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axioms | A Wisdom Archive on axioms | | axioms A selection of articles related to axioms | |
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axioms, Axiom, Axiom - Etymology, Axiom - Mathematics, Axiom - <span id=role>Role in mathematical logic</span>, Axiom - Further discussion, Axiom - Logical axioms, Axiom - Non-logical axioms, Axiomatic system, Peano axioms, Axiom of choice, Axiom of countability, Axiomatic set theory, Parallel postulate, Continuum hypothesis, Axiomatization, List of axioms
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| ARTICLES RELATED TO axioms | | | |  axioms: Encyclopedia II - Elementary group theory - R#* is a groupThe real numbers without 0 (R#) are a group under multiplication (*).
Closure: Clear; multiplying any two numbers gives another number.
Associativity: Clear; for any a, b, c in R, (a*b)*c=a*(b*c).
Identity: 1. For any a in R, a*1=a. (Hence the denotation 1 for identity)
Inverses: For any a in R, a -1*a=1. (Hence the denotation a ...
See also:Elementary group theory, Elementary group theory - R+ is a group, Elementary group theory - R* is not a group, Elementary group theory - R#* is a group, Elementary group theory - Inverses work on either side, Elementary group theory - An identity works on either side, Elementary group theory - Latin square property, Elementary group theory - The identity is unique, Elementary group theory - Inverses are unique, Elementary group theory - Inverting twice gets you back where you started, Elementary group theory - The inverse of ab, Elementary group theory - Cancellation, Elementary group theory - Repeated use of * Read more here: » Elementary group theory: Encyclopedia II - Elementary group theory - R#* is a group |
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| | | | | axioms: Encyclopedia II - First-order logic - Comparison with other logics
Most of these logics are in some sense extensions of first order logic: they include all the quantifiers and logical operators of first order logic with the same meanings. Lindstrom showed first order logic has no extensions (other than itself) that satisfy both the compactness theorem and the downward Lowenheim-Skolem theorem. A precise statement of this theorem requires listing several pages of technical conditions that the logic is assumed to satisfy; for example, changing the symbols of a language should make no essential differen ...
See also:First-order logic, First-order logic - Defining first-order logic, First-order logic - Vocabulary, First-order logic - Formation rules, First-order logic - Equality, First-order logic - Inference rules, First-order logic - Quantifier axioms, First-order logic - The predicate calculus, First-order logic - Metalogical theorems of first-order logic, First-order logic - Comparison with other logics Read more here: » First-order logic: Encyclopedia II - First-order logic - Comparison with other logics |
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| | | | | axioms: Encyclopedia II - Mathematics - Major themes in mathematics An alphabetical and subclassified list of mathematics articles is available. The following list of themes and links gives just one possible view. For a fuller treatment, see areas of mathematics or the list of mathematics lists.
Mathematics - Quantity.
Quantity starts with counting and measurement.
Natural numbers
...
See also:Mathematics, Mathematics - History, Mathematics - Inspiration pure and applied mathematics and aesthetics, Mathematics - Notation language and rigor, Mathematics - Is mathematics a science?, Mathematics - Overview of fields of mathematics, Mathematics - Major themes in mathematics, Mathematics - Quantity, Mathematics - Structure, Mathematics - Space, Mathematics - Change, Mathematics - Foundations and methods, Mathematics - Discrete mathematics, Mathematics - Applied mathematics, Mathematics - Important theorems, Mathematics - Important conjectures, Mathematics - History and the world of mathematicians, Mathematics - Mathematics and other fields, Mathematics - Mathematical tools, Mathematics - Common misconceptions Read more here: » Mathematics: Encyclopedia II - Mathematics - Major themes in mathematics |
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| | | | | | | | | axioms: Encyclopedia II - Logical graph - In lieu of a beginning In medias res, as always, we nevertheless need a quantum of formal matter to keep the topical momentum going. A game try at supplying that least bit of motivation may be found in this duo of transformations between the indicated forms of enclosure:
o-----------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `( ) ( )` ` ` = ` ` ` `( )` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o------- ...
See also:Logical graph, Logical graph - Abstract point of view, Logical graph - In lieu of a beginning, Logical graph - Duality logical and topological, Logical graph - Computational representation, Logical graph - Quick tour of the neighborhood, Logical graph - Primary arithmetic as semiotic system, Logical graph - Primary algebra as pattern calculus, Logical graph - Formal development, Logical graph - Axioms, Logical graph - Frequently used theorems, Logical graph - Exemplary proofs Read more here: » Logical graph: Encyclopedia II - Logical graph - In lieu of a beginning |
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| | | | | axioms: Encyclopedia II - Logical graph - Computational representation The parse graphs that we've been looking at so far are one step toward the pointer graphs that it takes to make trees live in computer memory, but they are still a couple of steps too abstract to properly suggest in much concrete detail the species of dynamic data structures that we need. The time has come to flesh out the skeleton that we've drawn up to this point.
Nodes in a graph depict records in computer memory. A record is a collection of data that can be thought to reside at a specific address. For semioticians, a ...
See also:Logical graph, Logical graph - Abstract point of view, Logical graph - In lieu of a beginning, Logical graph - Duality logical and topological, Logical graph - Computational representation, Logical graph - Quick tour of the neighborhood, Logical graph - Primary arithmetic as semiotic system, Logical graph - Primary algebra as pattern calculus, Logical graph - Formal development, Logical graph - Axioms, Logical graph - Frequently used theorems, Logical graph - Exemplary proofs Read more here: » Logical graph: Encyclopedia II - Logical graph - Computational representation |
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| | | | | axioms: Encyclopedia II - Logical graph - Duality logical and topological There are two types of duality that have to be kept separately mind in the use of logical graphs, logical duality and topological duality.
There is a standard way that graphs of the order that Peirce considered, those embedded in a continuous manifold like that commonly represented by a plane sheet of paper — with or without the paper bridges that Peirce used to augment its topological genus — can be represented in linear text as what are called parse strings or traversal strings< ...
See also:Logical graph, Logical graph - Abstract point of view, Logical graph - In lieu of a beginning, Logical graph - Duality logical and topological, Logical graph - Computational representation, Logical graph - Quick tour of the neighborhood, Logical graph - Primary arithmetic as semiotic system, Logical graph - Primary algebra as pattern calculus, Logical graph - Formal development, Logical graph - Axioms, Logical graph - Frequently used theorems, Logical graph - Exemplary proofs Read more here: » Logical graph: Encyclopedia II - Logical graph - Duality logical and topological |
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| | | | | | axioms: Encyclopedia II - Mathematics - Inspiration pure and applied mathematics and aesthetics Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were found in commerce, land measurement and later astronomy; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. Newton invented infinitesimal calculus and Feynman his Feynman path integral using a combination of reasoning and physical insight, and today's string theory also inspires new mathematics. Some mathematics is only relevant in the area that ins ...
See also:Mathematics, Mathematics - History, Mathematics - Inspiration pure and applied mathematics and aesthetics, Mathematics - Notation language and rigor, Mathematics - Is mathematics a science?, Mathematics - Overview of fields of mathematics, Mathematics - Major themes in mathematics, Mathematics - Quantity, Mathematics - Structure, Mathematics - Space, Mathematics - Change, Mathematics - Foundations and methods, Mathematics - Discrete mathematics, Mathematics - Applied mathematics, Mathematics - Important theorems, Mathematics - Important conjectures, Mathematics - History and the world of mathematicians, Mathematics - Mathematics and other fields, Mathematics - Mathematical tools, Mathematics - Common misconceptions Read more here: » Mathematics: Encyclopedia II - Mathematics - Inspiration pure and applied mathematics and aesthetics |
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| | | | | axioms: Encyclopedia II - Mathematics - Overview of fields of mathematics As noted above, the major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the relationships between numbers, to measure land, and to predict astronomical events. These four needs can be roughly related to the broad subdivision of mathematics into the study of quantity, structure, space, and change (i.e., arithmetic, algebra, geometry and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations) and to the emp ...
See also:Mathematics, Mathematics - History, Mathematics - Inspiration pure and applied mathematics and aesthetics, Mathematics - Notation language and rigor, Mathematics - Is mathematics a science?, Mathematics - Overview of fields of mathematics, Mathematics - Major themes in mathematics, Mathematics - Quantity, Mathematics - Structure, Mathematics - Space, Mathematics - Change, Mathematics - Foundations and methods, Mathematics - Discrete mathematics, Mathematics - Applied mathematics, Mathematics - Important theorems, Mathematics - Important conjectures, Mathematics - History and the world of mathematicians, Mathematics - Mathematics and other fields, Mathematics - Mathematical tools, Mathematics - Common misconceptions Read more here: » Mathematics: Encyclopedia II - Mathematics - Overview of fields of mathematics |
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| | | | | axioms: Encyclopedia II - Group mathematics - Notation for groups Usually the operation, whatever it really is, is thought of as an analogue of multiplication, and the group operations are therefore written multiplicatively. That is:
We write "a · b" or even "ab" for a * b and call it the product of a and b;
We write "1" (or "e") for the identity element and call it the unit element;
We write "a−1" for the inverse of a< ...
See also:Group mathematics, Group mathematics - History, Group mathematics - Basic definitions, Group mathematics - Notation for groups, Group mathematics - Some elementary examples and nonexamples, Group mathematics - An abelian group: the integers under addition, Group mathematics - Not a group: the integers under multiplication, Group mathematics - An abelian group: the nonzero rational numbers under multiplication, Group mathematics - A finite nonabelian group: permutations of a set, Group mathematics - Further examples, Group mathematics - Simple theorems, Group mathematics - Constructing new groups from given ones Read more here: » Group mathematics: Encyclopedia II - Group mathematics - Notation for groups |
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| | | | | axioms: Encyclopedia II - Group mathematics - Basic definitions A group (G, * ) is a nonempty set G together with a binary operation * : G × G → G, satisfying the group axioms below. "a * b" represents the result of applying the operation * to the ordered pair (a, b) of elements of G. The group axioms are the following:
Associativity: For all a, b and c in G, (a * b) * c = a * (b * c).
Neutral element: There is an element < ...
See also:Group mathematics, Group mathematics - History, Group mathematics - Basic definitions, Group mathematics - Notation for groups, Group mathematics - Some elementary examples and nonexamples, Group mathematics - An abelian group: the integers under addition, Group mathematics - Not a group: the integers under multiplication, Group mathematics - An abelian group: the nonzero rational numbers under multiplication, Group mathematics - A finite nonabelian group: permutations of a set, Group mathematics - Further examples, Group mathematics - Simple theorems, Group mathematics - Constructing new groups from given ones Read more here: » Group mathematics: Encyclopedia II - Group mathematics - Basic definitions |
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| | | | | | axioms: Encyclopedia II - David Hilbert - Later years Hilbert lived to see the Nazis purge many of the prominent faculty members at University of Göttingen, in 1933. [1]. Among those forced out were Hermann Weyl, who had taken Hilbert's chair when he retired in 1930, Emmy Noether and Edmund Landau. One of those who had to leave Germany was Paul Bernays, Hilbert's collaborator in mathematical logic, and co-author with him of the important book Grundlagen der Mathematik (which eventually appeared in two volumes, in 1934 and 1939). This was a sequel to the Hilbert-Ackermann book P ...
See also:David Hilbert, David Hilbert - Major contributions, David Hilbert - Miscellaneous talks essays and contributions, David Hilbert - Hilbert's program, David Hilbert - Later years, David Hilbert - Notes Read more here: » David Hilbert: Encyclopedia II - David Hilbert - Later years |
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| | | | | axioms: Encyclopedia II - Mathematics - Notation language and rigor Most of the mathematical notation we use today was not invented until the 16th Century. Before that, mathematics was written out in words, a painstaking process that limited mathematical discovery. Modern notation makes mathematics much easier for the professional, but beginners often find it daunting. It is extremely compressed: a few symbols contain a great deal of information. Like musical notation, modern mathematical notation has a strict grammar (under the influence of ...
See also:Mathematics, Mathematics - History, Mathematics - Inspiration pure and applied mathematics and aesthetics, Mathematics - Notation language and rigor, Mathematics - Is mathematics a science?, Mathematics - Overview of fields of mathematics, Mathematics - Major themes in mathematics, Mathematics - Quantity, Mathematics - Structure, Mathematics - Space, Mathematics - Change, Mathematics - Foundations and methods, Mathematics - Discrete mathematics, Mathematics - Applied mathematics, Mathematics - Important theorems, Mathematics - Important conjectures, Mathematics - History and the world of mathematicians, Mathematics - Mathematics and other fields, Mathematics - Mathematical tools, Mathematics - Common misconceptions Read more here: » Mathematics: Encyclopedia II - Mathematics - Notation language and rigor |
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| | | | | axioms: Encyclopedia II - Mathematics - Is mathematics a science? Carl Friedrich Gauss referred to mathematics as "the Queen of the Sciences".
If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science. An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist, J. M. Ziman, proposed that science is < ...
See also:Mathematics, Mathematics - History, Mathematics - Inspiration pure and applied mathematics and aesthetics, Mathematics - Notation language and rigor, Mathematics - Is mathematics a science?, Mathematics - Overview of fields of mathematics, Mathematics - Major themes in mathematics, Mathematics - Quantity, Mathematics - Structure, Mathematics - Space, Mathematics - Change, Mathematics - Foundations and methods, Mathematics - Discrete mathematics, Mathematics - Applied mathematics, Mathematics - Important theorems, Mathematics - Important conjectures, Mathematics - History and the world of mathematicians, Mathematics - Mathematics and other fields, Mathematics - Mathematical tools, Mathematics - Common misconceptions Read more here: » Mathematics: Encyclopedia II - Mathematics - Is mathematics a science? |
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| | | | | axioms: Encyclopedia II - Controversy over Cantor's theory - Footnote The quote "Later generations will regard set theory as a disease from which one has recovered" is from Kline[1982], and is apparently his translation of a quote from Poincaré's speech "The future of mathematics" given in 1908. There has been considerable dispute about what Poincaré actually intended to imply. Another translation reads "I think, [...] that it is important never to introduce any conception which may not be completely defined by a finite number of words. Whatever may be the remedy adopted, we can promise ourselves the joy of ...
See also:Controversy over Cantor's theory, Controversy over Cantor's theory - Preface, Controversy over Cantor's theory - Introduction, Controversy over Cantor's theory - Cantor's argument, Controversy over Cantor's theory - Reception of the argument, Controversy over Cantor's theory - Naïve objections, Controversy over Cantor's theory - Objections to Cantor's theorem, Controversy over Cantor's theory - Objections to Hume's principle, Controversy over Cantor's theory - Objection to the axiom of infinity, Controversy over Cantor's theory - Objections to the power set axiom, Controversy over Cantor's theory - Footnote Read more here: » Controversy over Cantor's theory: Encyclopedia II - Controversy over Cantor's theory - Footnote |
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| | | | | axioms: Encyclopedia II - Controversy over Cantor's theory - Objection to the axiom of infinity One of the most common (and also the most respectable) objections to Cantor's theory of infinite number involves the axiom of infinity. It is generally recognised view by all logicians that this axiom is not a logical truth. Indeed, as Mark Sainsbury (1979, p.305) has argued "there is room for doubt about whether it is a contingent truth, since it is an open question whether the universe is finite or infinite". Bertrand Russell for many years tried to establish a foundation for mathematics that did not rely on this axiom. Mayberry (2000, p.1 ...
See also:Controversy over Cantor's theory, Controversy over Cantor's theory - Preface, Controversy over Cantor's theory - Introduction, Controversy over Cantor's theory - Cantor's argument, Controversy over Cantor's theory - Reception of the argument, Controversy over Cantor's theory - Naïve objections, Controversy over Cantor's theory - Objections to Cantor's theorem, Controversy over Cantor's theory - Objections to Hume's principle, Controversy over Cantor's theory - Objection to the axiom of infinity, Controversy over Cantor's theory - Objections to the power set axiom, Controversy over Cantor's theory - Footnote Read more here: » Controversy over Cantor's theory: Encyclopedia II - Controversy over Cantor's theory - Objection to the axiom of infinity |
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