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Topic: Peano axioms

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Peano postulates

Natural number

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Giuseppe Peano

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	Online Encyclopedia and Dictionary - Peano axioms
		In mathematics, the Peano axioms (or Peano postulates) are a set of first-order axioms proposed by Giuseppe Peano which determine the theory of Peano arithmetic (also known as first-order arithmetic).
		This theory constitutes a fundamental formalism for arithmetic, and the Peano axioms form a basis for the formalisation of stronger theories, such as second-order arithmetic.
		The Peano axioms may be interpreted in the general context of category theory.
	fact-archive.com /encyclopedia/Peano_axioms (1904 words)

	Giuseppe Peano - Wikipedia, the free encyclopedia
		Born on a farm near the village of Spinetta in Piedmont, Italy, Peano enrolled at the nearby University of Turin in 1876.
		Peano played a key role in the axiomatization of mathematics and was a leading pioneer in the development of mathematical logic.
		After his mother died in 1910, Peano divided his time between teaching, working on texts aimed for secondary schooling including a dictionary of mathematics, and developing and promoting his and other artificial languages, becoming a revered member of the international auxiliary language movement.
	en.wikipedia.org /wiki/Peano (1186 words)

	Jules Henri Poincaré [Internet Encyclopedia of Philosophy]
		From a general point of view, an axiom system can be conceived of as an implicit definition only if it is possible to prove the existence of at least one object that satisfies all the axioms.
		Proving this is not an easy task, for the number of consequences of Peano axioms is infinite and so a direct inspection of each consequence is not possible.
		Axioms of geometry are neither synthetic a priori judgments nor analytic ones; they are conventions or 'disguised' definitions.
	www.iep.utm.edu /p/poincare.htm (3565 words)

	Peano axioms - Wikipedia, the free encyclopedia
		In mathematics, the Peano axioms (or Peano postulates) are a set of first-order axioms proposed by Giuseppe Peano which determine the theory of Peano arithmetic (also known as first-order arithmetic).
		This theory constitutes a fundamental formalism for arithmetic, and the Peano axioms form a basis for the formalisation of stronger theories, such as second-order arithmetic.
		The Peano axioms may be interpreted in the general context of category theory.
	en.wikipedia.org /wiki/Peano_axioms (1864 words)

	Peano, Giuseppe (1858-1932)
		Giuseppe Peano was one of the pioneers in mathematical logic and axiomatization of mathematics.
		Giuseppe Peano was born to a poor farming family in Spinetta, Italy, on August 27, 1858.
		Peano's greatest contributions, however, were in the fields of axiomatization of mathematics and mathematical logic.
	www.geocities.com /Athens/Olympus/2948/pgolba.html (881 words)

	Peano (Site not responding. Last check: )
		Peano was appointed assistant to Genocchi for 1881-82 and it was in 1882 that Peano made a discovery that would be typical of his style for many years, he discovered an error in a standard definition.
		Peano was about to teach the students about the area of a curved surface when he realised that the definition in Serret's book, which was the standard text for the course, was incorrect.
		Peano received his qualification to be a university professor in December 1884 and he continued to teach further courses, some for Genocchi whose health had not recovered sufficiently to allow him to return to the University.
	www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Peano.html (2345 words)

	Peano
		Peano's first work on logic (1888) showed that the calculus of classes and the propositional calculus were, up to notation, the same.
		Peano skipped over any attempt to define the natural numbers in logic, thus bypassing certain philosophical issues that mathematicians tend to view as being incapable of precise formulation, and concentrated on the manipulation of symbols, something mathematicians find most agreeable.
		Peano's response was that the ability to give brief and precise form to mathematical theorems would make the importance of his work clear.
	www.math.uwaterloo.ca /~snburris/htdocs/scav/peano/peano.html (650 words)

	Σελίδες για τον Bertrand Russell (Site not responding. Last check: )
		Peano was the founder of symbolic logic and his interests centred on the foundations of mathematics and on the development of a formal logical language.
		Peano introduced the basic elements of geometric calculus and gave new definitions for the length of an arc and for the area of a curved surface.
		Although Peano is a founder of mathematical logic, the German mathematical philosopher Gottlob Frege (1848-1925) is considered the father of mathematical logic.
	sfr.ee.teiath.gr /htmSELIDES/Russell/Peano.htm (275 words)

	PlanetMath: Peano arithmetic
		Peano's axioms are a definition of the set of natural numbers, denoted
		From these axioms Peano arithmetic on natural numbers can be derived.
		This is version 5 of Peano arithmetic, born on 2002-03-10, modified 2004-02-25.
	planetmath.org /encyclopedia/PeanoArithmetic.html (140 words)

	Peano axioms (Site not responding. Last check: )
		In mathematics, the Peano axioms (or Peano postulates) are a set of axioms proposed by Giuseppe Peano as a foundation for the natural numbers.
		Using the Peano axioms, one can construct most of the number systems and structures of modern mathematics.
		Two Peano systems (X, x, f) and (Y, y, g) are said to be isomorphic if there is a bijection φ : X → Y such that φ(x) = y and φf = gφ.
	www.sciencedaily.com /encyclopedia/peano_axioms (1927 words)

	PlanetMath: PA
		Peano Arithmetic (PA) is the restriction of Peano's axioms to a first order theory of arithmetic.
		The only change is that the induction axiom is replaced by induction restricted to arithmetic formulas:
		Note that this replaces the single, second-order, axiom of induction with a countably infinite schema of axioms.
	planetmath.org /encyclopedia/PeanoArithmetic2.html (141 words)

	0603-232 Computer Science 2
		The axioms are introduced by Guiseppe Peano and they are therefore called Peano axioms.
		Peano Giuseppe Peano; born: 27 Aug 1858 in Cuneo, Piemonte, Italy.
		Peano was the founder of symbolic logic and his interests centered on the foundations of mathematics and on the development of a formal logical language.
	www.cs.rit.edu /~hpb/Lectures/99_CS2_Java/all-inOne-9.html (753 words)

Biogragpy of Giuseppe Peano </a></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> He realized that <b>Peano</b> Giuseppe was a very talented child and he took him to Turin in 1870 for his secondary schooling, and to prepare him for university studies. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> <b>Peano</b> took exams at Ginnasio Cavour in 1873 and then was a student at Liceo Cavour from where he graduated in 1876. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> It was a triumph for <b>Peano</b> and Russell.</td></tr> <tr><td></td><td colspan=2><font color=gray>www.andrews.edu /~calkins/math/biograph/biopeano.htm</font> (1680 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td> </td> <td> <table > <tr><td> </td><td colspan=2><a href="http://miser-theory.info/astraendo/pn">Numbering Peano</a></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> I have been very neglectful of Numbering <b>Peano</b> and the other elements of the Miser Project. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> A Fine Kettle of Fish (2004-06-05) I also have a "specification" of Num.java that nails down the contract and also ties it to the <b>Peano</b> <b>axioms</b> for <a href="/topics/Arithmetic" title="Arithmetic" class=fl>arithmetic</a>. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> This is the final post that will be made to Numbering <b>Peano</b> at the location http://nfocentrale.net/miser/astraendo/pn.</td></tr> <tr><td></td><td colspan=2><font color=gray>miser-theory.info /astraendo/pn</font> (4611 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td> </td> <td> <table > <tr><td> </td><td colspan=2><u>Russell’s reduction of mathematics to logic</u> <i>(Site not responding. Last check: )</i></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> Russell’s paradox and the <b>axiom</b> schema of comprehension </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> Typically, the <b>axioms</b> of the theory to be reduced are given in different vocabulary from the <b>axioms</b> of the reducing theory. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> And it is: by <b>axiom</b> 3, the <b>Axiom</b> of Infinity.</td></tr> <tr><td></td><td colspan=2><font color=gray>www.arts.mcgill.ca /philo/speaks/370/logicism.html</font> (4343 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td> </td> <td> <table > <tr><td> </td><td colspan=2><a href="http://www-groups.dcs.st-and.ac.uk/history/Search/historysearch.cgi?SUGGESTION=Peano&CONTEXT=1">Search Results for Peano</a></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> <b>Peano</b> was appointed as assistant to D'Ovidio at Turin in 1880 and Vailati was among the first <a href="/topics/Group-%28mathematics%29" title="Group %28mathematics%29" class=fl>group</a> of students for whom he had to care. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> <b>Peano's</b> 1888 book Calcolo geometrico secondo l'Ausdehnungslehre di H. It was many years before this notation was to become accepted, in fact <b>Peano's</b> book seems to have had very little influence for many years. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> <b>Peano</b> goes on to state the existence of a zero object 0 and says that 0a = 0, that a - b means a + (-b) and states it is easy to show that a - a = 0 and 0 + a = a.</td></tr> <tr><td></td><td colspan=2><font color=gray>www-groups.dcs.st-and.ac.uk /history/Search/historysearch.cgi?SUGGESTION=Peano&CONTEXT=1</font> (3040 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td> </td> <td> <table > <tr><td> </td><td colspan=2><u>[No title]</u> <i>(Site not responding. Last check: )</i></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> Giuseppe <b>Peano</b> proposed the following five <b>axioms</b> for the <a href="/topics/Natural-number" title="Natural number" class=fl>natural numbers</a>; they have come to be known as the <b>Peano</b> <b>axioms</b> or <b>Peano</b> postulates. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> These <b>axioms</b> are sometimes paraphrased differently, starting at 1 instead of 0. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> for each <a href="/topics/First_order-logic" title="First_order logic" class=fl>first order</a> property P(x) (an infinite number of <b>axioms</b>) then although <a href="/topics/Natural-number" title="Natural number" class=fl>natural numbers</a> satisfy these <b>axioms</b>, there are other, nonstandard models of arbitrary <a href="/topics/Large-cardinal" title="Large cardinal" class=fl>large cardinality </a>- by the <a href="/topics/Compactness-theorem" title="Compactness theorem" class=fl>compactness theorem</a> the existence of infinite <a href="/topics/Natural-number" title="Natural number" class=fl>natural numbers</a> cannot be excluded in any axiomatization; by an "upward <a href="/topics/L%C3%B6wenheim_Skolem-theorem" title="L%C3%B6wenheim_Skolem theorem" class=fl>Löwenheim-Skolem theorem</a>", there exist models of all <a href="/topics/Cardinality" title="Cardinality" class=fl>cardinalities</a>.</td></tr> <tr><td></td><td colspan=2><font color=gray>www.informationgenius.com /encyclopedia/p/pe/peano_axioms.html</font> (704 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td> </td> <td> <table > <tr><td> </td><td colspan=2><a href="http://www.utm.edu/research/iep/p/poincare.htm">Jules Henri Poincaré [Internet Encyclopedia of Philosophy]</a></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> From a general point of view, an <b>axiom</b> system can be conceived of as an implicit definition only if it is possible to prove the existence of at least one object that satisfies all the <b>axioms</b>. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> Proving this is not an easy task, for the number of consequences of <b>Peano</b> <b>axioms</b> is infinite and so a direct inspection of each consequence is not possible. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> <b>Axioms</b> of geometry are neither synthetic a priori judgments nor analytic ones; they are conventions or 'disguised' definitions.</td></tr> <tr><td></td><td colspan=2><font color=gray>www.utm.edu /research/iep/p/poincare.htm</font> (3565 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td> </td> <td> <table > <tr><td> </td><td colspan=2><a href="http://users.skynet.be/TGMDev/curvepeano.htm">Fractals: Peano Curves</a></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> On 29 September 1880 <b>Peano</b> graduated as doctor of <a href="/topics/Mathematics" title="Mathematics" class=fl>mathematics</a> joined the staff at the University of Turin in 1880. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> In 1889 <b>Peano</b> published (in Latin !!!) his famous <b>axioms</b>, called <b>Peano</b> <b>axioms</b>, which defined the <a href="/topics/Natural-number" title="Natural number" class=fl>natural numbers</a> in terms of <a href="/topics/Category-of-sets" title="Category of sets" class=fl>sets</a>. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> The project was <a href="/topics/G%C3%B6del%27s-completeness-theorem" title="G%C3%B6del%27s completeness theorem" class=fl>completed</a> in 1908 and one has to admire what <b>Peano</b> achieved but although the work contained a mine of information it was little used.</td></tr> <tr><td></td><td colspan=2><font color=gray>users.skynet.be /TGMDev/curvepeano.htm</font> (651 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td> </td> <td> <table > <tr><td> </td><td colspan=2><a href="http://www.mtholyoke.edu/courses/barring/232/lecture/10.htm">Lecture 10 for Math 232</a></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> The most interesting of these <b>axioms</b> is the fifth, because it touches on what we mean by "counting forever", the essence of the <a href="/topics/Number-system" title="Number system" class=fl>number system</a>. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> Any system satisfying the first four <b>Peano</b> <b>axioms</b> must have a version of each of the non-negative <a href="/topics/Integer" title="Integer" class=fl>integers </a>-- for example, it must have a "three" because it must have a "successor of the successor of the successor of zero". </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> The "closed under successor" version of the fifth <b>axiom</b> is pretty clearly a restatement of the "no other numbers than those obtained by counting from zero" version.</td></tr> <tr><td></td><td colspan=2><font color=gray>www.mtholyoke.edu /courses/barring/232/lecture/10.htm</font> (1585 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td> </td> <td> <table > <tr><td> </td><td colspan=2><u>Construction of sets and Peano's Axioms</u> <i>(Site not responding. Last check: )</i></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> <b>Peano's</b> five <b>axioms</b> define the <a href="/topics/Natural-number" title="Natural number" class=fl>natural numbers</a> starting with just 0 and s, the <a href="/topics/Successor-function" title="Successor function" class=fl>successor function</a>. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> From these <b>axioms</b>, and simple logical reasoning, all properties of the <a href="/topics/Natural-number" title="Natural number" class=fl>natural numbers</a> can be deduced, including the fact that N has no maximum, and the <a href="/topics/Well_order" title="Well_order" class=fl>well-ordering</a> theorem: </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> The <b>Peano</b> Postulates -- Proving the properties of <a href="/topics/Natural-number" title="Natural number" class=fl>natural numbers</a> using the <b>Peano</b> Postulates, which have been formulated so that zero is not included in the <a href="/topics/Category-of-sets" title="Category of sets" class=fl>set</a> of <a href="/topics/Natural-number" title="Natural number" class=fl>natural numbers</a>.</td></tr> <tr><td></td><td colspan=2><font color=gray>mcraeclan.com /MathHelp/BasicSetConstruction.htm</font> (632 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td> </td> <td> <table > <tr><td> </td><td colspan=2><u>Peano, Giuseppe</u> <i>(Site not responding. Last check: )</i></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> <b>Peano</b> studied <a href="/topics/Mathematics" title="Mathematics" class=fl>mathematics</a> at the University of Turin and joined the faculty there (1880), becoming a professor in 1890. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> In 1889 <b>Peano</b> published his famous postulates, called <b>Peano</b> <b>axioms</b>, which defined the <a href="/topics/Natural-number" title="Natural number" class=fl>natural numbers</a> in terms of a <a href="/topics/Category-of-sets" title="Category of sets" class=fl>set</a> of elements. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> <b>Peano</b> was also interested in universal, or international, languages and created the artificial language Interlingua (see LANGUAGES, ARTIFICIAL).</td></tr> <tr><td></td><td colspan=2><font color=gray>euler.ciens.ucv.ve /English/mathematics/peano.html</font> (129 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td> </td> <td> <table > <tr><td> </td><td colspan=2><u>Archimedes Plutonium</u> <i>(Site not responding. Last check: )</i></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> The correcting and overhauling of the <b>Peano</b> <b>axioms</b> for Number theory involves a realization that (1) there is no valid concept of "finite" but that all numbers have a component of infinity. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> (2) <a href="/topics/Mathematical-induction" title="Mathematical induction" class=fl>Mathematical Induction</a> <b>axiom</b> is false (3) and the <b>axiom</b> that 0 has no predecessor is false For geometry <b>axioms</b>: (1) the <b>axiom</b> of a point are false because in physics the lowest item in the universe is a atom and a atom has internal parts. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> The answer is that p-adic strings fit the <b>Peano</b> <b>axiom</b> system the best although there are predecessors to 0 and <a href="/topics/Mathematical-induction" title="Mathematical induction" class=fl>mathematical induction</a> does not hold.</td></tr> <tr><td></td><td colspan=2><font color=gray>www.iw.net /~a_plutonium/File102.html</font> (626 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td> </td> <td> <table > <tr><td> </td><td colspan=2><a href="http://www.britannica.com/eb/article-9058868">Peano, Giuseppe -- Encyclopædia Britannica</a></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> Italian mathematician and a founder of symbolic logic whose interests centred on the foundations of <a href="/topics/Mathematics" title="Mathematics" class=fl>mathematics</a> and on the development of a formal logical language. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> <b>Peano</b> became a lecturer of infinitesimal calculus at the University of Turin in 1884 and a professor in 1890. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> Interlingua was originally developed in 1903 by the Italian mathematician Giuseppe <b>Peano</b>, but lack of clarity as to what parts of Latin were to be retained and what were to be discarded led to numerous “dialects” of Interlingua, confusion, and its dying out among enthusiasts.</td></tr> <tr><td></td><td colspan=2><font color=gray>www.britannica.com /eb/article-9058868</font> (734 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td> </td> <td> <table > <tr><td> </td><td colspan=2><a href="http://nrich.maths.org/askedNRICH/edited/4906.html">nrich.maths.org::Mathematics Enrichment::NRICH</a></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> There are various <a href="/topics/Category-of-sets" title="Category of sets" class=fl>sets</a> of <b>axioms</b> covering different <a href="/topics/Mathematics" title="Mathematics" class=fl>mathematical</a> objects which in the end can all be deduced using a suitable <a href="/topics/Category-of-sets" title="Category of sets" class=fl>set</a> of <b>axioms</b> for <a href="/topics/Category-of-sets" title="Category of sets" class=fl>set</a> theory. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> Given this <a href="/topics/Category-of-sets" title="Category of sets" class=fl>set</a> of <b>axioms</b> you can define what multiplication and <a href="/topics/Addition-in-N" title="Addition in N" class=fl>addition</a> of <a href="/topics/Natural-number" title="Natural number" class=fl>natural numbers</a> means and solve quite a few problems in number theory (although not all of them). </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> However, there are other <a href="/topics/Category-of-sets" title="Category of sets" class=fl>sets</a> of <b>axioms</b> out there for different <a href="/topics/Mathematics" title="Mathematics" class=fl>mathematical</a> objects of interest in algebra like the <a href="/topics/Group-%28mathematics%29" title="Group %28mathematics%29" class=fl>group</a> <b>axioms</b> (things like (ab)c=a(bc)), ring <b>axioms</b> (things like (a+b)c=ac+bc), field <b>axioms</b> (fields are rings with multiplicative inverses), and you also get <a href="/topics/Commutative" title="Commutative" class=fl>commutative</a> <a href="/topics/Group-%28mathematics%29" title="Group %28mathematics%29" class=fl>groups</a>, rings and fields (<a href="/topics/Commutative" title="Commutative" class=fl>commutative</a> means that ab=ba).</td></tr> <tr><td></td><td colspan=2><font color=gray>nrich.maths.org /askedNRICH/edited/4906.html</font> (829 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td> </td> <td> <table > <tr><td> </td><td colspan=2><a href="http://www.specware.org/documentation/4.1/tutorial/x67.html">Specification Components</a></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> A morphism must be such that each <b>axiom</b> of the source spec maps to a theorem in the target spec: in other words, the translation of the <b>axiom</b> (according to the mapping expressed by the morphism) must be provable from the <b>axioms</b> in the target spec. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> The colimit operation produces a spec whose types, ops, and <b>axioms</b> are the disjoint union of the types, ops, and <b>axioms</b> of the specs in the diagram. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> The colimit operation produces a spec containing all the types, ops, and <b>axioms</b> of the specs in the diagram, but all the types or ops that are linked, directly or indirectly, through the morphisms, are identified (i.e., they are the same type or op).</td></tr> <tr><td></td><td colspan=2><font color=gray>www.specware.org /documentation/4.1/tutorial/x67.html</font> (1853 words)</td></tr> </table> </td> </tr> </table><script language="JavaScript">  </script> <script language="JavaScript">  </script> <script language="JavaScript" src="http://pagead2.googlesyndication.com/pagead/show_ads.js"> </script> <br> <p style="margin-left:30px;font-size:13px;"><b>Try your search on: <a href="http://www.qwika.com/find/Peano axioms">Qwika</a> (all wikis)</b></p> <form action=http://www.factbites.com/search.php><table width="100%" cellspacing=0 cellpadding=0 border=0><tr><td background="/images/f1.gif"><table cellspacing=0 cellpadding=0 border=0 background="/images/b.gif"><tr><td><img src="/images/f2.gif" width=38 height=37 alt=" "/></td><td><table cellspacing=0 cellpadding=0 border=0><tr><td><a href="/"><img src="/images/f3.gif" width=95 height=37 alt="Factbites" border=0 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