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Set-theoretic definition of = natural=20 numbers

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Several ways have been proposed to define the natural numbers = using set=20 theory.

1 The = contemporary=20 standard=20
2 The oldest=20 definition=20
3 Problem=20
4 See = also=20
5 References=20
6 External = links=20

[edit]=20 The = contemporary=20 standard

In standard (ZF)= set=20 theory the natural = numbers are=20 defined recursively by 0 =3D {} (the empty set) and n + 1 =3D = n =E2=88=AA=20 {n}. Then n =3D {0, 1, ..., n =E2=88=92 1} for each = natural number=20 n. The first few numbers defined this way are 0 =3D {}, 1 =3D = {0}, 2 =3D=20 {0,1}.

The set N of natural numbers is defined as the smallest set = containing=20 0 and closed under the successor function S defined by = S(n) =3D=20 n =E2=88=AA {n}. (For the existence of such a set we need = an Axiom of = Infinity.)=20 The structure (N,0,S) is a model of Peano = arithmetic.

The set N and its elements, when constructed this way, are = examples of=20 von=20 Neumann ordinals.

[edit]=20 The oldest=20 definition

Frege and Bertrand = Russell each=20 proposed the following definition. Informally, each natural number = n is=20 defined as the set whose members each have n elements. More = formally, a=20 natural number is the equivalence = class of=20 all sets under the relation= of equinumerosity. = This may=20 appear circular but is not.

Even more formally, first define 0 as $3D\{\varnothing\}=20$ =20 (this is the set whose only element is the empty set). Then = given any set=20 A, define:

=CF=83(A) as $3D"\{x$ =20

=CF=83(A) is the set obtained by adding a new element y = to every=20 member x of A. =CF=83 is a = set-theoretic=20 operationalization of the successor = function.=20 With the function =CF=83 in hand, we can say 1 =3D =CF=83(0), 2 =3D=20 =CF=83(1), 3 =3D =CF=83(2), and so=20 forth. This definition has the desired effect: the 3 we have just = defined=20 actually is the set whose members all have three elements.

This definition works in naive set = theory, type=20 theory, and in set theories that grew out of type theory, such as New=20 Foundations and related systems. But it does not work in the axiomatic set = theory ZFC and related systems, = because in=20 such systems the equivalence = classes=20 under equinumerosity = are "too=20 large" to be sets. For that matter, there is no universal = set=20 V in ZFC, under pain of the Russell = paradox.

Hatcher (1982) derives Peano's axioms from several foundational = systems,=20 including ZFC and category = theory. Most=20 curious is his meticulous derivation of these axioms from the system of = Frege's = Grundgesetze using=20 modern notation and natural = deduction. The=20 Russell = paradox proved=20 this system inconsistent, of course, but George Boolos = (1998) and=20 Anderson and Zalta (2004) show how to repair it.

[edit]=20 Problem

A consequence of Kurt = G=C3=B6del's work on incompleteness = is that in=20 any axiomatization of number theory = (ie. one=20 containing minimal arithmetic), there will be true statements of number = theory=20 which cannot be proven in that system. So trivially it follows that ZFC = or any=20 other formal system = cannot=20 capture entirely what a number is.

Whether this is a problem or not depends on whether you were seeking = a formal=20 definition of the concept of number. For people such as Bertrand = Russell (who=20 thought number theory, and hence mathematics, was a branch of logic and = number=20 was something to be defined in terms of formal logic) it was an = insurmountable=20 problem. But if you take the concept of number as an absolutely = fundamental and=20 irreducible one, it is to be expected. After all, if any concept is to = be left=20 formally undefined in mathematics, it might as well be one which = everyone=20 understands.

Poincar=C3=A9, amongst others (Bernays, Wittgenstein), held that any = attempt to=20 define natural number as it is endeavoured to do so above is = doomed to=20 failure by circularity. Informally, G=C3=B6del's theorem shows that a = formal=20 axiomatic definition is impossible (incompleteness), Poincar=C3=A9 = claims that no=20 definition, formal or informal, is possible (circularity). As such, they = give=20 two separate reasons why purported definitions of number must fail to = define=20 number. A quote from Poincar=C3=A9: "The definitions of number are very = numerous and=20 of great variety, and I will not attempt to enumerate their names and = their=20 authors. We must not be surprised that there are so many. If any of them = were=20 satisfactory we should not get any new ones." A quote from Wittgenstein: = "This=20 is not a definition. This is nothing but the arithmetical calculus with = frills=20 tacked on." A quote from Bernays: "Thus in spite of the possibility of=20 incorporating arithmetic into logistic, arithmetic constitutes the more = abstract=20 ('purer') schema; and this appears paradoxical only because of a = traditional,=20 but on closer examination unjustified view according to which logical = generality=20 is in every respect the highest generality."

Specifically, there are at least four points:

Zero is defined to be the number of things satisfying a condition = which is=20 satisfied in no case. It is not clear that a great deal of progress = has been=20 made.=20
It would be quite a challenge to enumerate the instances where = Russell (or=20 anyone else reading the definition out loud) refers to "an object" or = "the=20 class", phrases which are incomprehensible if one does not know that = the=20 speaker is speaking of one thing and one thing only.=20
The use of the concept of a relation, of any sort, presupposes the = concept=20 of two. For the idea of a relation is incomprehensible without the = idea of two=20 terms; that they must be two and only two.=20
Wittgenstein's "frills-tacked on comment". It is not at all clear = how one=20 would interpret the definitions at hand if one could not count. =

These problems with defining number disappear if one takes, as = Poincar=C3=A9 did,=20 the concept of number as basic ie. preliminary to and implicit in any = logical=20 thought whatsoever. Note that from such a viewpoint, set theory does not = precede=20 number = theory.

[edit]=20 See also

[edit]=20 References

Anderson, D. J., and Edward Zalta, = 2004,=20 "Frege, Boolos, and Logical Objects," Journal of Philosophical = Logic=20 33: 1-26.=20
George Boolos, = 1998.=20 Logic, Logic, and Logic.=20
Hatcher, William S., 1982. The Logical Foundations of = Mathematics.=20 Pergamon. In this text, S refers to the Peano axioms.=20
Holmes, Randall, 1998. Elementary Set Theory with a Universal Set.=20 Academia-Bruylant. The publisher has graciously consented to permit = diffusion=20 of this introduction to NFU via the = web.=20 Copyright is reserved.=20
Patrick = Suppes, 1972=20 (1960). Axiomatic Set Theory. Dover.

[edit]=20 External links

Stanford=20 Encyclopedia of Philosophy:=20
- Quine's New=20 Foundations =E2=80=94 by Thomas Forster.=20
- Alternative axiomatic set theories =E2=80=94 by = Randall Holmes.=20
McGuire, Gary, "What are the Natural Numbers?"=20
Randall Holmes: New=20 Foundations Home Page.

Retrieved from "http://en.wikipedia.org/wiki/Set-theoretic_definition_of_natural= _numbers"

Categories: = Basic=20 concepts in infinite set theory | Formal=20 theories of arithmetic

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------=_NextPart_000_0061_01CB5411.67B7FF40 Content-Type: application/octet-stream Content-Transfer-Encoding: quoted-printable Content-Location: http://bits.wikimedia.org/skins-1.5/common/ajax.js?283r // remote scripting library=0A= // (c) copyright 2005 modernmethod, inc=0A= var sajax_debug_mode =3D false;=0A= var sajax_request_type =3D "GET";=0A= =0A= /**=0A= * if sajax_debug_mode is true, this function outputs given the message = into =0A= * the element with id =3D sajax_debug; if no such element exists in the = document, =0A= * it is injected.=0A= */=0A= function sajax_debug(text) {=0A= if (!sajax_debug_mode) return false;=0A= =0A= var e=3D document.getElementById('sajax_debug');=0A= =0A= if (!e) {=0A= e=3D document.createElement("p");=0A= e.className=3D 'sajax_debug';=0A= e.id=3D 'sajax_debug';=0A= =0A= var b=3D document.getElementsByTagName("body")[0];=0A= =0A= if (b.firstChild) b.insertBefore(e, b.firstChild);=0A= else b.appendChild(e);=0A= }=0A= =0A= var m=3D document.createElement("div");=0A= m.appendChild( document.createTextNode( text ) );=0A= =0A= e.appendChild( m );=0A= =0A= return true;=0A= }=0A= =0A= /**=0A= * compatibility wrapper for creating a new XMLHttpRequest object.=0A= */=0A= function sajax_init_object() {=0A= sajax_debug("sajax_init_object() called..")=0A= var A;=0A= try {=0A= // Try the new style before ActiveX so we don't=0A= // unnecessarily trigger warnings in IE 7 when=0A= // set to prompt about ActiveX usage=0A= A =3D new XMLHttpRequest();=0A= } catch (e) {=0A= try {=0A= A=3Dnew ActiveXObject("Msxml2.XMLHTTP");=0A= } catch (e) {=0A= try {=0A= A=3Dnew ActiveXObject("Microsoft.XMLHTTP");=0A= } catch (oc) {=0A= A=3Dnull;=0A= }=0A= }=0A= }=0A= if (!A)=0A= sajax_debug("Could not create connection object.");=0A= =0A= return A;=0A= }=0A= =0A= /**=0A= * Perform an ajax call to mediawiki. Calls are handeled by = AjaxDispatcher.php=0A= * func_name - the name of the function to call. Must be registered in = $wgAjaxExportList=0A= * args - an array of arguments to that function=0A= * target - the target that will handle the result of the call. If this = is a function,=0A= * if will be called with the XMLHttpRequest as a parameter; = if it's an input=0A= * element, its value will be set to the resultText; if it's = another type of=0A= * element, its innerHTML will be set to the resultText.=0A= *=0A= * Example:=0A= * sajax_do_call('doFoo', [1, 2, 3], = document.getElementById("showFoo"));=0A= *=0A= * This will call the doFoo function via MediaWiki's AjaxDispatcher, with=0A= * (1, 2, 3) as the parameter list, and will show the result in the = element=0A= * with id =3D showFoo=0A= */=0A= function sajax_do_call(func_name, args, target) {=0A= var i, x, n;=0A= var uri;=0A= var post_data;=0A= uri =3D wgServer +=0A= ((wgScript =3D=3D null) ? 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Error: " + x.status + = " " + x.statusText + " (" + x.responseText + ")

";=0A= }=0A= }=0A= else {=0A= alert("bad target for sajax_do_call: not a function or object: " + = target);=0A= }=0A= =0A= return;=0A= }=0A= =0A= sajax_debug(func_name + " uri =3D " + uri + " / post =3D " + post_data);=0A= x.send(post_data);=0A= sajax_debug(func_name + " waiting..");=0A= delete x;=0A= =0A= return true;=0A= }=0A= =0A= /**=0A= * @return boolean whether the browser supports XMLHttpRequest=0A= */=0A= function wfSupportsAjax() {=0A= var request =3D sajax_init_object();=0A= var supportsAjax =3D request ? true : false;=0A= delete request;=0A= return supportsAjax;=0A= }=0A= =0A= ------=_NextPart_000_0061_01CB5411.67B7FF40 Content-Type: application/octet-stream Content-Transfer-Encoding: quoted-printable Content-Location: http://bits.wikimedia.org/skins-1.5/common/mwsuggest.js?283r /*=0A= * OpenSearch ajax suggestion engine for MediaWiki=0A= *=0A= * uses core MediaWiki open search support to fetch suggestions=0A= * and show them below search boxes and other inputs=0A= *=0A= * by Robert Stojnic (April 2008)=0A= */=0A= =0A= // search_box_id -> Results object=0A= var os_map =3D {};=0A= // cached data, url -> json_text=0A= var os_cache =3D {};=0A= // global variables for suggest_keypress=0A= var os_cur_keypressed =3D 0;=0A= var os_keypressed_count =3D 0;=0A= // type: Timer=0A= var os_timer =3D null;=0A= // tie mousedown/up events=0A= var os_mouse_pressed =3D false;=0A= var os_mouse_num =3D -1;=0A= // if true, the last change was made by mouse (and not keyboard)=0A= var os_mouse_moved =3D false;=0A= // delay between keypress and suggestion (in ms)=0A= var os_search_timeout =3D 250;=0A= // these pairs of inputs/forms will be autoloaded at startup=0A= var os_autoload_inputs =3D new Array('searchInput', 'searchInput2', = 'powerSearchText', 'searchText');=0A= var os_autoload_forms =3D new Array('searchform', 'searchform2', = 'powersearch', 'search' );=0A= // if we stopped the service=0A= var os_is_stopped =3D false;=0A= // max lines to show in suggest table=0A= var os_max_lines_per_suggest =3D 7;=0A= // number of steps to animate expansion/contraction of container width=0A= var os_animation_steps =3D 6;=0A= // num of pixels of smallest step=0A= var os_animation_min_step =3D 2;=0A= // delay between steps (in ms)=0A= var os_animation_delay =3D 30;=0A= // max width of container in percent of normal size (1 =3D=3D 100%)=0A= var os_container_max_width =3D 2;=0A= // currently active animation timer=0A= var os_animation_timer =3D null;=0A= /**=0A= * is a new HTML5 element that allows you to manually supply=0A= * suggestion lists and have them rendered according to the right = platform=0A= * conventions. However, the only shipping browser as of early 2010 is = Opera,=0A= * and that has a fatal problem: the suggestion lags behind what the = user types=0A= * by one keypress. (Reported as DSK-276870 to Opera's secret bug = tracker.)=0A= * The code here otherwise seems to work, though, so this can be flipped = on=0A= * (maybe with a UA check) when some browser has a better implementation.=0A= */=0A= // var os_use_datalist =3D 'list' in document.createElement( 'input' );=0A= var os_use_datalist =3D false;=0A= =0A= /** Timeout timer class that will fetch the results */=0A= function os_Timer( id, r, query ) {=0A= this.id =3D id;=0A= this.r =3D r;=0A= this.query =3D query;=0A= }=0A= =0A= /** Property class for single search box */=0A= function os_Results( name, formname ) {=0A= this.searchform =3D formname; // id of the searchform=0A= this.searchbox =3D name; // id of the searchbox=0A= this.container =3D name + 'Suggest'; // div that holds results=0A= this.resultTable =3D name + 'Result'; // id base for the result table = (+num =3D table row)=0A= this.resultText =3D name + 'ResultText'; // id base for the spans = within result tables (+num)=0A= this.toggle =3D name + 'Toggle'; // div that has the toggle = (enable/disable) link=0A= this.query =3D null; // last processed query=0A= this.results =3D null; // parsed titles=0A= this.resultCount =3D 0; // number of results=0A= this.original =3D null; // query that user entered=0A= this.selected =3D -1; // which result is selected=0A= this.containerCount =3D 0; // number of results visible in container=0A= this.containerRow =3D 0; // height of result field in the container=0A= this.containerTotal =3D 0; // total height of the container will all = results=0A= this.visible =3D false; // if container is visible=0A= this.stayHidden =3D false; // don't try to show if lost focus=0A= }=0A= =0A= /** Timer user to animate expansion/contraction of container width */=0A= function os_AnimationTimer( r, target ) {=0A= this.r =3D r;=0A= var current =3D document.getElementById(r.container).offsetWidth;=0A= this.inc =3D Math.round( ( target - current ) / os_animation_steps );=0A= if( this.inc < os_animation_min_step && this.inc >=3D0 ) {=0A= this.inc =3D os_animation_min_step; // minimal animation step=0A= }=0A= if( this.inc > -os_animation_min_step && this.inc < 0 ) {=0A= this.inc =3D -os_animation_min_step;=0A= }=0A= this.target =3D target;=0A= }=0A= =0A= /******************=0A= * Initialization=0A= ******************/=0A= =0A= /** Initialization, call upon page onload */=0A= function os_MWSuggestInit() {=0A= for( i =3D 0; i < os_autoload_inputs.length; i++ ) {=0A= var id =3D os_autoload_inputs[i];=0A= var form =3D os_autoload_forms[i];=0A= element =3D document.getElementById( id );=0A= if( element !=3D null ) {=0A= os_initHandlers( id, form, element );=0A= }=0A= }=0A= }=0A= =0A= /** Init Result objects and event handlers */=0A= function os_initHandlers( name, formname, element ) {=0A= var r =3D new os_Results( name, formname );=0A= // event handler=0A= os_hookEvent( element, 'keyup', function( event ) { os_eventKeyup( = event ); } );=0A= os_hookEvent( element, 'keydown', function( event ) { os_eventKeydown( = event ); } );=0A= os_hookEvent( element, 'keypress', function( event ) { = os_eventKeypress( event ); } );=0A= if ( !os_use_datalist ) {=0A= // These are needed for the div hack to hide it if the user blurs.=0A= os_hookEvent( element, 'blur', function( event ) { os_eventBlur( event = ); } );=0A= os_hookEvent( element, 'focus', function( event ) { os_eventFocus( = event ); } );=0A= // We don't want browser auto-suggestions interfering with our div, but=0A= // autocomplete must be on for datalist to work (at least in Opera=0A= // 10.10).=0A= element.setAttribute( 'autocomplete', 'off' );=0A= }=0A= // stopping handler=0A= os_hookEvent( document.getElementById( formname ), 'submit', function( = event ) { return os_eventOnsubmit( event ); } );=0A= os_map[name] =3D r;=0A= // toggle link=0A= if( document.getElementById( r.toggle ) =3D=3D null ) {=0A= // TODO: disable this while we figure out a way for this to work in = all browsers=0A= /* if( name =3D=3D 'searchInput' ) {=0A= // special case: place above the main search box=0A= var t =3D os_createToggle( r, 'os-suggest-toggle' );=0A= var searchBody =3D document.getElementById( 'searchBody' );=0A= var first =3D = searchBody.parentNode.firstChild.nextSibling.appendChild(t);=0A= } else {=0A= // default: place below search box to the right=0A= var t =3D os_createToggle( r, 'os-suggest-toggle-def' );=0A= var top =3D element.offsetTop + element.offsetHeight;=0A= var left =3D element.offsetLeft + element.offsetWidth;=0A= t.style.position =3D 'absolute';=0A= t.style.top =3D top + 'px';=0A= t.style.left =3D left + 'px';=0A= element.parentNode.appendChild( t );=0A= // only now width gets calculated, shift right=0A= left -=3D t.offsetWidth;=0A= t.style.left =3D left + 'px';=0A= t.style.visibility =3D 'visible';=0A= } */=0A= }=0A= =0A= }=0A= =0A= function os_hookEvent( element, hookName, hookFunct ) {=0A= if ( element.addEventListener ) {=0A= element.addEventListener( hookName, hookFunct, false );=0A= } else if ( window.attachEvent ) {=0A= element.attachEvent( 'on' + hookName, hookFunct );=0A= }=0A= }=0A= =0A= /********************=0A= * Keyboard events=0A= ********************/=0A= =0A= /** Event handler that will fetch results on keyup */=0A= function os_eventKeyup( e ) {=0A= var targ =3D os_getTarget( e );=0A= var r =3D os_map[targ.id];=0A= if( r =3D=3D null ) {=0A= return; // not our event=0A= }=0A= =0A= // some browsers won't generate keypressed for arrow keys, catch it=0A= if( os_keypressed_count =3D=3D 0 ) {=0A= os_processKey( r, os_cur_keypressed, targ );=0A= }=0A= var query =3D targ.value;=0A= os_fetchResults( r, query, os_search_timeout );=0A= }=0A= =0A= /** catch arrows up/down and escape to hide the suggestions */=0A= function os_processKey( r, keypressed, targ ) {=0A= if ( keypressed =3D=3D 40 && !r.visible && os_timer =3D=3D null ) {=0A= // If the user hits the down arrow, fetch results immediately if none=0A= // are already displayed.=0A= r.query =3D '';=0A= os_fetchResults( r, targ.value, 0 );=0A= }=0A= // Otherwise, if we're not using datalist, we need to handle scrolling = and=0A= // so on.=0A= if ( os_use_datalist ) {=0A= return;=0A= }=0A= if ( keypressed =3D=3D 40 ) { // Arrow Down=0A= if ( r.visible ) {=0A= os_changeHighlight( r, r.selected, r.selected + 1, true );=0A= }=0A= } else if ( keypressed =3D=3D 38 ) { // Arrow Up=0A= if ( r.visible ) {=0A= os_changeHighlight( r, r.selected, r.selected - 1, true );=0A= }=0A= } else if( keypressed =3D=3D 27 ) { // Escape=0A= document.getElementById( r.searchbox ).value =3D r.original;=0A= r.query =3D r.original;=0A= os_hideResults( r );=0A= } else if( r.query !=3D document.getElementById( r.searchbox ).value ) {=0A= // os_hideResults( r ); // don't show old suggestions=0A= }=0A= }=0A= =0A= /** When keys is held down use a timer to output regular events */=0A= function os_eventKeypress( e ) {=0A= var targ =3D os_getTarget( e );=0A= var r =3D os_map[targ.id];=0A= if( r =3D=3D null ) {=0A= return; // not our event=0A= }=0A= =0A= var keypressed =3D os_cur_keypressed;=0A= =0A= os_keypressed_count++;=0A= os_processKey( r, keypressed, targ );=0A= }=0A= =0A= /** Catch the key code (Firefox bug) */=0A= function os_eventKeydown( e ) {=0A= if ( !e ) {=0A= e =3D window.event;=0A= }=0A= var targ =3D os_getTarget( e );=0A= var r =3D os_map[targ.id];=0A= if( r =3D=3D null ) {=0A= return; // not our event=0A= }=0A= =0A= os_mouse_moved =3D false;=0A= =0A= os_cur_keypressed =3D ( e.keyCode =3D=3D undefined ) ? e.which : = e.keyCode;=0A= os_keypressed_count =3D 0;=0A= }=0A= =0A= =0A= /** When the form is submitted hide everything, cancel updates... */=0A= function os_eventOnsubmit( e ) {=0A= var targ =3D os_getTarget( e );=0A= =0A= os_is_stopped =3D true;=0A= // kill timed requests=0A= if( os_timer !=3D null && os_timer.id !=3D null ) {=0A= clearTimeout( os_timer.id );=0A= os_timer =3D null;=0A= }=0A= // Hide all suggestions=0A= for( i =3D 0; i < os_autoload_inputs.length; i++ ) {=0A= var r =3D os_map[os_autoload_inputs[i]];=0A= if( r !=3D null ) {=0A= var b =3D document.getElementById( r.searchform );=0A= if( b !=3D null && b =3D=3D targ ) {=0A= // set query value so the handler won't try to fetch additional = results=0A= r.query =3D document.getElementById( r.searchbox ).value;=0A= }=0A= os_hideResults( r );=0A= }=0A= }=0A= return true;=0A= }=0A= =0A= =0A= =0A= /** Hide results from the user, either making the div = visibility=3Dhidden or=0A= * detaching the datalist from the input. */=0A= function os_hideResults( r ) {=0A= if ( os_use_datalist ) {=0A= document.getElementById( r.searchbox ).setAttribute( 'list', '' );=0A= } else {=0A= var c =3D document.getElementById( r.container );=0A= if ( c !=3D null ) {=0A= c.style.visibility =3D 'hidden';=0A= }=0A= }=0A= r.visible =3D false;=0A= r.selected =3D -1;=0A= }=0A= =0A= function os_decodeValue( value ) {=0A= if ( decodeURIComponent ) {=0A= return decodeURIComponent( value );=0A= }=0A= if( unescape ) {=0A= return unescape( value );=0A= }=0A= return null;=0A= }=0A= =0A= function os_encodeQuery( value ) {=0A= if ( encodeURIComponent ) {=0A= return encodeURIComponent( value );=0A= }=0A= if( escape ) {=0A= return escape( value );=0A= }=0A= return null;=0A= }=0A= =0A= /** Handles data from XMLHttpRequest, and updates the suggest results */=0A= function os_updateResults( r, query, text, cacheKey ) {=0A= os_cache[cacheKey] =3D text;=0A= r.query =3D query;=0A= r.original =3D query;=0A= if( text =3D=3D '' ) {=0A= r.results =3D null;=0A= r.resultCount =3D 0;=0A= os_hideResults( r );=0A= } else {=0A= try {=0A= var p =3D eval( '(' + text + ')' ); // simple json parse, could do a = safer one=0A= if( p.length < 2 || p[1].length =3D=3D 0 ) {=0A= r.results =3D null;=0A= r.resultCount =3D 0;=0A= os_hideResults( r );=0A= return;=0A= }=0A= if ( os_use_datalist ) {=0A= os_setupDatalist( r, p[1] );=0A= } else {=0A= os_setupDiv( r, p[1] );=0A= }=0A= } catch( e ) {=0A= // bad response from server or such=0A= os_hideResults( r );=0A= os_cache[cacheKey] =3D null;=0A= }=0A= }=0A= }=0A= =0A= /**=0A= * Create and populate a .=0A= *=0A= * @param r os_Result object=0A= * @param results Array of the new results to replace existing ones=0A= */=0A= function os_setupDatalist( r, results ) {=0A= var s =3D document.getElementById( r.searchbox );=0A= var c =3D document.getElementById( r.container );=0A= if ( c =3D=3D null ) {=0A= c =3D document.createElement( 'datalist' );=0A= c.setAttribute( 'id', r.container );=0A= document.body.appendChild( c );=0A= } else {=0A= c.innerHTML =3D '';=0A= }=0A= s.setAttribute( 'list', r.container );=0A= =0A= r.results =3D new Array();=0A= r.resultCount =3D results.length;=0A= r.visible =3D true;=0A= for ( i =3D 0; i < results.length; i++ ) {=0A= var title =3D os_decodeValue( results[i] );=0A= var opt =3D document.createElement( 'option' );=0A= opt.value =3D title;=0A= r.results[i] =3D title;=0A= c.appendChild( opt );=0A= }=0A= }=0A= =0A= /** Fetch namespaces from checkboxes or hidden fields in the search form,=0A= if none defined use wgSearchNamespaces global */=0A= function os_getNamespaces( r ) {=0A= var namespaces =3D '';=0A= var elements =3D document.forms[r.searchform].elements;=0A= for( i =3D 0; i < elements.length; i++ ) {=0A= var name =3D elements[i].name;=0A= if( typeof name !=3D 'undefined' && name.length > 2 && name[0] =3D=3D = 'n' &&=0A= name[1] =3D=3D 's' && (=0A= ( elements[i].type =3D=3D 'checkbox' && elements[i].checked ) ||=0A= ( elements[i].type =3D=3D 'hidden' && elements[i].value =3D=3D '1' )=0A= )=0A= ) {=0A= if( namespaces !=3D '' ) {=0A= namespaces +=3D '|';=0A= }=0A= namespaces +=3D name.substring( 2 );=0A= }=0A= }=0A= if( namespaces =3D=3D '' ) {=0A= namespaces =3D wgSearchNamespaces.join('|');=0A= }=0A= return namespaces;=0A= }=0A= =0A= /** Update results if user hasn't already typed something else */=0A= function os_updateIfRelevant( r, query, text, cacheKey ) {=0A= var t =3D document.getElementById( r.searchbox );=0A= if( t !=3D null && t.value =3D=3D query ) { // check if response is = still relevant=0A= os_updateResults( r, query, text, cacheKey );=0A= }=0A= r.query =3D query;=0A= }=0A= =0A= /** Fetch results after some timeout */=0A= function os_delayedFetch() {=0A= if( os_timer =3D=3D null ) {=0A= return;=0A= }=0A= var r =3D os_timer.r;=0A= var query =3D os_timer.query;=0A= os_timer =3D null;=0A= var path =3D wgMWSuggestTemplate.replace( "{namespaces}", = os_getNamespaces( r ) )=0A= .replace( "{dbname}", wgDBname )=0A= .replace( "{searchTerms}", os_encodeQuery( query ) );=0A= =0A= // try to get from cache, if not fetch using ajax=0A= var cached =3D os_cache[path];=0A= if( cached !=3D null && cached !=3D undefined ) {=0A= os_updateIfRelevant( r, query, cached, path );=0A= } else {=0A= var xmlhttp =3D sajax_init_object();=0A= if( xmlhttp ) {=0A= try {=0A= xmlhttp.open( 'GET', path, true );=0A= xmlhttp.onreadystatechange =3D function() {=0A= if ( xmlhttp.readyState =3D=3D 4 && typeof os_updateIfRelevant = =3D=3D 'function' ) {=0A= os_updateIfRelevant( r, query, xmlhttp.responseText, path );=0A= }=0A= };=0A= xmlhttp.send( null );=0A= } catch ( e ) {=0A= if ( window.location.hostname =3D=3D 'localhost' ) {=0A= alert( "Your browser blocks XMLHttpRequest to 'localhost', try = using a real hostname for development/testing." );=0A= }=0A= throw e;=0A= }=0A= }=0A= }=0A= }=0A= =0A= /** Init timed update via os_delayedUpdate() */=0A= function os_fetchResults( r, query, timeout ) {=0A= if( query =3D=3D '' ) {=0A= r.query =3D '';=0A= os_hideResults( r );=0A= return;=0A= } else if( query =3D=3D r.query ) {=0A= return; // no change=0A= }=0A= =0A= os_is_stopped =3D false; // make sure we're running=0A= =0A= // cancel any pending fetches=0A= if( os_timer !=3D null && os_timer.id !=3D null ) {=0A= clearTimeout( os_timer.id );=0A= }=0A= // schedule delayed fetching of results=0A= if( timeout !=3D 0 ) {=0A= os_timer =3D new os_Timer( setTimeout( "os_delayedFetch()", timeout ), = r, query );=0A= } else {=0A= os_timer =3D new os_Timer( null, r, query );=0A= os_delayedFetch(); // do it now!=0A= }=0A= }=0A= =0A= /** Find event target */=0A= function os_getTarget( e ) {=0A= if ( !e ) {=0A= e =3D window.event;=0A= }=0A= if ( e.target ) {=0A= return e.target;=0A= } else if ( e.srcElement ) {=0A= return e.srcElement;=0A= } else {=0A= return null;=0A= }=0A= }=0A= =0A= /** Check if x is a valid integer */=0A= function os_isNumber( x ) {=0A= if( x =3D=3D '' || isNaN( x ) ) {=0A= return false;=0A= }=0A= for( var i =3D 0; i < x.length; i++ ) {=0A= var c =3D x.charAt( i );=0A= if( !( c >=3D '0' && c <=3D '9' ) ) {=0A= return false;=0A= }=0A= }=0A= return true;=0A= }=0A= =0A= /** Call this to enable suggestions on input (id=3DinputId), on a form = (name=3DformName) */=0A= function os_enableSuggestionsOn( inputId, formName ) {=0A= os_initHandlers( inputId, formName, document.getElementById( inputId ) = );=0A= }=0A= =0A= /** Call this to disable suggestios on input box (id=3DinputId) */=0A= function os_disableSuggestionsOn( inputId ) {=0A= r =3D os_map[inputId];=0A= if( r !=3D null ) {=0A= // cancel/hide results=0A= os_timer =3D null;=0A= os_hideResults( r );=0A= // turn autocomplete on !=0A= document.getElementById( inputId ).setAttribute( 'autocomplete', 'on' = );=0A= // remove descriptor=0A= os_map[inputId] =3D null;=0A= }=0A= =0A= // Remove the element from the os_autoload_* arrays=0A= var index =3D os_autoload_inputs.indexOf( inputId );=0A= if ( index >=3D 0 ) {=0A= os_autoload_inputs[index] =3D os_autoload_forms[index] =3D '';=0A= }=0A= }=0A= =0A= /************************************************=0A= * Div-only functions (irrelevant for datalist)=0A= ************************************************/=0A= =0A= /** Event: loss of focus of input box */=0A= function os_eventBlur( e ) {=0A= var targ =3D os_getTarget( e );=0A= var r =3D os_map[targ.id];=0A= if( r =3D=3D null ) {=0A= return; // not our event=0A= }=0A= if( !os_mouse_pressed ) {=0A= os_hideResults( r );=0A= // force canvas to stay hidden=0A= r.stayHidden =3D true;=0A= // cancel any pending fetches=0A= if( os_timer !=3D null && os_timer.id !=3D null ) {=0A= clearTimeout( os_timer.id );=0A= }=0A= os_timer =3D null;=0A= }=0A= }=0A= =0A= /** Event: focus (catch only when stopped) */=0A= function os_eventFocus( e ) {=0A= var targ =3D os_getTarget( e );=0A= var r =3D os_map[targ.id];=0A= if( r =3D=3D null ) {=0A= return; // not our event=0A= }=0A= r.stayHidden =3D false;=0A= }=0A= =0A= /**=0A= * Create and populate a

, for non--supporting browsers.=0A= *=0A= * @param r os_Result object=0A= * @param results Array of the new results to replace existing ones=0A= */=0A= function os_setupDiv( r, results ) {=0A= var c =3D document.getElementById( r.container );=0A= if ( c =3D=3D null ) {=0A= c =3D os_createContainer( r );=0A= }=0A= c.innerHTML =3D os_createResultTable( r, results );=0A= // init container table sizes=0A= var t =3D document.getElementById( r.resultTable );=0A= r.containerTotal =3D t.offsetHeight;=0A= r.containerRow =3D t.offsetHeight / r.resultCount;=0A= os_fitContainer( r );=0A= os_trimResultText( r );=0A= os_showResults( r );=0A= }=0A= =0A= /** Create the result table to be placed in the container div */=0A= function os_createResultTable( r, results ) {=0A= var c =3D document.getElementById( r.container );=0A= var width =3D c.offsetWidth - os_operaWidthFix( c.offsetWidth );=0A= var html =3D '';=0A= r.results =3D new Array();=0A= r.resultCount =3D results.length;=0A= for( i =3D 0; i < results.length; i++ ) {=0A= var title =3D os_decodeValue( results[i] );=0A= r.results[i] =3D title;=0A= html +=3D '';=0A= }=0A= html +=3D '

' + title + = '

';=0A= return html;=0A= }=0A= =0A= /** Show results div */=0A= function os_showResults( r ) {=0A= if( os_is_stopped ) {=0A= return;=0A= }=0A= if( r.stayHidden ) {=0A= return;=0A= }=0A= os_fitContainer( r );=0A= var c =3D document.getElementById( r.container );=0A= r.selected =3D -1;=0A= if( c !=3D null ) {=0A= c.scrollTop =3D 0;=0A= c.style.visibility =3D 'visible';=0A= r.visible =3D true;=0A= }=0A= }=0A= =0A= function os_operaWidthFix( x ) {=0A= // For browsers that don't understand overflow-x, estimate scrollbar = width=0A= if( typeof document.body.style.overflowX !=3D 'string' ) {=0A= return 30;=0A= }=0A= return 0;=0A= }=0A= =0A= /** Brower-dependent functions to find window inner size, and scroll = status */=0A= function f_clientWidth() {=0A= return f_filterResults(=0A= window.innerWidth ? window.innerWidth : 0,=0A= document.documentElement ? document.documentElement.clientWidth : 0,=0A= document.body ? document.body.clientWidth : 0=0A= );=0A= }=0A= =0A= function f_clientHeight() {=0A= return f_filterResults(=0A= window.innerHeight ? window.innerHeight : 0,=0A= document.documentElement ? document.documentElement.clientHeight : 0,=0A= document.body ? document.body.clientHeight : 0=0A= );=0A= }=0A= =0A= function f_scrollLeft() {=0A= return f_filterResults(=0A= window.pageXOffset ? window.pageXOffset : 0,=0A= document.documentElement ? document.documentElement.scrollLeft : 0,=0A= document.body ? document.body.scrollLeft : 0=0A= );=0A= }=0A= =0A= function f_scrollTop() {=0A= return f_filterResults(=0A= window.pageYOffset ? window.pageYOffset : 0,=0A= document.documentElement ? document.documentElement.scrollTop : 0,=0A= document.body ? document.body.scrollTop : 0=0A= );=0A= }=0A= =0A= function f_filterResults( n_win, n_docel, n_body ) {=0A= var n_result =3D n_win ? n_win : 0;=0A= if ( n_docel && ( !n_result || ( n_result > n_docel ) ) ) {=0A= n_result =3D n_docel;=0A= }=0A= return n_body && ( !n_result || ( n_result > n_body ) ) ? n_body : = n_result;=0A= }=0A= =0A= /** Get the height available for the results container */=0A= function os_availableHeight( r ) {=0A= var absTop =3D document.getElementById( r.container ).style.top;=0A= var px =3D absTop.lastIndexOf( 'px' );=0A= if( px > 0 ) {=0A= absTop =3D absTop.substring( 0, px );=0A= }=0A= return f_clientHeight() - ( absTop - f_scrollTop() );=0A= }=0A= =0A= /** Get element absolute position {left,top} */=0A= function os_getElementPosition( elemID ) {=0A= var offsetTrail =3D document.getElementById( elemID );=0A= var offsetLeft =3D 0;=0A= var offsetTop =3D 0;=0A= while ( offsetTrail ) {=0A= offsetLeft +=3D offsetTrail.offsetLeft;=0A= offsetTop +=3D offsetTrail.offsetTop;=0A= offsetTrail =3D offsetTrail.offsetParent;=0A= }=0A= if ( navigator.userAgent.indexOf('Mac') !=3D -1 && typeof = document.body.leftMargin !=3D 'undefined' ) {=0A= offsetLeft +=3D document.body.leftMargin;=0A= offsetTop +=3D document.body.topMargin;=0A= }=0A= return { left:offsetLeft, top:offsetTop };=0A= }=0A= =0A= /** Create the container div that will hold the suggested titles */=0A= function os_createContainer( r ) {=0A= var c =3D document.createElement( 'div' );=0A= var s =3D document.getElementById( r.searchbox );=0A= var pos =3D os_getElementPosition( r.searchbox );=0A= var left =3D pos.left;=0A= var top =3D pos.top + s.offsetHeight;=0A= c.className =3D 'os-suggest';=0A= c.setAttribute( 'id', r.container );=0A= document.body.appendChild( c );=0A= =0A= // dynamically generated style params=0A= // IE workaround, cannot explicitely set "style" attribute=0A= c =3D document.getElementById( r.container );=0A= c.style.top =3D top + 'px';=0A= c.style.left =3D left + 'px';=0A= c.style.width =3D s.offsetWidth + 'px';=0A= =0A= // mouse event handlers=0A= c.onmouseover =3D function( event ) { os_eventMouseover( r.searchbox, = event ); };=0A= c.onmousemove =3D function( event ) { os_eventMousemove( r.searchbox, = event ); };=0A= c.onmousedown =3D function( event ) { return os_eventMousedown( = r.searchbox, event ); };=0A= c.onmouseup =3D function( event ) { os_eventMouseup( r.searchbox, event = ); };=0A= return c;=0A= }=0A= =0A= /** change container height to fit to screen */=0A= function os_fitContainer( r ) {=0A= var c =3D document.getElementById( r.container );=0A= var h =3D os_availableHeight( r ) - 20;=0A= var inc =3D r.containerRow;=0A= h =3D parseInt( h / inc ) * inc;=0A= if( h < ( 2 * inc ) && r.resultCount > 1 ) { // min: two results=0A= h =3D 2 * inc;=0A= }=0A= if( ( h / inc ) > os_max_lines_per_suggest ) {=0A= h =3D inc * os_max_lines_per_suggest;=0A= }=0A= if( h < r.containerTotal ) {=0A= c.style.height =3D h + 'px';=0A= r.containerCount =3D parseInt( Math.round( h / inc ) );=0A= } else {=0A= c.style.height =3D r.containerTotal + 'px';=0A= r.containerCount =3D r.resultCount;=0A= }=0A= }=0A= =0A= /** If some entries are longer than the box, replace text with "..." */=0A= function os_trimResultText( r ) {=0A= // find max width, first see if we could expand the container to fit it=0A= var maxW =3D 0;=0A= for( var i =3D 0; i < r.resultCount; i++ ) {=0A= var e =3D document.getElementById( r.resultText + i );=0A= if( e.offsetWidth > maxW ) {=0A= maxW =3D e.offsetWidth;=0A= }=0A= }=0A= var w =3D document.getElementById( r.container ).offsetWidth;=0A= var fix =3D 0;=0A= if( r.containerCount < r.resultCount ) {=0A= fix =3D 20; // give 20px for scrollbar=0A= } else {=0A= fix =3D os_operaWidthFix( w );=0A= }=0A= if( fix < 4 ) {=0A= fix =3D 4; // basic padding=0A= }=0A= maxW +=3D fix;=0A= =0A= // resize container to fit more data if permitted=0A= var normW =3D document.getElementById( r.searchbox ).offsetWidth;=0A= var prop =3D maxW / normW;=0A= if( prop > os_container_max_width ) {=0A= prop =3D os_container_max_width;=0A= } else if( prop < 1 ) {=0A= prop =3D 1;=0A= }=0A= var newW =3D Math.round( normW * prop );=0A= if( w !=3D newW ) {=0A= w =3D newW;=0A= if( os_animation_timer !=3D null ) {=0A= clearInterval( os_animation_timer.id );=0A= }=0A= os_animation_timer =3D new os_AnimationTimer( r, w );=0A= os_animation_timer.id =3D setInterval( "os_animateChangeWidth()", = os_animation_delay );=0A= w -=3D fix; // this much is reserved=0A= }=0A= =0A= // trim results=0A= if( w < 10 ) {=0A= return;=0A= }=0A= for( var i =3D 0; i < r.resultCount; i++ ) {=0A= var e =3D document.getElementById( r.resultText + i );=0A= var replace =3D 1;=0A= var lastW =3D e.offsetWidth + 1;=0A= var iteration =3D 0;=0A= var changedText =3D false;=0A= while( e.offsetWidth > w && ( e.offsetWidth < lastW || iteration < 2 ) = ) {=0A= changedText =3D true;=0A= lastW =3D e.offsetWidth;=0A= var l =3D e.innerHTML;=0A= e.innerHTML =3D l.substring( 0, l.length - replace ) + '...';=0A= iteration++;=0A= replace =3D 4; // how many chars to replace=0A= }=0A= if( changedText ) {=0A= // show hint for trimmed titles=0A= document.getElementById( r.resultTable + i ).setAttribute( 'title', = r.results[i] );=0A= }=0A= }=0A= }=0A= =0A= /** Invoked on timer to animate change in container width */=0A= 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Set-theoretic definition of = natural=20 numbers

Contents

[edit]=20 The = contemporary=20 standard

[edit]=20 The oldest=20 definition

[edit]=20 Problem

[edit]=20 See also

[edit]=20 References

[edit]=20 External links

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