Cesare Burali-Forti was the first to state a paradox: Very soon thereafter, Bertrand Russell discovered Russell's paradox in , and Jules Richard discovered Richard's paradox. Zermelo b provided the first set of axioms for set theory. These axioms, together with the additional axiom of replacement proposed by Abraham Fraenkel , are now called Zermelo—Fraenkel set theory ZF. Zermelo's axioms incorporated the principle of limitation of size to avoid Russell's paradox. This seminal work developed the theory of functions and cardinality in a completely formal framework of type theory , which Russell and Whitehead developed in an effort to avoid the paradoxes.
Fraenkel proved that the axiom of choice cannot be proved from the axioms of Zermelo's set theory with urelements. Later work by Paul Cohen showed that the addition of urelements is not needed, and the axiom of choice is unprovable in ZF. Cohen's proof developed the method of forcing , which is now an important tool for establishing independence results in set theory.
Skolem realized that this theorem would apply to first-order formalizations of set theory, and that it implies any such formalization has a countable model.
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This counterintuitive fact became known as Skolem's paradox. These results helped establish first-order logic as the dominant logic used by mathematicians. It showed the impossibility of providing a consistency proof of arithmetic within any formal theory of arithmetic. Hilbert, however, did not acknowledge the importance of the incompleteness theorem for some time. This leaves open the possibility of consistency proofs that cannot be formalized within the system they consider. Gentzen proved the consistency of arithmetic using a finitistic system together with a principle of transfinite induction.
Gentzen's result introduced the ideas of cut elimination and proof-theoretic ordinals , which became key tools in proof theory. Alfred Tarski developed the basics of model theory. Beginning in , a group of prominent mathematicians collaborated under the pseudonym Nicolas Bourbaki to publish a series of encyclopedic mathematics texts. These texts, written in an austere and axiomatic style, emphasized rigorous presentation and set-theoretic foundations. Terminology coined by these texts, such as the words bijection , injection , and surjection , and the set-theoretic foundations the texts employed, were widely adopted throughout mathematics.
Kleene introduced the concepts of relative computability, foreshadowed by Turing , and the arithmetical hierarchy. Kleene later generalized recursion theory to higher-order functionals. Kleene and Kreisel studied formal versions of intuitionistic mathematics, particularly in the context of proof theory. At its core, mathematical logic deals with mathematical concepts expressed using formal logical systems. These systems, though they differ in many details, share the common property of considering only expressions in a fixed formal language. The systems of propositional logic and first-order logic are the most widely studied today, because of their applicability to foundations of mathematics and because of their desirable proof-theoretic properties.
First-order logic is a particular formal system of logic. Its syntax involves only finite expressions as well-formed formulas, while its semantics are characterized by the limitation of all quantifiers to a fixed domain of discourse. Early results from formal logic established limitations of first-order logic. This shows that it is impossible for a set of first-order axioms to characterize the natural numbers, the real numbers, or any other infinite structure up to isomorphism.
Grigori Mints (Author of A Short Introduction to Intuitionistic Logic)
As the goal of early foundational studies was to produce axiomatic theories for all parts of mathematics, this limitation was particularly stark. It shows that if a particular sentence is true in every model that satisfies a particular set of axioms, then there must be a finite deduction of the sentence from the axioms. It says that a set of sentences has a model if and only if every finite subset has a model, or in other words that an inconsistent set of formulas must have a finite inconsistent subset.
The completeness and compactness theorems allow for sophisticated analysis of logical consequence in first-order logic and the development of model theory , and they are a key reason for the prominence of first-order logic in mathematics. The first incompleteness theorem states that for any consistent, effectively given defined below logical system that is capable of interpreting arithmetic, there exists a statement that is true in the sense that it holds for the natural numbers but not provable within that logical system and which indeed may fail in some non-standard models of arithmetic which may be consistent with the logical system.
Here a logical system is said to be effectively given if it is possible to decide, given any formula in the language of the system, whether the formula is an axiom, and one which can express the Peano axioms is called "sufficiently strong. The second incompleteness theorem states that no sufficiently strong, consistent, effective axiom system for arithmetic can prove its own consistency, which has been interpreted to show that Hilbert's program cannot be completed.
Many logics besides first-order logic are studied. These include infinitary logics , which allow for formulas to provide an infinite amount of information, and higher-order logics , which include a portion of set theory directly in their semantics. In this logic, quantifiers may only be nested to finite depths, as in first-order logic, but formulas may have finite or countably infinite conjunctions and disjunctions within them. Higher-order logics allow for quantification not only of elements of the domain of discourse, but subsets of the domain of discourse, sets of such subsets, and other objects of higher type.
The semantics are defined so that, rather than having a separate domain for each higher-type quantifier to range over, the quantifiers instead range over all objects of the appropriate type.
The logics studied before the development of first-order logic, for example Frege's logic, had similar set-theoretic aspects. Although higher-order logics are more expressive, allowing complete axiomatizations of structures such as the natural numbers, they do not satisfy analogues of the completeness and compactness theorems from first-order logic, and are thus less amenable to proof-theoretic analysis.
Another type of logics are fixed-point logic s that allow inductive definitions , like one writes for primitive recursive functions. One can formally define an extension of first-order logic — a notion which encompasses all logics in this section because they behave like first-order logic in certain fundamental ways, but does not encompass all logics in general, e.
Modal logics include additional modal operators, such as an operator which states that a particular formula is not only true, but necessarily true. Intuitionistic logic was developed by Heyting to study Brouwer's program of intuitionism, in which Brouwer himself avoided formalization. Intuitionistic logic specifically does not include the law of the excluded middle , which states that each sentence is either true or its negation is true. Kleene's work with the proof theory of intuitionistic logic showed that constructive information can be recovered from intuitionistic proofs.
For example, any provably total function in intuitionistic arithmetic is computable ; this is not true in classical theories of arithmetic such as Peano arithmetic. Algebraic logic uses the methods of abstract algebra to study the semantics of formal logics. A fundamental example is the use of Boolean algebras to represent truth values in classical propositional logic, and the use of Heyting algebras to represent truth values in intuitionistic propositional logic.
Stronger logics, such as first-order logic and higher-order logic, are studied using more complicated algebraic structures such as cylindric algebras. Set theory is the study of sets , which are abstract collections of objects. Many of the basic notions, such as ordinal and cardinal numbers, were developed informally by Cantor before formal axiomatizations of set theory were developed.
The first such axiomatization , due to Zermelo b , was extended slightly to become Zermelo—Fraenkel set theory ZF , which is now the most widely used foundational theory for mathematics. New Foundations takes a different approach; it allows objects such as the set of all sets at the cost of restrictions on its set-existence axioms.
The system of Kripke—Platek set theory is closely related to generalized recursion theory. Two famous statements in set theory are the axiom of choice and the continuum hypothesis. The axiom of choice, first stated by Zermelo , was proved independent of ZF by Fraenkel , but has come to be widely accepted by mathematicians. It states that given a collection of nonempty sets there is a single set C that contains exactly one element from each set in the collection.
The set C is said to "choose" one element from each set in the collection. While the ability to make such a choice is considered obvious by some, since each set in the collection is nonempty, the lack of a general, concrete rule by which the choice can be made renders the axiom nonconstructive. Stefan Banach and Alfred Tarski showed that the axiom of choice can be used to decompose a solid ball into a finite number of pieces which can then be rearranged, with no scaling, to make two solid balls of the original size.
This theorem, known as the Banach—Tarski paradox , is one of many counterintuitive results of the axiom of choice.
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The continuum hypothesis, first proposed as a conjecture by Cantor, was listed by David Hilbert as one of his 23 problems in In , Paul Cohen showed that the continuum hypothesis cannot be proven from the axioms of Zermelo—Fraenkel set theory Cohen This independence result did not completely settle Hilbert's question, however, as it is possible that new axioms for set theory could resolve the hypothesis. Recent work along these lines has been conducted by W. Hugh Woodin , although its importance is not yet clear Woodin Contemporary research in set theory includes the study of large cardinals and determinacy.
Large cardinals are cardinal numbers with particular properties so strong that the existence of such cardinals cannot be proved in ZFC. The existence of the smallest large cardinal typically studied, an inaccessible cardinal , already implies the consistency of ZFC. Despite the fact that large cardinals have extremely high cardinality , their existence has many ramifications for the structure of the real line.
Determinacy refers to the possible existence of winning strategies for certain two-player games the games are said to be determined. The existence of these strategies implies structural properties of the real line and other Polish spaces. Model theory studies the models of various formal theories. Here a theory is a set of formulas in a particular formal logic and signature , while a model is a structure that gives a concrete interpretation of the theory.
Model theory is closely related to universal algebra and algebraic geometry , although the methods of model theory focus more on logical considerations than those fields. The set of all models of a particular theory is called an elementary class ; classical model theory seeks to determine the properties of models in a particular elementary class, or determine whether certain classes of structures form elementary classes. The method of quantifier elimination can be used to show that definable sets in particular theories cannot be too complicated.
Tarski established quantifier elimination for real-closed fields , a result which also shows the theory of the field of real numbers is decidable. He also noted that his methods were equally applicable to algebraically closed fields of arbitrary characteristic. A modern subfield developing from this is concerned with o-minimal structures. Morley's categoricity theorem , proved by Michael D. Morley , states that if a first-order theory in a countable language is categorical in some uncountable cardinality, i. A trivial consequence of the continuum hypothesis is that a complete theory with less than continuum many nonisomorphic countable models can have only countably many.
Vaught's conjecture , named after Robert Lawson Vaught , says that this is true even independently of the continuum hypothesis.
Many special cases of this conjecture have been established. Recursion theory , also called computability theory , studies the properties of computable functions and the Turing degrees , which divide the uncomputable functions into sets that have the same level of uncomputability. Recursion theory also includes the study of generalized computability and definability. Classical recursion theory focuses on the computability of functions from the natural numbers to the natural numbers.
More advanced results concern the structure of the Turing degrees and the lattice of recursively enumerable sets. Generalized recursion theory extends the ideas of recursion theory to computations that are no longer necessarily finite. Contemporary research in recursion theory includes the study of applications such as algorithmic randomness , computable model theory , and reverse mathematics , as well as new results in pure recursion theory.
An important subfield of recursion theory studies algorithmic unsolvability; a decision problem or function problem is algorithmically unsolvable if there is no possible computable algorithm that returns the correct answer for all legal inputs to the problem. The first results about unsolvability, obtained independently by Church and Turing in , showed that the Entscheidungsproblem is algorithmically unsolvable. Turing proved this by establishing the unsolvability of the halting problem , a result with far-ranging implications in both recursion theory and computer science.
There are many known examples of undecidable problems from ordinary mathematics. The word problem for groups was proved algorithmically unsolvable by Pyotr Novikov in and independently by W. Hilbert's tenth problem asked for an algorithm to determine whether a multivariate polynomial equation with integer coefficients has a solution in the integers. The algorithmic unsolvability of the problem was proved by Yuri Matiyasevich in Davis Proof theory is the study of formal proofs in various logical deduction systems.
These proofs are represented as formal mathematical objects, facilitating their analysis by mathematical techniques. Several deduction systems are commonly considered, including Hilbert-style deduction systems , systems of natural deduction , and the sequent calculus developed by Gentzen. The study of constructive mathematics , in the context of mathematical logic, includes the study of systems in non-classical logic such as intuitionistic logic, as well as the study of predicative systems.
An early proponent of predicativism was Hermann Weyl , who showed it is possible to develop a large part of real analysis using only predicative methods Weyl Because proofs are entirely finitary, whereas truth in a structure is not, it is common for work in constructive mathematics to emphasize provability. The relationship between provability in classical or nonconstructive systems and provability in intuitionistic or constructive, respectively systems is of particular interest.
Recent developments in proof theory include the study of proof mining by Ulrich Kohlenbach and the study of proof-theoretic ordinals by Michael Rathjen. Skolem , but also to physics R. Fevrier , to biology J. Tarski , to psychology F. Hempel , to law and morals K. Oppenheim , to economics J. Morgenstern , to practical questions E.
Stamm , and even to metaphysics J. Its applications to the history of logic have proven extremely fruitful J. The study of computability theory in computer science is closely related to the study of computability in mathematical logic. There is a difference of emphasis, however.
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Computer scientists often focus on concrete programming languages and feasible computability , while researchers in mathematical logic often focus on computability as a theoretical concept and on noncomputability. The theory of semantics of programming languages is related to model theory , as is program verification in particular, model checking. The Curry—Howard isomorphism between proofs and programs relates to proof theory , especially intuitionistic logic.
Formal calculi such as the lambda calculus and combinatory logic are now studied as idealized programming languages. Computer science also contributes to mathematics by developing techniques for the automatic checking or even finding of proofs, such as automated theorem proving and logic programming. Descriptive complexity theory relates logics to computational complexity. Guenthner, Kluwer 2nd edition , but little changed from 1st edition. Read first twenty short sections. That should smooth the path to reading the main text: Beginning set theory Start with Herbert B.
Extras On second-order logic: Stewart Shapiro, Foundations without Foundationalism: Chs 1—6 of this very scarily titled book give a particularly attractive stand-alone introduction to the semantics and a proof-system for intuitionist logic. December 1, at 3: Hello, Thank you very much for this list and the guide — it has been very helpful. December 4, at 5: