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...nster pointed out, the axioms can be stated without references to category theory. ...ory that emphasizes sets as abstract structures (as opposed to collections of elements). ...

{{Short description|Schema of axioms in set theory}} ...tion of the usual [[axiom schema of separation]] in [[Zermelo–Fraenkel set theory]]. ...

{{Short description|Axiomatic set theory}} ...eory]], without substantially weakening the [[axiom of comprehension|axiom of unrestricted comprehension]]. ...

{{Short description|Principle in set theory}} ...= ''x'' ∪ {''y''} given by "adjoining" the set ''y'' to the set ''x''. It is stated as ...

...mathematics)|graph]] whose [[Vertex (graph theory)|vertices]] are a subset of the [[real numbers]]: two different versions were introduced by {{harvtxt|A ...h pair of elements of ''X'' is colored either black or white, with the set of white pairs being open in the [[complete graph]] on ''X''. The open colorin ...

==Statement of the problem== ...h>, multiplication <math>(\cdot)</math>, and exponentiation to be standard axioms taught in high school: ...

{{Short description|System of mathematical set theory}} {{DISPLAYTITLE:<b style="font-family:sans-serif">S</b> (set theory)}} ...

...enkel axioms of set theory]] are isomorphic to a member of a certain class of sets. ...Zermelo-Fraenkel set theory, but with a second-order version of the axiom of replacement formulated as follows:<ref>S. Shapiro, ''Foundations Without Fo ...

{{short description|Property of pairs of points in a cycle}} ...pairs. This '''point-pair separation''' is an invariant of projectivities of the line. ...

...onal dependencies]] on a [[Relation (database)|relation scheme]]) is a set of dependencies such that F [[logical implication|logically implies]] all depe The [[set (mathematics)|set]] <math>F_c</math> has two important properties: ...

...48-02/S0002-9947-1940-0002894-7/home.html Polyadic groups], ''Transactions of the American Mathematical Society'' 48 (1940), 208&ndash;350.</ref> ==Axioms== ...

{{Short description|Axiom in set theory}} ...a)_{\alpha \in A}</math> is a family of [[empty set|non-empty]] ''[[finite set]]s'', then ...

...y related axiom is '''Streicher's axiom K''', which asserts that any proof of an equality <math>x = x</math> is equal to the trivial reflexivity proof. These axioms thus have the respective types: ...

'''Q₀''' is [[Peter B. Andrews|Peter Andrews]]' formulation of the [[Simply typed lambda calculus|simply-typed lambda calculus]], ...ides a foundation for mathematics comparable to first-order logic plus set theory. ...

...qual to <math> 2^{\aleph_0}\!</math>, the cardinality of the [[power set]] of the [[natural numbers]]. ...saying that no [[cardinality]] lies between that of the continuum and that of the [[natural numbers]], <math>\aleph_0</math>, or alternatively, that <mat ...

{{Short description|Decidable theory of equality}} ...nsisting of only the equality relation symbol, and includes no non-logical axioms at all.<ref>{{Cite book| last1=Monk | first1=J. Donald | title=Mathematical ...

...as assignment, iteration and composition instructions and from the axioms of the [[data structure]]s in question see {{harvtxt|Mirkowska|Salwicki|1987}} ...ogic}\end{array}\right ] \subset \left [\begin{array}{l}\mathrm{Calculus\ of\ programs}\\or \\ \mbox{Algorithmic logic} \end{array}\right ] </math><br / ...

...comprehensive mathematical exposition in the first volume of ''Foundations of Measurement'' (Krantz, Luce, Suppes & Tversky, 1971), which Krantz and Tver ...easurement lies in the quantification of natural attributes in the absence of concatenation operations. Polynomial conjoint measurement differs from the ...

...eneralization of [[set theory]], which allows only a limited consideration of classes. ...class term, which formally correctly describes classes through a property of their elements. Today the class term is denoted in the form {x|A(x)}, where ...

...[[Julius Richard Büchi]]. The [[signature (mathematical logic)|signature]] of Büchi arithmetic contains only the addition operation, <math>V_k</math> and ...age of Büchi arithmetic, whether that sentence is provable from the axioms of Büchi arithmetic. ...