognises and is intimately concerned with a duality: it examines semantical elements meaning and truth by means of syntactical elements formulas and proofs of a corresponding language.

Model theory developed rapidly during the s, and a more modern definition is provided by Wilfrid Hodges :. Other nearby areas of mathematics include combinatorics, number theory, arithmetic dynamics, analytic functions, and non-standard analysis.

In a similar way to proof theory, model theory is situated in an area of interdisciplinarity among mathematics, philosophy, and computer science.

The most prominent professional organization in the field of model theory is the Association for Symbolic Logic. Computability theory. Computability theory, also called recursion theory, is a branch of mathematical logic, of computer science, and of the theory of computation that originated in the s with the study of computable functions and Turing degrees. The field has since grown to include the study of generalized computability and definability.

In these areas, recursion theory overlaps with proof theory and effective descriptive set theory. The answers to these questions have led to a rich theory that is still being actively researched. Recursion theorists in mathematical logic often study the theory of relative computability, reducibility notions and degree structures described in this article.

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This contrasts with the theory of subrecursive hierarchies, formal methods and formal languages that is common in the study of computability theory in computer science. There is considerable overlap in knowledge and methods between these two research communities, however, and no firm line can be drawn between them. Proof theory. Proof theory is a major branch[1] of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques.

Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of the logical system. As such, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature.

Some of the major areas of proof theory include structural proof theory, ordinal analysis, provability logic, reverse mathematics, proof mining, automated theorem proving, and proof complexity. Much research also focuses on applications in computer science, linguistics, and philosophy. Type theory. In mathematics, logic, and computer science, a type theory is any of a class of formal systems, some of which can serve as alternatives to set theory as a foundation for all mathematics.

Type theory is closely related to and in some cases overlaps with type systems, which are a programming language feature used to reduce bugs. Type theory was created to avoid paradoxes in a variety of formal logics and rewrite systems. Sequent calculus.This is an illustrated basic course in mathematical logic. We invite everyone who wants to be creative in mathematics and programming.

mathematical logic theory and math

Enrol now and get started! This course offers basic knowledge in mathematical logic. Upon completion of the course, students will have acquired fundamental knowledge that is valuable in itself and will serve as the foundation for other studies.

For example, software engineers strongly rely on logic-mathematical theories in their work. No need to reinvent the wheel. The language of mathematical logic offers a great opportunity to practice this translation between languages and is used as a powerful formalised tool for transmission of information between distant languages. Most of the course content will be understandable for students with only a high school level of education.

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Some minor sections of the course will require knowledge of imperative programming and elements of mathematical analysis.

Goals, objectives, methods. Relation between mathematics and mathematical logic. Examples of logical errors, sophisms and paradoxes. Brief history of mathematical logic, discussing how problems mathematical logic faced and solved in its development, and how mathematical logic integrates further and further into programming.

The language of propositional logic has limited tools, so we talk about more complex languages based on predicate logic. The language of predicate logic offers tools for full and exact description of any formal notions and statements. The axiomatic method makes it possible to solve many logical problems, errors and paradoxes.

It is widely used in today's mathematics and the knowledge of it is vital for anyone using functional and logical programming languages. To learn about the possibilities of the algorithmic approach and the limitations of calculations, one must know the rigorous definition of algorithms and computability. The module offers these definitions and defines algorithmically unsolvable problems. The module introduces the concept of algorithm complexity, which is an important factor when selecting algorithms to solve problems.Mathematical logic investigates the power of mathematical reasoning itself.

The various subfields of this area are connected through their study of foundational notions: sets, proof, computation, and models. The period from the s thru the s saw great progress in logic. MIT was a major center in the field from the s through the s. The exciting and active areas of logic today are set theory, model theory and connections with computer science. Set theory addresses various ways to axiomatize mathematics, with implications for understanding the properties of sets having large infinite cardinalities and connections with the axiomatization of mathematics.

Model theory investigates particular mathematical theories such as complex algebraic geometry, and has been used to settle open questions in these areas. NP are being pursued with techniques from logic. Massachusetts Institute of Technology Department of Mathematics. For website help or updates, please email Accessibility.

Faculty Bjorn Poonen. Massachusetts Institute of Technology Department of Mathematics Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics. It bears close connections to metamathematicsthe foundations of mathematicsand theoretical computer science. Mathematical logic is often divided into the fields of set theorymodel theoryrecursion theoryand proof theory. These areas share basic results on logic, particularly first-order logicand definability.

In computer science particularly in the ACM Classification mathematical logic encompasses additional topics not detailed in this article; see Logic in computer science for those. Since its inception, mathematical logic has both contributed to, and has been motivated by, the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometryarithmeticand analysis. In the early 20th century it was shaped by David Hilbert 's program to prove the consistency of foundational theories.

Work in set theory showed that almost all ordinary mathematics can be formalized in terms of sets, although there are some theorems that cannot be proven in common axiom systems for set theory.

Contemporary work in the foundations of mathematics often focuses on establishing which parts of mathematics can be formalized in particular formal systems as in reverse mathematics rather than trying to find theories in which all of mathematics can be developed. The Handbook of Mathematical Logic [2] in makes a rough division of contemporary mathematical logic into four areas:.

Each area has a distinct focus, although many techniques and results are shared among multiple areas.

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The borderlines amongst these fields, and the lines separating mathematical logic and other fields of mathematics, are not always sharp. The method of forcing is employed in set theory, model theory, and recursion theory, as well as in the study of intuitionistic mathematics.

The mathematical field of category theory uses many formal axiomatic methods, and includes the study of categorical logicbut category theory is not ordinarily considered a subfield of mathematical logic.

Because of its applicability in diverse fields of mathematics, mathematicians including Saunders Mac Lane have proposed category theory as a foundational system for mathematics, independent of set theory. These foundations use toposeswhich resemble generalized models of set theory that may employ classical or nonclassical logic.

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The first half of the 20th century saw an explosion of fundamental results, accompanied by vigorous debate over the foundations of mathematics. Theories of logic were developed in many cultures in history, including ChinaIndiaGreece and the Islamic world. Greek methods, particularly Aristotelian logic or term logic as found in the Organonfound wide application and acceptance in Western science and mathematics for millennia. In 18th-century Europe, attempts to treat the operations of formal logic in a symbolic or algebraic way had been made by philosophical mathematicians including Leibniz and Lambertbut their labors remained isolated and little known.

In the middle of the nineteenth century, George Boole and then Augustus De Morgan presented systematic mathematical treatments of logic.

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Charles Sanders Peirce built upon the work of Boole to develop a logical system for relations and quantifiers, which he published in several papers from to Gottlob Frege presented an independent development of logic with quantifiers in his Begriffsschriftpublished ina work generally considered as marking a turning point in the history of logic. Frege's work remained obscure, however, until Bertrand Russell began to promote it near the turn of the century.

The two-dimensional notation Frege developed was never widely adopted and is unused in contemporary texts. This work summarized and extended the work of Boole, De Morgan, and Peirce, and was a comprehensive reference to symbolic logic as it was understood at the end of the 19th century.

Concerns that mathematics had not been built on a proper foundation led to the development of axiomatic systems for fundamental areas of mathematics such as arithmetic, analysis, and geometry. In logic, the term arithmetic refers to the theory of the natural numbers. Peano was unaware of Frege's work at the time. Around the same time Richard Dedekind showed that the natural numbers are uniquely characterized by their induction properties.

Dedekind proposed a different characterization, which lacked the formal logical character of Peano's axioms. Dedekind's work, however, proved theorems inaccessible in Peano's system, including the uniqueness of the set of natural numbers up to isomorphism and the recursive definitions of addition and multiplication from the successor function and mathematical induction.

In the midth century, flaws in Euclid's axioms for geometry became known Katzp.

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In addition to the independence of the parallel postulateestablished by Nikolai Lobachevsky in Lobachevskymathematicians discovered that certain theorems taken for granted by Euclid were not in fact provable from his axioms.Logic and Mathematics Stephen G.

This article is an overview of logic and the philosophy of mathematics. It is intended for the general reader.

Mathematical Logic, Learning Notes

Contents Logic Aristotelean logic The predicate calculus Foundations of mathematics The geometry of Euclid Formal theories for mathematics Philosophy of mathematics Plato and Aristotle The 20th century The future Bibliography Logic Logic is the science of formal principles of reasoning or correct inference.

Historically, logic originated with the ancient Greek philosopher Aristotle. Logic was further developed and systematized by the Stoics and by the medieval scholastic philosophers. In the late 19th and 20th centuries, logic saw explosive growth, which has continued up to the present. One may ask whether logic is part of philosophy or independent of it. However, Aristotle did go to great pains to formulate the basic concepts of logic terms, premises, syllogisms, etc.

Thus Aristotle seems to have viewed logic not as part of philosophy but rather as a tool or instrument 1 to be used by philosophers and scientists alike. This attitude about logic is in agreement with the modern view, according to which the predicate calculus see 1. Logic is the science of correct reasoning.

What then is reasoning? According to Aristotle [ 13Topics, a25], reasoning is any argument in which certain assumptions or premises are laid down and then something other than these necessarily follows. Thus logic is the science of necessary inference. However, when logic is applied to specific subject matter, it is important to note that not all logical inference constitutes a scientifically valid demonstration.

This is because a piece of formally correct reasoning is not scientifically valid unless it is based on a true and primary starting point.

mathematical logic theory and math

Furthermore, any decisions about what is true and primary do not pertain to logic but rather to the specific subject matter under consideration. In this way we limit the scope of logic, maintaining a sharp distinction between logic and the other sciences.

All reasoning, both scientific and non-scientific, must take place within the logical framework, but it is only a framework, nothing more. This is what is meant by saying that logic is a formal science.

For example, consider the following inference: Some real estate will increase in value. Anything that will increase in value is a good investment. Therefore, some real estate is a good investment. Yet this same inference may not be a demonstration of its conclusion, because one or both of the premises may be faulty.

Thus logic can help us to clarify our reasoning, but it can only go so far.When interest rates are low, the dividend adjustment outweighs the financing cost, so fair value for index futures is typically lower than the index value.

Whenever the index futures price moves away from fair value, it creates a trading opportunity called index arbitrage.

As soon as the index futures price premium, or discount to fair value, covers their transaction costs (clearing, settlement, commissions and expected market impact) plus a small profit margin, the computers jump in, either selling index futures and buying the underlying stocks if futures trade at a premium, or the reverse if futures trade at a discount. Index Futures Trading Hours Index arbitrage keeps the index futures price close to fair value, but only when both index futures and the underlying stocks are trading at the same time.

Liquidity in index futures drops outside stock exchange trading hours because the index arbitrage players can no longer ply their trade. If the futures price gets out of whack, they cannot hedge an index futures purchase or sale through an offsetting sale or purchase of the underlying stocks.

But other market participants are still active. Index Futures Predict the Opening Direction Suppose good news comes out abroad overnight - the ECB cuts interest rates, or China reports stronger than expected growth in GDP. The local equity markets will probably rise, and investors may anticipate a stronger U. If they buy index futures, the price will go up.

And with index arbitrageurs on the sidelines until the U.

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As soon as New York opens, though, the index arbitrageurs will execute whatever trades are needed to bring the index futures price back in line - in this example, by buying the component stocks and selling index futures. Investors cannot just check whether the futures price is above or below its closing value on the previous day, though. The dividend adjustments to index futures fair value change overnight (they are constant during each day), and the indicated market direction depends on the price of index futures relative to fair value regardless of the preceding close.

On a day when several big index constituents go ex-dividend, index futures may trade above the prior close but still imply a lower opening. Trading is typically volatile at the opening, which accounts for a disproportionate amount of total trading volume. If an institutional investor weighs in with a large buy or sell program in multiple stocks, the market impact can overwhelm whatever price movement the index futures indicate.

Institutional traders do watch futures prices, of course, but the bigger the orders they have to execute, the less important the index futures direction signal becomes. Late openings can also disrupt index arbitrage activity. Although the market opens at 9:30am, not every stock starts to trade at once.

The opening price is set through an auction procedure, and if the bids and offers do not overlap, the stock remains closed until matching orders come in. The longer index arbitrageurs stay on the sidelines, the greater the chances that other market activity will negate the index futures direction signal.


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