Based on predicate calculus various logico-mathematical theories have been constructed see Logico-mathematical calculus , representing the formalization of interesting mathematical theories: arithmetic, analysis, set theory, group theory, etc. Side-by-side with elementary theories cf. Elementary theory , higher-order theories were also considered.
In these one also admits quantifiers over predicates, predicates over predicates, etc.
The traditional questions studied in these formal logical systems were the investigation of the structure of the deductions in the system, derivability of various formulas, and questions of consistency and completeness. According to this theorem, if a formal system containing arithmetic is consistent, then the assertion of its consistency expressed in the system cannot be proved by formalization within it.
This means that with questions on foundations of mathematics the matter is not as simple as first desired or believed by Hilbert. Similar proofs of the consistency of arithmetic were obtained by G. Gentzen and P. Novikov cf.
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As a result of the analysis of Cantor's set theory and the related paradoxes, various systems of axiomatic set theory were constructed with various restrictions on the formation of sets, to exclude the known inconsistencies. Within these axiomatic systems suitably extensive parts of mathematics could be developed.
The consistency question for fairly rich axiomatic systems of set theory remains open. Cohen on the independence of these axioms from the Zermelo—Fraenkel axioms. One should note that these two axiom systems, and , are equiconsistent. For the construction of models of the set theory in which the negation of the continuum hypothesis or the axiom of choice holds, Cohen introduced the so-called forcing method , which subsequently became a fundamental method for constructing models of set theory with various properties cf.
One of the most remarkable achievements of mathematical logic was the development of the notion of a general recursive function and the formulation of the Church thesis , asserting that the notion of a general recursive function makes precise the intuitive notion of an algorithm. Of the equivalent elaborations of the notion of an algorithm the most widely used are the idea of a Turing machine and a Markov normal algorithm.
In essence all mathematics is connected with some algorithm or other. But the possibility of identifying undecidable algorithmic problems cf. Algorithmic problem in mathematics appeared only with the refinement of the notion of an algorithm. Undecidable algorithmic problems were discovered in many areas of mathematics algebra, number theory, topology, probability theory, etc. Moreover, it turned out that they could be connected with very widespread and fundamental ideas in mathematics. Research into algorithmic problems in various areas of mathematics, as a rule, is accompanied by the penetration of the ideas and methods of mathematical logic into the area, which then leads to the solution of other problems no longer of an algorithmic nature.
The development of a precise notion of an algorithm made it possible to refine the notion of effectiveness and to develop on that basis a refinement of the constructive directions in mathematics see Constructive mathematics , which embodies certain features of intuitionism, but is essentially different from the latter.
The foundations of constructive analysis, constructive topology, constructive probability theory, etc. In the theory of algorithms itself it is possible to pick out research in the domain of recursive arithmetic, containing various classifications of recursive and recursively-enumerable sets, degrees of undecidability of recursively-enumerable sets, research into the complexity of description of algorithms and the complexity of algorithmic calculations in time and extent, see Algorithm, computational complexity of an ; Algorithm, complexity of description of an.
An extensively developing area in the theory of algorithms is the theory of enumeration. As noted above, the axiomatic method exerted a major influence on the development of many areas of mathematics. Of special significance was the penetration of this method into algebra. Thus at the junction of mathematical logic and algebra, the general theory of algebraic systems cf. Algebraic system , or model theory , arose. The foundations of this theory were laid by A. Mal'tsev, A. Tarski and their followers.
Here one should note research into the elementary theory of classes of models, in particular, decidability questions in these theories, axiomatizability of classes of models, isomorphism of models, and questions of categoricity and completeness of classes of models.
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An important place in model theory is occupied by studies on non-standard models of arithmetic and analysis. Even at the dawn of differential calculus, in the work of Leibniz and I. Newton, infinitely-small and infinitely-large quantities were regarded as numbers. Later the notion of a variable quantity appeared, and mathematicians turned away from the use of infinitely-small numbers, the modulus of which was different from zero and less than any positive real number, since their use required the loss of the Archimedean axiom.
Only after three centuries, as a result of the development of the methods of mathematical logic, was it possible to establish that non-standard analysis with infinitely-small and infinitely-large numbers is consistent relative to the usual standard analysis of real numbers cf. Non-standard analysis.
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One could not conclude without mentioning the influence of the axiomatic method on intuitionistic mathematics. Thus, as long ago as , A. Heyting introduced formal systems of intuitionistic logic of propositions and predicates constructive propositional and predicate calculi.
Later, formal systems of intuitionistic analysis were introduced see, for example, . Much of the research in intuitionistic logic and mathematics is concerned with formal systems. Special study was made of so-called intermediate logics cf. Intermediate logic ; also called super-intuitionistic logics , that is, logics lying between classical and intuitionistic logics. The notion of Kleene realizability of formulas is an attempt to interpret the idea of intuitionistic truth from the point of view of classical mathematics.
However, it turned out that not every realizable formula of propositional calculus was derivable in intuitionistic constructive propositional calculus. Modal logic has also been formalized. However, in spite of the large number of papers on formal systems of modal logic and its semantics Kripke models , this can still be said to be an accumulation of uncoordinated facts.
Mathematical logic has a more applied value too; with each year there is a deeper penetration of the ideas and methods of mathematical logic into cybernetics, computational mathematics and structural linguistics. A basic reference for mathematical logic as a whole is [a8]. The pioneering seminal work in non-standard analysis is due to A. Robinson, [a10] , cf. Log in. Namespaces Page Discussion. Views View View source History. Jump to: navigation , search. How to Cite This Entry: Mathematical logic. Adyan originator , Encyclopedia of Mathematics. This page was last modified on 7 February , at Hilbert, P.
Bernays, "Grundlagen der Mathematik" , 1—2 , Springer — Ershov, E. Palyutin, "Mathematical logic" , Moscow In Russian. Novikov, "Constructive mathematical logic from a classical point of view" , Moscow In Russian. Kleene, R. Vesley, "The foundations of intuitionistic mathematics: especially in relation to recursive functions" , North-Holland Mathematical logic. Number theory.
Probability theory , Moscow In Russian. Manin, "A course in mathematical logic" , Springer Translated from Russian. Troelstra, D. She can never complete an infinite construction, even though she can complete arbitrarily large finite initial parts of it. This entails that intuitionism resolutely rejects the existence of the actual or completed infinite; only potentially infinite collections are given in the activity of construction.
A basic example is the successive construction in time of the individual natural numbers. From these general considerations about the nature of mathematics, based on the condition of the human mind Moore , intuitionists infer to a revisionist stance in logic and mathematics. They find non-constructive existence proofs unacceptable.
Non-constructive existence proofs are proofs that purport to demonstrate the existence of a mathematical entity having a certain property without even implicitly containing a method for generating an example of such an entity. The characteristic feature of non-constructive existence proofs is that they make essential use of the principle of excluded third.
In classical logic, these principles are valid. The logic of intuitionistic mathematics is obtained by removing the principle of excluded third and its equivalents from classical logic. This of course leads to a revision of mathematical knowledge. For instance, the classical theory of elementary arithmetic, Peano Arithmetic , can no longer be accepted.
Instead, an intuitionistic theory of arithmetic called Heyting Arithmetic is proposed which does not contain the principle of excluded third.
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Although intuitionistic elementary arithmetic is weaker than classical elementary arithmetic, the difference is not all that great. There exists a simple syntactical translation which translates all classical theorems of arithmetic into theorems which are intuitionistically provable.
In the first decades of the twentieth century, parts of the mathematical community were sympathetic to the intuitionistic critique of classical mathematics and to the alternative that it proposed. This situation changed when it became clear that in higher mathematics, the intuitionistic alternative differs rather drastically from the classical theory. For instance, intuitionistic mathematical analysis is a fairly complicated theory, and it is very different from classical mathematical analysis. This dampened the enthusiasm of the mathematical community for the intuitionistic project.
David Hilbert agreed with the intuitionists that there is a sense in which the natural numbers are basic in mathematics. But unlike the intuitionists, Hilbert did not take the natural numbers to be mental constructions.
Instead, he argued that the natural numbers can be taken to be symbols. Symbols are strictly speaking abstract objects. Nonetheless, it is essential to symbols that they can be embodied by concrete objects, so we may call them quasi-concrete objects Parsons , chapter 1.
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Perhaps physical entities could play the role of the natural numbers. For instance, we may take a concrete ink trace of the form to be the number 0, a concretely realized ink trace to be the number 1, and so on. Hilbert thought it doubtful at best that higher mathematics could be directly interpreted in a similarly straightforward and perhaps even concrete manner.