Search results

[[Category:Logic and foundations]]

...rom the classical one, e.g. [[Modal logic|modal logic]], or intuitionistic logic if the meta-theory is built within the frame of [[Intuitionism|intuitionism ....g. by means of set theory. In this case, classical logic acts as the meta-logic.

...lign="top"> R. Fraïssé, ''Theory of Relations'', Studies in Logic and the Foundations of Mathematics, Elsevier (2011) {{ISBN|0080960413}}</TD></TR> [[Category:Logic and foundations]]

...ishment of derivability. One of the most important results in mathematical logic is the theorem stating that the cut rule is sound (see [[Gentzen formal sys ...matical principles such as consistency, $\omega$-consistency, completeness and reflection principles.

...uccessful. As K. Gödel showed [[#References|[3]]], no formalized system of logic can be an adequate basis for mathematics (cf. [[Gödel incompleteness theor ...align="top">[5]</TD> <TD valign="top"> A.A. Fraenkel, Y. Bar-Hillel, "Foundations of set theory" , North-Holland (1958)</TD></TR></table>

...ssing an object of the theory and depending on parameters $x_1,\dots,x_n$ (and possibly also on other parameters), then ...op">[2]</TD> <TD valign="top"> H.B. Curry, "Foundations of mathematical logic" , McGraw-Hill (1963)</TD></TR></table>

''in mathematical logic'' ...the consistency of the axiom of choice can be regarded as the syntactical and finitistically-provable assertion that if the Zermelo–Fraenkel formal the

A program for the foundations of mathematics initiated by D. Hilbert. The aim of this program was to prov ...ssertions of the theory are changed to finite sequences of definite signs, and the logical methods of inference — to formal rules for generating new for

...place predicate symbol, i.e. a subset of the Cartesian power $M^n$ of $M$, and an $n$-place function $M^n\to M$ to an $n$-place function symbol. The set $ <TR><TD valign="top">[1]</TD> <TD valign="top"> S.C. Kleene, "Mathematical logic" , Wiley (1967)</TD></TR>

...y representatives of the intuitionistic and constructive directions in the foundations of mathematics (cf. [[Intuitionism|Intuitionism]]; [[Constructive mathemati

* R. Fraïssé, ''Theory of Relations'', Studies in Logic and the Foundations of Mathematics, Elsevier (2011) p.44. {{ISBN|0080960413}}

...et of truth values (cf. [[Truth value|Truth value]]) $\{\text T,\text F\}$ and taking values in this set. With every [[Logical operation|logical operation ...in the set $\{\text T,\text F\}$. Such functions are used in mathematical logic as an analogue of the concept of a [[Predicate|predicate]].

[[Category:Logic and foundations]]

...hus, if $f$ is a variable with as values integrable real-valued functions, and $x$, $a$, $b$ are variables whose values are real numbers, then the express ...ules, independent of the semantics of the language, for constructing terms and distinguishing free variables in them. In many-sorted languages there are a

* Peter T. Johnstone; ''Sketches of an elephant'', ser. Oxford Logic Guides (2002) Oxford University Press. {{ISBN|0198534256}} pp. 491-492 {{ ...en; ''Categorical Foundations: Special Topics in Order, Topology, Algebra, and Sheaf Theory'', (2004) Cambridge University Press {{ISBN|0-521-83414

A methodological point of view, due to D. Hilbert, as to what objects and methods of argument in mathematics should be counted as absolutely reliable ...xample the written form of natural numbers, formulas in symbolic language, and finite collections of them;

[[Category:Logic and foundations]]

...he equation $x+3=2$", then one is using $x$ as the name of the letter $x$, and is using "$x+3=2$" as the name of the expression $x+3=2$. If one says "12 ...the meaning of the term with the term itself: it is both the object itself and the name by which it is denoted.

Please remove this comment and the {{TEX|auto}} line below, and $ {} _ {-} ^ {*} $,

...general scheme: After an alphabet $A$ not containing the letter $\bullet$ and a natural number $l$ have been selected, a derivation rule with $l$-premise .../TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J.B. Rosser, "Logic for mathematicians" , McGraw-Hill (1953)</TD></TR></table>

...gative integers, if the following two conditions hold: 1) $P(0)$ is valid; and 2) for any $x$, the truth of $P(x)$ implies that of $P(x+1)$. The induction ...m, $P(x)$ is called the induction predicate, or the induction proposition, and $x$ is called the induction variable, induction parameter or the variable w

[[Category:Logic and foundations]]

...with the property $\sigma$. A local theorem holds for a property $\sigma$ (and a corresponding class of models) if every model locally having property $\s ...vestigated separately for each $\sigma$, has thus been reduced to a common and quite "grammatical" question: Is it possible to describe $\sigma$ by univ

...tems and Post canonical systems. Post production systems were used by Post and A.A. Markov (1947) to construct the first examples of an [[Associative calc ...p">[a1]</TD> <TD valign="top"> H.B. Curry, "Foundations of mathematical logic" , McGraw-Hill (1963)</TD></TR></table>

...as to employ only model-theoretic ideas (cf. [[Model (in logic)|Model (in logic)]]; [[Model theory|Model theory]]). ...it contains. This interpretation is called a Henkin model for the maximal and term-complete set.

A certain set, fixed within the framework of a given fundamental theory and containing as members all objects considered in this theory. For example, i The set of all sets forms an object of study in the theory of sets and classes. In this theory one considers along with sets, (proper) classes —

...the reasoning usual in mathematics and logic. Criteria for the naturalness and quality of a deduction cannot be specified with complete precision, but the ...nal advantages — the [[Subformulation property|subformulation property]] — and so form a basis for further advances in the problem of constructing natural

...ained by primitive recursion from an $n$-place function $g(x_1,\dots,x_n)$ and an $(n+2)$-place function $h(x_1,\dots,x_n,y,z)$ if for all natural number and

...each formula of the infinite sequence $A(\bar0),\ldots,A(\bar n),\ldots,$ and the formula $\neg\forall x\ A(x)$ are provable, where $\bar 0$ is a constan [[Category:Logic and foundations]]

...s" (cf. [[#References|[4]]]). This description of a constructive process, and thus of a constructive object, cannot, of course, have the pretension of be ...[[Algorithm|Algorithm]]; [[Calculus|Calculus]]), finitely-presented groups and other analogous mathematical objects may be defined as constructive objects

Please remove this comment and the {{TEX|auto}} line below, ...re originally conceived as formalizations of parts of intuitionistic logic and mathematics.

...a R b$ for all $a,b \in A$, the equality relation, $a R b$ for $a=b \in A$ and the empty (nil) relation are transitive. ...lign="top"> R. Fraïssé, ''Theory of Relations'', Studies in Logic and the Foundations of Mathematics, Elsevier (2011) {{ISBN|0080960413}}</TD></TR>

...lus]]) which has remarkable properties otherwise not shared by first-order logic. It consists of Horn clauses or [[Quasi-identity|quasi-identities]], formul where the $R_i$ and $R$ are atomic formulas (and $R$ could also be $x_p = x_q$ or $\texttt{false}$. (Strict) propositional H

...ion $A|A$; the [[Disjunction|disjunction]] $A\lor B$ of two assertions $A$ and $B$ is expressed as: ...nd the [[Implication|implication]] $A\to B$ are expressed as $(A|B)|(A|B)$ and $A|(B|B)$, respectively. Sheffer's stroke was first considered by H. Sheffe

Please remove this comment and the {{TEX|auto}} line below, ...alization method, which is one of the most central methods in mathematical logic.

Please remove this comment and the {{TEX|auto}} line below, A spread (cf. [[Spread (in intuitionistic logic)|Spread (in intuitionistic logic)]]) $ \pi $

[[Category:Logic and foundations]]

Please remove this comment and the {{TEX|auto}} line below, if and only if the formula

...operation of concatenation of words, etc.) are transformed into relations and operations having a simple algorithmic nature (e.g. are primitive recursive ...ration of words arising from arithmetization is called a Gödel enumeration and the number corresponding to a word is called its Gödel number.

...descriptive definitions (definitions modelling the extensions of concepts and notations in constructing a mathematical theory). The available apparatus o [[Category:Logic and foundations]]

...ico-mathematical language be regarded as ranging over the natural numbers, and let $\phi(x)$ be a formula of this language. If each formula in the infinit ...lign="top"> J. Barwise, "Infinitary logics" E. Agazzi (ed.) , ''Modern logic—a survey'' , Reidel (1981) pp. 93–112</TD></TR></table>

...A truth table has the form of the table below, in which T denotes "true" and F denotes "false". In it, $A_1,\dots,A_n$ are propositional variables, $\d [[Propositional formula|propositional formula]], and the truth value of $\fA(A_1,\dots,A_n)$ is determined by the truth values o

...ivity arose in connection with the discovery of set-theoretical paradoxes, and the term itself is due to H. Poincaré (1906), who was first to raise objec ...for the property $\phi(M)$ one takes the formula $M=A\lor M=B$, where $A$ and $B$ are certain fixed sets, then \eqref{*} is equivalent to the predicative

...in their content. 1) Let $\Psi$ be a [[Normal algorithm|normal algorithm]] and let $P$ be a word over its alphabet. If the proposition that $\Psi$ is not ...the partial recursive function with Gödel number $z$ is defined at $x$ if and only if $\exists yT_1(z,x,y)$ (cf. [[#References|[3]]]). 3) If a recursivel

...pical abstractions in mathematics are "pure" abstractions, idealizations and their various multi-layered superpositions (see [[#References|[5]]]). ...he interrelationships between them, with the exclusion of other properties and relationships which are considered as irrelevant. The result of such an act

Please remove this comment and the {{TEX|auto}} line below, ...based on set-theoretic foundations, such infinitesimal reals do not exist, and the synthetic methods cannot be made mathematically rigorous in a direct wa

...efinable if there are an $L$-formula $\phi(x_1,\ldots,x_n,y_1,\ldots,y_k)$ and $b_1,\ldots,b_k \in M$ such that ...den Dries aims to explain the other finiteness phenomena in real algebraic and real analytic geometry as consequences of $o$-minimality (cf. [[#References

$#C+1 = 13 : ~/encyclopedia/old_files/data/M062/M.0602660 Mathematical logic, Please remove this comment and the {{TEX|auto}} line below,

Please remove this comment and the {{TEX|auto}} line below, ...of the forms of non-classical [[Interpretation|interpretation]] of logical and logico-mathematical languages. Various interpretations of realizability typ

Please remove this comment and the {{TEX|auto}} line below, ...y reflect to some extent the usual methods of handling logical connectives and assumptions.

Please remove this comment and the {{TEX|auto}} line below, and $ t $

$#C+1 = 118 : ~/encyclopedia/old_files/data/L110/L.1100170 Logic programming, Please remove this comment and the {{TEX|auto}} line below,

...9 : ~/encyclopedia/old_files/data/S086/S.0806890 Spread (in intuitionistic logic) Please remove this comment and the {{TEX|auto}} line below,

and Definition 3 in Section 5.3 of Chapter IX in {{Cite|Bo}}). An open set of a ...on|zero-dimensional]], [[Totally-disconnected space|totally disconnected]] and with no isolated points. Observe that the Baire space is the [[Topological

...mplete first-order theory (cf. also [[Logical calculus|Logical calculus]]) and let $n(T)$ be the number of countable models of $T$, up to isomorphism (cf. Variants of this conjecture have been formulated for incomplete theories, and for sentences in $L_{\omega_{1}\omega}$. In 1970, M. Morley [[#References|[

...hout obtaining a contradiction. In other words, an axiom is independent if and only if there is an [[Interpretation|interpretation]] of the theory in whic In relation to formal systems, and calculi (cf. [[Calculus|Calculus]]) in general, it is meaningful to speak o

Please remove this comment and the {{TEX|auto}} line below, ...e lattice]] with a largest element "1" , the unit of the Boolean algebra, and a smallest element "0" , the zero of the Boolean algebra, that contains to

Please remove this comment and the {{TEX|auto}} line below, ...s and with the applications of this concept in various branches of science and technology.

...s of $A$ into $B$ is also closed with respect to the triple multiplication and is a generalized heap [[#References|[4]]], i.e. a semi-heap with the identi Generalized heaps find application in the foundations of [[Differential geometry|differential geometry]] in the study of coordina

...nt to the property that "there is no formula $\phi$ such that both $\phi$ and $\neg\phi$ are provable". A class of formulas of a given formal system is c Any consistency proof uses tools of one mathematical theory or other and so only reduces the question of the consistency of one theory to that of an

...h a circle $K'$ touching $K$ on the outside at $z=1$, with centre $\alpha$ and radius $\rho$. Under the mapping $w=\lambda(z)$, the image of $K'$ is a clo ...e exterior of $L'$. To obtain a Zhukovskii profile of a more general shape and disposition, the generalized Zhukovskii function is applied (see [[#Referen

...d the other. In fact, it may be claimed that, at a very basic level, logic and category theory are the same. ...s. Logic has also been used for presenting the foundations of mathematics, and here too category theory has something to say.

...any non-empty domain of objects with arbitrary predicates (i.e. properties and relations) given on these objects. ...d. Further, if $P$ is an $n$-place predicate symbol of $\Omega$, $n\geq0$, and if $t_1,\dots,t_n$ are terms, then $P(t_1,\dots,t_n)$ is, by definition, an

...and the set $A\times A$ itself are called, respectively, the nil relation and the universal relation in the set $A$. The diagonal of the set $A\times A$, ...nion]] $R_1\cup R_2$, [[Intersection of sets|intersection]] $R_1\cap R_2$, and negation or [[Relative complement|complementation]] $R'=(A\times A)\setminu

...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...ice|complete lattice]] with bottom element $\perp$ and top element $\top$, and the [[Tensor product|tensor product]] $\otimes : L \times L \rightarrow L$

...ng the conditions that each element is right-sided ($a \otimes 1_Q \le a$) and idempotent ($a \otimes a = a$)) led certain authors to restrict the term " ...by closed linear sum, product given by closed linear product of subspaces, and involution by involution within the $C^*$-algebra. The right-sided elements

Please remove this comment and the {{TEX|auto}} line below, ...ical concepts, as well as certain ways of reasoning customary in classical logic.

...ct $\mathcal P(1)$ serves as the natural domain of values of propositional logic in the topos $\mathcal C$. .../TD> <TD valign="top"> R.I. Goldblatt, "Topoi: the categorical analysis of logic" , North-Holland (1979)</TD></TR></table>

...ough revision of the theory as a whole, attracts attention to new effects, and ultimately stimulates further studies. This feature of antinomies attracted ...ite period of time; therefore, Achilles will never arrive at the point $B$ and will never overtake the tortoise.

...space $X$, the lattice $\mathcal{O}(X)$ of open subsets of $X$ is complete and satisfies the infinite distributive law ...erminology: they redefine "space" to mean what is called a locale above, and use "locale" to mean a frame in the terminology above. The sense in which

...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...regarded as the basic framework onto which further axioms can be adjoined and investigated. A modern presentation of ZFC follows.

$#C+1 = 83 : ~/encyclopedia/old_files/data/F040/F.0400960 Foundations of geometry Please remove this comment and the {{TEX|auto}} line below,

Please remove this comment and the {{TEX|auto}} line below, ...l set|Borel set]]) by repeated application of the operations of projection and taking complements. Projective sets are classified in classes forming the p

...hereby, to [[Algebraic logic#Abstract algebraic logic|''Abstract algebraic logic'']]. ...L}$ of symbols $p_0, p_1,... p_{\gamma}...$ called propositional variables and a finite set $\Pi$ of symbols $F_0, F_1,..., F_n$ called logical connective

...andard real-valued function $f$ is continuous at a standard point $x_0$ if and only if $f(x)$ is infinitely close to $f(x_0)$ for all (non-standard) point ...to the infinite. Non-standard analysis places the ideas of G.&nbsp;Leibniz and his followers, about the existence of infinitely small non-zero quantities,

Please remove this comment and the {{TEX|auto}} line below, A theorem from the theories of sense and synonymy. This theory provides a solution for the treatment of hyperintensi

...with computers and computer science. The enormous development of computers and the resulting profound changes in scientific methodology have opened new ho ...se observations refers to the growing importance of discrete mathematics — and only the very beginning of the influence of discrete mathematics is witness

Please remove this comment and the {{TEX|auto}} line below, ...A. Markov [[#References|[9]]], [[#References|[10]]], [[#References|[11]]], and others. In particular, Markov rendered the concept of an algorithm more pre

Please remove this comment and the {{TEX|auto}} line below, ...such as the theory of elementary functions, the theory of series, Riemann and Lebesgue integration, the theory of functions of a complex variable, the th

"a history of Hilbert’s program for the foundations of mathematics, initiated by his Problems Address given in Paris, 1900" -- ...required the demonstration of the ''independence'', the ''completeness'', and the ''consistency'' of its axioms. More specifically with respect to geomet

Please remove this comment and the {{TEX|auto}} line below, ...set theory and in mathematical logic — in the study of "infinite sets" , and in other branches of mathematics.

...tical logic studying mathematical models (cf. [[Model (in logic)|Model (in logic)]]). The origins of model theory go back to the 1920's and 1930's, when the following two fundamental theorems were proved.

Please remove this comment and the {{TEX|auto}} line below, ...the influence of the ideas of Hilbert). One of the most generally followed and definitive approaches to the development of constructive mathematics on thi

Please remove this comment and the {{TEX|auto}} line below, ...f theory|proof theory]] as one of the main chapters of modern mathematical logic.

...uthor(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Alike Lic interest in logical argument, probability and statistics, and his fertility

...uthor(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Alike Lic Von Mises was principally known for his work on the foundations of

If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category. ...(or generation), takes some grammatical or meaning representation as input and produces (written or spoken) natural language surface expressions as output

Please remove this comment and the {{TEX|auto}} line below, ..., was the first to use the letters of the alphabet to denote the constants and the variables in a problem. Most of the present-day symbols of algebra were

...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...said to learn from experience $E$ with respect to some class of tasks $T$ and performance measure $\pi$, if its performance at tasks $t \in T$, as measur

Please remove this comment and the {{TEX|auto}} line below, ...s with fragments of the informal theory of sets by methods of mathematical logic. Usually, to this end, these fragments of set theory are formulated as a fo

...technology and natural science, the accumulation of quantitative relations and spatial forms studied in mathematics is continuously expanding; so this gen ...with very simple aspects of economic life. The first problems of mechanics and physics were already in this collection of fundamental mathematical ideas.

...ierarchy of types (not necessarily linear, and not necessarily countable), and the presence of type-theoretic comprehension axioms (or their equivalents). ...ots \sigma _{n} ) $ . Thus, in a type-theoretic system a property $ y $ and objects $ x _{1} \dots x _{n} $ which satisfy this property belong to di

...f strategies (cf. [[Strategy (in game theory)|Strategy (in game theory)]]) and the same pay-off function. (Important texts in this area are [[#References| ...//www.encyclopediaofmath.org/legacyimages/e/e110/e110130/e1101301.png" />, and a pay-off function <img align="absmiddle" border="0" src="https://www.encyc

Please remove this comment and the {{TEX|auto}} line below, ...ts without recourse to probability theory, so that the concepts of entropy and quantity of information might be applicable to individual objects. It also

...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category. ...connection. It is very strong in classical logic and this gives classical logic its distinctly algebraic character.

Please remove this comment and the {{TEX|auto}} line below, ...) under the condition that these collections contain the identity mappings and are closed with respect to successive composition (or product) of mappings.

...uthor(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Alike Lic to mean values and variability and association between statistical variates,

...od for studying many problems in contemporary algebra, geometry, topology, and analysis. ...orphism. The space $\cF$ together with the stalk-wise algebraic operations and the projection $p$ is called the sheaf of Abelian groups (rings, etc.) over

Please remove this comment and the {{TEX|auto}} line below, ...son with classical (analytical) mathematical methods, considerably broader and encompasses practically all sciences. In the simplest case, a cybernetic sy

...iques. These approaches borrow mathematical techniques from lattice theory and logical systems. These have mainly been developed for imperative, functional and object-oriented programs. For a given [[Program|program]] $ p $

...uthor(s), the article has been donated to ''Encyclopedia of Mathematics'', and its further issues are under ''Creative Commons Attribution Share-Alike Lic the Sorbonne by Émile Picard and Élie Cartan,

...ssociated with it and concerning the notion of 'quantum computation' exist and are still unsolved. Other properties of the quantum Turing machine like the ...physical system. A calculation is the execution of a physical experiment, and its result is provided by the observation of the experiment.

Please remove this comment and the {{TEX|auto}} line below, ...was created in the early 20th century by the studies of E. Borel, R. Baire and H. Lebesgue in connection with the measurability of sets. Borel-measurable

...to Hilbert’s statement of his 2nd problem, initiating his program for the foundations of mathematics'' -- see [[Hilbert problems]] ...had developed deductively a large part of analysis using the real numbers and their properties as a starting point.

Please remove this comment and the {{TEX|auto}} line below, ...erent kinds, as well as analysis of the axiomatic structure of number sets and the properties of numbers. When referring to the logical analysis of the co

Please remove this comment and the {{TEX|auto}} line below, ...pment of mathematics. The practical activities of mankind on the one hand, and the internal demands of mathematics on the other, have determined the devel

...formalizable was seen as "vague", "intuitive" (as opposed to "rigorous"), and imprecise.... This movement is later called 'the discretization program'".< ...atization program came to signify the set-theoretic definition of function and the set-theoretic construction of the real line

Please remove this comment and the {{TEX|auto}} line below, ...pplications. Of particular significance is the connection with the science and practice of economics.