27:
184:
It is straightforward to show that a
Dedekind cut among the real numbers is uniquely defined by the corresponding cut among the rational numbers. Similarly, every cut of reals is identical to the cut produced by a specific real number (which can be identified as the smallest element of the
2637:
An incommensurable number can be defined only by indicating how the magnitude it expresses can be formed by means of unity. In what follows, we suppose that this definition consists of indicating which are the commensurable numbers smaller or larger than it
2930:
Jun-Iti Nagata, Modern
General Topology, Second revised edition, Theorem VIII.2, p. 461. Actually, the theorem holds in the setting of generalized ordered spaces, but in this more general setting pseudo-gaps should be taken into
125:
contains every rational number greater than or equal to the cut. An irrational cut is equated to an irrational number which is in neither set. Every real number, rational or not, is equated to one and only one cut of rationals.
1107:
1183:, and further, by defining properly arithmetic operators over these sets (addition, subtraction, multiplication, and division), these sets (together with these arithmetic operations) form the familiar real numbers.
2909:
1030:
1637:
1889:
1330:
1843:
1514:
382:
615:
does determine the other. It can be a simplification, in terms of notation if nothing more, to concentrate on one "half" â say, the lower one â and call any downward-closed set
325:
1701:
1566:
2701:
number, which we regard as completely defined by this cut ... . From now on, therefore, to every definite cut there corresponds a definite rational or irrational number ....
2805:
919:
232:
1984:
1669:
1181:
1137:
777:
745:
1786:
1462:
1383:
2366:
1250:
2783:
2444:
2418:
2392:
2210:
2184:
1753:
1727:
1429:
561:
535:
509:
460:
434:
408:
2730:
2334:
2158:
2046:
951:
2750:
2831:
2277:
2257:
2230:
2126:
2106:
2086:
2066:
2014:
1949:
1929:
1909:
1534:
1403:
1350:
1270:
1224:
1204:
1157:
581:
480:
345:
295:
272:
252:
3038:
1036:
882:, which might not have had the least-upper-bound property, within a (usually larger) linearly ordered set that does have this useful property.
2838:
2667:
959:
2589:
2501:
2978:
2950:
755:
would contain every positive rational number whose square is greater than or equal to 2. Even though there is no rational value for
2212:, with the lower cut and the upper cut being given by projections. This corresponds exactly to the set of intervals approximating
891:
59:
1571:
3031:
3006:
1848:
1275:
105:
has a smallest element among the rationals, the cut corresponds to that rational. Otherwise, that cut defines a unique
3001:
178:
2630:
870:
The set of all
Dedekind cuts is itself a linearly ordered set (of sets). Moreover, the set of Dedekind cuts has the
1791:
878:
upper bound. Thus, constructing the set of
Dedekind cuts serves the purpose of embedding the original ordered set
871:
3104:
3024:
588:
1252:(please refer to the link above for the precise definition of how the multiplication of cuts is defined), is
2482:
1467:
2687:
651:
855:). In this way, set inclusion can be used to represent the ordering of numbers, and all other relations (
353:
3109:
3099:
3078:
2511:
2280:
302:
198:
26:
2608:; it is ordered by inclusion. The Dedekind-MacNeille completion is the smallest complete lattice with
688:
complete. The cut itself can represent a number not in the original collection of numbers (most often
2487:
1674:
1539:
2236:
130:
2996:
2788:
2469:. A linearly ordered set endowed with the order topology is compact if and only if it has no gap.
902:
215:
2481:. The relevant notion in this case is a Cuesta-Dutari cut, named after the Spanish mathematician
1965:
1642:
1162:
1118:
758:
726:
67:
1758:
1434:
1355:
1159:, which is the set of all rational numbers whose squares are less than 2, to "represent" number
2477:
A construction resembling
Dedekind cuts is used for (one among many possible) constructions of
2339:
1229:
2974:
2946:
2663:
2585:
1113:
788:
106:
71:
31:
2755:
2655:
2523:
2423:
2397:
2371:
2189:
2163:
1962:
1732:
1706:
1408:
720:
540:
514:
488:
439:
413:
387:
201:
133:
by defining a
Dedekind cut as a partition of a totally ordered set into two non-empty parts
94:
51:
2715:
2307:
2131:
2019:
924:
3068:
3047:
2735:
2235:
This allows the basic arithmetic operations on the real numbers to be defined in terms of
1536:
is indeed a cut. Now armed with the multiplication between cuts, it is easy to check that
897:
689:
63:
55:
2810:
2697:
Whenever, then, we have to do with a cut produced by no rational number, we create a new
2833:
is already forbidden by the first condition. This results in the equivalent expression
2478:
2262:
2242:
2215:
2111:
2091:
2071:
2051:
1999:
1934:
1914:
1894:
1519:
1388:
1335:
1255:
1209:
1189:
1142:
751:, along with every non-negative rational number whose square is less than 2; similarly
566:
465:
330:
280:
257:
237:
3093:
20:
3073:
194:
190:
43:
35:
684:
The important purpose of the
Dedekind cut is to work with number sets that are
2659:
19:
For the
American record producer known professionally as Dedekind Cut, see
2239:. This property and its relation with real numbers given only in terms of
2982:
1102:{\displaystyle B=\{b\in \mathbb {Q} :b^{2}\geq 2{\text{ and }}b\geq 0\}.}
3058:
2904:{\displaystyle B=\{b\in \mathbb {Q} :b^{2}>2{\text{ and }}b>0\}.}
1025:{\displaystyle A=\{a\in \mathbb {Q} :a^{2}<2{\text{ or }}a<0\},}
2973:, "Continuity and Irrational Numbers," Dover Publications: New York,
2553:
1272:(note that rigorously speaking this number 2 is represented by a cut
892:
Construction of the real numbers § Construction by
Dedekind cuts
2556:. A related completion that preserves all existing sups and infs of
1139:
in
Dedekind's construction. The essential idea is that we use a set
3016:
874:, i.e., every nonempty subset of it that has any upper bound has a
1332:). To show the first part, we show that for any positive rational
129:
Dedekind cuts can be generalized from the rational numbers to any
25:
1206:
really is a cut (according to the definition) and the square of
587:
By omitting the first two requirements, we formally obtain the
3020:
101:
may or may not have a smallest element among the rationals. If
2560:
is obtained by the following construction: For each subset
1632:{\displaystyle x\times y\leq 2,\forall x,y\in A,x,y\geq 0}
867:, and so on) can be similarly created from set relations.
2541:
generalizes the least-upper-bound property of the reals.
2296:
In the general case of an arbitrary linearly ordered set
2128:, it can be equivalently represented as the set of pairs
1911:
constructed above, this means that we have a sequence in
2279:
is particularly important in weaker foundations such as
121:
contains every rational number less than the cut, and
2841:
2813:
2791:
2758:
2738:
2718:
2426:
2400:
2374:
2342:
2310:
2265:
2245:
2218:
2192:
2166:
2134:
2114:
2094:
2074:
2054:
2022:
2002:
1968:
1937:
1917:
1897:
1851:
1794:
1761:
1735:
1709:
1677:
1645:
1574:
1542:
1522:
1470:
1437:
1411:
1391:
1358:
1338:
1278:
1258:
1232:
1212:
1192:
1165:
1145:
1121:
1039:
962:
927:
905:
761:
729:
569:
543:
517:
491:
468:
442:
416:
390:
356:
333:
305:
283:
260:
240:
218:
696:, even though the numbers contained in the two sets
2752:without any difference as there is no solution for
2943:Foundations of Analysis over Surreal Number Fields
2903:
2825:
2799:
2777:
2744:
2724:
2438:
2412:
2386:
2360:
2328:
2271:
2251:
2224:
2204:
2178:
2152:
2120:
2100:
2080:
2060:
2040:
2008:
1996:Given a Dedekind cut representing the real number
1978:
1943:
1923:
1903:
1884:{\displaystyle 2-y^{2}\leq {\frac {\epsilon }{2}}}
1883:
1837:
1780:
1747:
1721:
1695:
1663:
1631:
1560:
1528:
1508:
1456:
1423:
1397:
1377:
1344:
1325:{\displaystyle \{x\ |\ x\in \mathbb {Q} ,x<2\}}
1324:
1264:
1244:
1218:
1198:
1175:
1151:
1131:
1101:
1024:
945:
913:
771:
739:
575:
555:
529:
503:
474:
454:
428:
402:
376:
339:
319:
289:
266:
246:
226:
109:which, loosely speaking, fills the "gap" between
779:, if the rational numbers are partitioned into
212:A Dedekind cut is a partition of the rationals
197:is defined as a Dedekind cut of rationals is a
3032:
2446:. Some authors add the requirement that both
1931:whose square can become arbitrarily close to
1838:{\displaystyle x>0,2-x^{2}=\epsilon >0}
787:this way, the partition itself represents an
747:by putting every negative rational number in
8:
2895:
2848:
1319:
1279:
1093:
1046:
1016:
969:
626:is complete, then, for every Dedekind cut (
619:without greatest element a "Dedekind cut".
3039:
3025:
3017:
2945:. Mathematics Studies 141. North-Holland.
607:) notation for Dedekind cuts, but each of
145:is closed downwards (meaning that for all
16:Method of construction of the real numbers
2881:
2869:
2858:
2857:
2840:
2812:
2793:
2792:
2790:
2763:
2757:
2737:
2717:
2425:
2399:
2373:
2341:
2309:
2264:
2244:
2217:
2191:
2165:
2133:
2113:
2093:
2073:
2053:
2021:
2001:
1969:
1967:
1936:
1916:
1896:
1871:
1862:
1850:
1817:
1793:
1766:
1760:
1734:
1708:
1676:
1644:
1573:
1541:
1521:
1477:
1469:
1442:
1436:
1410:
1390:
1363:
1357:
1337:
1303:
1302:
1288:
1277:
1257:
1231:
1211:
1191:
1166:
1164:
1144:
1122:
1120:
1079:
1067:
1056:
1055:
1038:
1002:
990:
979:
978:
961:
926:
907:
906:
904:
762:
760:
730:
728:
568:
542:
516:
490:
467:
441:
415:
389:
370:
369:
355:
332:
313:
312:
304:
282:
259:
239:
220:
219:
217:
2621:
1703:, and it suffices to show that for any
177:contains no greatest element. See also
30:Dedekind used his cut to construct the
1186:To establish this, one must show that
1509:{\displaystyle y={\frac {2x+2}{x+2}}}
583:does not contain a greatest element.)
7:
2681:
2679:
2921:R. Engelking, General Topology, I.3
2465:has a minimum, the cut is called a
704:do not actually include the number
599:It is more symmetrical to use the (
377:{\displaystyle x,y\in \mathbb {Q} }
50:, named after German mathematician
2652:Eine kurze Geschichte der Analysis
2580:denote the set of lower bounds of
2572:denote the set of upper bounds of
1593:
692:). The cut can represent a number
320:{\displaystyle A\neq \mathbb {Q} }
14:
2689:Continuity and Irrational Numbers
666:, +â). In this case, we say that
70:of the rational numbers into two
2016:by splitting the rationals into
886:Construction of the real numbers
60:construction of the real numbers
2971:Essays on the Theory of Numbers
2292:Arbitrary linearly ordered sets
1992:Relation to interval arithmetic
1696:{\displaystyle A\times A\geq 2}
1561:{\displaystyle A\times A\leq 2}
2323:
2311:
2147:
2135:
2035:
2023:
1568:(essentially, this is because
1289:
940:
928:
896:A typical Dedekind cut of the
85:is less than every element of
54:(but previously considered by
1:
2590:DedekindâMacNeille completion
2502:DedekindâMacNeille completion
1788:. For this we notice that if
2800:{\displaystyle \mathbb {Q} }
1951:, which finishes the proof.
914:{\displaystyle \mathbb {Q} }
819:) (of the same superset) if
646:, hence we must have that
642:must have a minimal element
227:{\displaystyle \mathbb {Q} }
81:, such that each element of
3002:Encyclopedia of Mathematics
2529:with an order-embedding of
1979:{\displaystyle {\sqrt {2}}}
1664:{\displaystyle A\times A=2}
1176:{\displaystyle {\sqrt {2}}}
1132:{\displaystyle {\sqrt {2}}}
772:{\displaystyle {\sqrt {2}}}
740:{\displaystyle {\sqrt {2}}}
723:, they can still be cut at
708:that their cut represents.
179:completeness (order theory)
3126:
2941:Alling, Norman L. (1987).
2686:Dedekind, Richard (1872).
2584:. (These operators form a
2499:
1781:{\displaystyle x^{2}>r}
1639:). Therefore to show that
1457:{\displaystyle y^{2}<2}
1378:{\displaystyle x^{2}<2}
921:is given by the partition
889:
872:least-upper-bound property
204:without any further gaps.
189:set). In other words, the
18:
3054:
2660:10.1007/978-3-662-57816-2
2629:Bertrand, Joseph (1849).
2361:{\displaystyle A\cup B=X}
1245:{\displaystyle A\times A}
799:Regard one Dedekind cut (
589:extended real number line
2596:consists of all subsets
1112:This cut represents the
2778:{\displaystyle x^{2}=2}
1954:Note that the equality
482:is "closed downwards".)
173:is closed upwards, and
2905:
2827:
2801:
2779:
2746:
2726:
2650:Spalt, Detlef (2019).
2496:Partially ordered sets
2483:Norberto Cuesta Dutari
2440:
2439:{\displaystyle a<b}
2414:
2413:{\displaystyle b\in B}
2388:
2387:{\displaystyle a\in A}
2362:
2330:
2273:
2253:
2226:
2206:
2205:{\displaystyle b\in B}
2180:
2179:{\displaystyle a\in A}
2154:
2122:
2102:
2082:
2062:
2042:
2010:
1980:
1945:
1925:
1905:
1885:
1839:
1782:
1749:
1748:{\displaystyle x\in A}
1723:
1722:{\displaystyle r<2}
1697:
1665:
1633:
1562:
1530:
1510:
1458:
1425:
1424:{\displaystyle x<y}
1399:
1385:, there is a rational
1379:
1346:
1326:
1266:
1246:
1220:
1200:
1177:
1153:
1133:
1103:
1026:
947:
915:
831:is a proper subset of
823:is a proper subset of
811:another Dedekind cut (
773:
741:
577:
557:
556:{\displaystyle y>x}
531:
530:{\displaystyle y\in A}
511:, then there exists a
505:
504:{\displaystyle x\in A}
476:
456:
455:{\displaystyle x\in A}
430:
429:{\displaystyle y\in A}
404:
403:{\displaystyle x<y}
378:
341:
321:
291:
268:
248:
228:
66:. A Dedekind cut is a
39:
3079:Superparticular ratio
2985:at Project Gutenberg.
2906:
2828:
2802:
2780:
2747:
2727:
2725:{\displaystyle \geq }
2632:Traité d'Arithmétique
2512:partially ordered set
2441:
2415:
2389:
2363:
2331:
2329:{\displaystyle (A,B)}
2281:constructive analysis
2274:
2254:
2227:
2207:
2181:
2155:
2153:{\displaystyle (a,b)}
2123:
2103:
2083:
2063:
2043:
2041:{\displaystyle (A,B)}
2011:
1981:
1946:
1926:
1906:
1886:
1840:
1783:
1750:
1724:
1698:
1666:
1634:
1563:
1531:
1511:
1459:
1426:
1400:
1380:
1347:
1327:
1267:
1247:
1221:
1201:
1178:
1154:
1134:
1104:
1027:
948:
946:{\displaystyle (A,B)}
916:
861:less than or equal to
774:
742:
578:
558:
532:
506:
477:
457:
431:
405:
379:
342:
322:
292:
269:
249:
229:
29:
2839:
2811:
2789:
2756:
2745:{\displaystyle >}
2736:
2716:
2712:In the second line,
2552:subsets, ordered by
2424:
2398:
2372:
2340:
2308:
2263:
2243:
2216:
2190:
2164:
2132:
2112:
2092:
2072:
2052:
2020:
2000:
1966:
1935:
1915:
1895:
1849:
1792:
1759:
1733:
1707:
1675:
1643:
1572:
1540:
1520:
1468:
1435:
1409:
1389:
1356:
1336:
1276:
1256:
1230:
1210:
1190:
1163:
1143:
1119:
1037:
960:
925:
903:
759:
727:
567:
541:
515:
489:
466:
440:
414:
388:
354:
331:
303:
281:
258:
238:
216:
2969:Dedekind, Richard,
2826:{\displaystyle b=0}
2732:may be replaced by
2506:More generally, if
2461:has a maximum, nor
2237:interval arithmetic
2048:where rationals in
827:. Equivalently, if
622:If the ordered set
131:totally ordered set
58:), are Đ° method of
2901:
2823:
2797:
2775:
2742:
2722:
2548:is the set of its
2544:One completion of
2436:
2410:
2384:
2358:
2326:
2269:
2249:
2222:
2202:
2176:
2150:
2118:
2098:
2078:
2058:
2038:
2006:
1976:
1961:cannot hold since
1941:
1921:
1901:
1881:
1835:
1778:
1745:
1719:
1693:
1661:
1629:
1558:
1526:
1506:
1454:
1421:
1395:
1375:
1342:
1322:
1262:
1242:
1216:
1196:
1173:
1149:
1129:
1099:
1022:
943:
911:
769:
737:
573:
553:
527:
501:
472:
452:
426:
400:
374:
337:
317:
287:
264:
244:
224:
117:. In other words,
40:
3087:
3086:
2884:
2669:978-3-662-57815-5
2586:Galois connection
2550:downwardly closed
2272:{\displaystyle B}
2252:{\displaystyle A}
2225:{\displaystyle r}
2121:{\displaystyle r}
2108:are greater than
2101:{\displaystyle B}
2088:and rationals in
2081:{\displaystyle r}
2061:{\displaystyle A}
2009:{\displaystyle r}
1974:
1944:{\displaystyle 2}
1924:{\displaystyle A}
1904:{\displaystyle y}
1879:
1529:{\displaystyle A}
1504:
1398:{\displaystyle y}
1345:{\displaystyle x}
1295:
1287:
1265:{\displaystyle 2}
1219:{\displaystyle A}
1199:{\displaystyle A}
1171:
1152:{\displaystyle A}
1127:
1114:irrational number
1082:
1005:
789:irrational number
767:
735:
671:is represented by
576:{\displaystyle A}
475:{\displaystyle A}
340:{\displaystyle B}
290:{\displaystyle A}
267:{\displaystyle B}
247:{\displaystyle A}
234:into two subsets
107:irrational number
3117:
3105:Rational numbers
3048:Rational numbers
3041:
3034:
3027:
3018:
3010:
2957:
2956:
2938:
2932:
2928:
2922:
2919:
2913:
2910:
2908:
2907:
2902:
2885:
2882:
2874:
2873:
2861:
2832:
2830:
2829:
2824:
2806:
2804:
2803:
2798:
2796:
2784:
2782:
2781:
2776:
2768:
2767:
2751:
2749:
2748:
2743:
2731:
2729:
2728:
2723:
2710:
2704:
2703:
2694:
2683:
2674:
2673:
2647:
2641:
2640:
2626:
2612:embedded in it.
2539:complete lattice
2537:. The notion of
2524:complete lattice
2491:
2445:
2443:
2442:
2437:
2419:
2417:
2416:
2411:
2393:
2391:
2390:
2385:
2367:
2365:
2364:
2359:
2335:
2333:
2332:
2327:
2278:
2276:
2275:
2270:
2258:
2256:
2255:
2250:
2231:
2229:
2228:
2223:
2211:
2209:
2208:
2203:
2185:
2183:
2182:
2177:
2159:
2157:
2156:
2151:
2127:
2125:
2124:
2119:
2107:
2105:
2104:
2099:
2087:
2085:
2084:
2079:
2067:
2065:
2064:
2059:
2047:
2045:
2044:
2039:
2015:
2013:
2012:
2007:
1985:
1983:
1982:
1977:
1975:
1970:
1960:
1950:
1948:
1947:
1942:
1930:
1928:
1927:
1922:
1910:
1908:
1907:
1902:
1890:
1888:
1887:
1882:
1880:
1872:
1867:
1866:
1844:
1842:
1841:
1836:
1822:
1821:
1787:
1785:
1784:
1779:
1771:
1770:
1754:
1752:
1751:
1746:
1728:
1726:
1725:
1720:
1702:
1700:
1699:
1694:
1670:
1668:
1667:
1662:
1638:
1636:
1635:
1630:
1567:
1565:
1564:
1559:
1535:
1533:
1532:
1527:
1515:
1513:
1512:
1507:
1505:
1503:
1492:
1478:
1463:
1461:
1460:
1455:
1447:
1446:
1430:
1428:
1427:
1422:
1404:
1402:
1401:
1396:
1384:
1382:
1381:
1376:
1368:
1367:
1351:
1349:
1348:
1343:
1331:
1329:
1328:
1323:
1306:
1293:
1292:
1285:
1271:
1269:
1268:
1263:
1251:
1249:
1248:
1243:
1225:
1223:
1222:
1217:
1205:
1203:
1202:
1197:
1182:
1180:
1179:
1174:
1172:
1167:
1158:
1156:
1155:
1150:
1138:
1136:
1135:
1130:
1128:
1123:
1108:
1106:
1105:
1100:
1083:
1080:
1072:
1071:
1059:
1031:
1029:
1028:
1023:
1006:
1003:
995:
994:
982:
952:
950:
949:
944:
920:
918:
917:
912:
910:
898:rational numbers
795:Ordering of cuts
778:
776:
775:
770:
768:
763:
746:
744:
743:
738:
736:
731:
721:rational numbers
690:rational numbers
582:
580:
579:
574:
562:
560:
559:
554:
536:
534:
533:
528:
510:
508:
507:
502:
481:
479:
478:
473:
461:
459:
458:
453:
435:
433:
432:
427:
409:
407:
406:
401:
383:
381:
380:
375:
373:
346:
344:
343:
338:
326:
324:
323:
318:
316:
296:
294:
293:
288:
273:
271:
270:
265:
253:
251:
250:
245:
233:
231:
230:
225:
223:
95:greatest element
64:rational numbers
52:Richard Dedekind
3125:
3124:
3120:
3119:
3118:
3116:
3115:
3114:
3090:
3089:
3088:
3083:
3069:Dyadic rational
3050:
3045:
3014:
2995:
2992:
2966:
2961:
2960:
2953:
2940:
2939:
2935:
2929:
2925:
2920:
2916:
2883: and
2865:
2837:
2836:
2809:
2808:
2787:
2786:
2759:
2754:
2753:
2734:
2733:
2714:
2713:
2711:
2707:
2692:
2685:
2684:
2677:
2670:
2649:
2648:
2644:
2628:
2627:
2623:
2618:
2504:
2498:
2485:
2479:surreal numbers
2475:
2473:Surreal numbers
2422:
2421:
2396:
2395:
2370:
2369:
2338:
2337:
2306:
2305:
2294:
2289:
2287:Generalizations
2261:
2260:
2241:
2240:
2214:
2213:
2188:
2187:
2162:
2161:
2130:
2129:
2110:
2109:
2090:
2089:
2070:
2069:
2050:
2049:
2018:
2017:
1998:
1997:
1994:
1986:is not rational
1964:
1963:
1955:
1933:
1932:
1913:
1912:
1893:
1892:
1858:
1847:
1846:
1813:
1790:
1789:
1762:
1757:
1756:
1731:
1730:
1729:, there exists
1705:
1704:
1673:
1672:
1671:, we show that
1641:
1640:
1570:
1569:
1538:
1537:
1518:
1517:
1493:
1479:
1466:
1465:
1438:
1433:
1432:
1407:
1406:
1387:
1386:
1359:
1354:
1353:
1334:
1333:
1274:
1273:
1254:
1253:
1228:
1227:
1208:
1207:
1188:
1187:
1161:
1160:
1141:
1140:
1117:
1116:
1081: and
1063:
1035:
1034:
986:
958:
957:
923:
922:
901:
900:
894:
888:
797:
757:
756:
725:
724:
711:For example if
597:
595:Representations
565:
564:
539:
538:
513:
512:
487:
486:
464:
463:
438:
437:
412:
411:
386:
385:
352:
351:
329:
328:
327:(equivalently,
301:
300:
279:
278:
256:
255:
236:
235:
214:
213:
210:
56:Joseph Bertrand
24:
17:
12:
11:
5:
3123:
3121:
3113:
3112:
3107:
3102:
3092:
3091:
3085:
3084:
3082:
3081:
3076:
3071:
3066:
3061:
3055:
3052:
3051:
3046:
3044:
3043:
3036:
3029:
3021:
3012:
3011:
2997:"Dedekind cut"
2991:
2990:External links
2988:
2987:
2986:
2965:
2962:
2959:
2958:
2951:
2933:
2923:
2914:
2912:
2911:
2900:
2897:
2894:
2891:
2888:
2880:
2877:
2872:
2868:
2864:
2860:
2856:
2853:
2850:
2847:
2844:
2822:
2819:
2816:
2795:
2774:
2771:
2766:
2762:
2741:
2721:
2705:
2695:. Section IV.
2675:
2668:
2642:
2620:
2619:
2617:
2614:
2500:Main article:
2497:
2494:
2474:
2471:
2454:are nonempty.
2435:
2432:
2429:
2409:
2406:
2403:
2383:
2380:
2377:
2357:
2354:
2351:
2348:
2345:
2325:
2322:
2319:
2316:
2313:
2293:
2290:
2288:
2285:
2268:
2248:
2221:
2201:
2198:
2195:
2175:
2172:
2169:
2149:
2146:
2143:
2140:
2137:
2117:
2097:
2077:
2068:are less than
2057:
2037:
2034:
2031:
2028:
2025:
2005:
1993:
1990:
1973:
1959: = 2
1940:
1920:
1900:
1878:
1875:
1870:
1865:
1861:
1857:
1854:
1834:
1831:
1828:
1825:
1820:
1816:
1812:
1809:
1806:
1803:
1800:
1797:
1777:
1774:
1769:
1765:
1744:
1741:
1738:
1718:
1715:
1712:
1692:
1689:
1686:
1683:
1680:
1660:
1657:
1654:
1651:
1648:
1628:
1625:
1622:
1619:
1616:
1613:
1610:
1607:
1604:
1601:
1598:
1595:
1592:
1589:
1586:
1583:
1580:
1577:
1557:
1554:
1551:
1548:
1545:
1525:
1502:
1499:
1496:
1491:
1488:
1485:
1482:
1476:
1473:
1453:
1450:
1445:
1441:
1420:
1417:
1414:
1394:
1374:
1371:
1366:
1362:
1341:
1321:
1318:
1315:
1312:
1309:
1305:
1301:
1298:
1291:
1284:
1281:
1261:
1241:
1238:
1235:
1215:
1195:
1170:
1148:
1126:
1110:
1109:
1098:
1095:
1092:
1089:
1086:
1078:
1075:
1070:
1066:
1062:
1058:
1054:
1051:
1048:
1045:
1042:
1032:
1021:
1018:
1015:
1012:
1009:
1004: or
1001:
998:
993:
989:
985:
981:
977:
974:
971:
968:
965:
942:
939:
936:
933:
930:
909:
887:
884:
796:
793:
766:
734:
662:the interval [
596:
593:
585:
584:
572:
552:
549:
546:
526:
523:
520:
500:
497:
494:
483:
471:
451:
448:
445:
425:
422:
419:
399:
396:
393:
372:
368:
365:
362:
359:
348:
336:
315:
311:
308:
298:
286:
263:
243:
222:
209:
206:
15:
13:
10:
9:
6:
4:
3:
2:
3122:
3111:
3108:
3106:
3103:
3101:
3098:
3097:
3095:
3080:
3077:
3075:
3072:
3070:
3067:
3065:
3062:
3060:
3057:
3056:
3053:
3049:
3042:
3037:
3035:
3030:
3028:
3023:
3022:
3019:
3015:
3008:
3004:
3003:
2998:
2994:
2993:
2989:
2984:
2980:
2979:0-486-21010-3
2976:
2972:
2968:
2967:
2963:
2954:
2952:0-444-70226-1
2948:
2944:
2937:
2934:
2927:
2924:
2918:
2915:
2898:
2892:
2889:
2886:
2878:
2875:
2870:
2866:
2862:
2854:
2851:
2845:
2842:
2835:
2834:
2820:
2817:
2814:
2772:
2769:
2764:
2760:
2739:
2719:
2709:
2706:
2702:
2700:
2691:
2690:
2682:
2680:
2676:
2671:
2665:
2661:
2657:
2653:
2646:
2643:
2639:
2634:
2633:
2625:
2622:
2615:
2613:
2611:
2607:
2603:
2599:
2595:
2591:
2587:
2583:
2579:
2575:
2571:
2567:
2563:
2559:
2555:
2551:
2547:
2542:
2540:
2536:
2532:
2528:
2525:
2521:
2517:
2513:
2509:
2503:
2495:
2493:
2489:
2484:
2480:
2472:
2470:
2468:
2464:
2460:
2455:
2453:
2449:
2433:
2430:
2427:
2407:
2404:
2401:
2381:
2378:
2375:
2355:
2352:
2349:
2346:
2343:
2320:
2317:
2314:
2303:
2299:
2291:
2286:
2284:
2282:
2266:
2246:
2238:
2233:
2219:
2199:
2196:
2193:
2173:
2170:
2167:
2144:
2141:
2138:
2115:
2095:
2075:
2055:
2032:
2029:
2026:
2003:
1991:
1989:
1987:
1971:
1958:
1952:
1938:
1918:
1898:
1876:
1873:
1868:
1863:
1859:
1855:
1852:
1832:
1829:
1826:
1823:
1818:
1814:
1810:
1807:
1804:
1801:
1798:
1795:
1775:
1772:
1767:
1763:
1742:
1739:
1736:
1716:
1713:
1710:
1690:
1687:
1684:
1681:
1678:
1658:
1655:
1652:
1649:
1646:
1626:
1623:
1620:
1617:
1614:
1611:
1608:
1605:
1602:
1599:
1596:
1590:
1587:
1584:
1581:
1578:
1575:
1555:
1552:
1549:
1546:
1543:
1523:
1500:
1497:
1494:
1489:
1486:
1483:
1480:
1474:
1471:
1464:. The choice
1451:
1448:
1443:
1439:
1418:
1415:
1412:
1392:
1372:
1369:
1364:
1360:
1339:
1316:
1313:
1310:
1307:
1299:
1296:
1282:
1259:
1239:
1236:
1233:
1213:
1193:
1184:
1168:
1146:
1124:
1115:
1096:
1090:
1087:
1084:
1076:
1073:
1068:
1064:
1060:
1052:
1049:
1043:
1040:
1033:
1019:
1013:
1010:
1007:
999:
996:
991:
987:
983:
975:
972:
966:
963:
956:
955:
954:
937:
934:
931:
899:
893:
885:
883:
881:
877:
873:
868:
866:
862:
858:
854:
850:
846:
842:
838:
834:
830:
826:
822:
818:
814:
810:
806:
802:
794:
792:
790:
786:
782:
764:
754:
750:
732:
722:
719:only contain
718:
714:
709:
707:
703:
699:
695:
691:
687:
682:
680:
676:
672:
669:
665:
661:
657:
653:
649:
645:
641:
637:
633:
629:
625:
620:
618:
614:
610:
606:
602:
594:
592:
590:
570:
550:
547:
544:
524:
521:
518:
498:
495:
492:
484:
469:
449:
446:
443:
423:
420:
417:
397:
394:
391:
366:
363:
360:
357:
349:
347:is nonempty).
334:
309:
306:
299:
284:
277:
276:
275:
261:
241:
207:
205:
203:
200:
196:
192:
188:
182:
180:
176:
172:
169:as well) and
168:
164:
161:implies that
160:
156:
152:
148:
144:
140:
136:
132:
127:
124:
120:
116:
112:
108:
104:
100:
96:
92:
88:
84:
80:
76:
73:
69:
65:
61:
57:
53:
49:
48:Dedekind cuts
45:
37:
33:
28:
22:
21:Fred Warmsley
3110:Real numbers
3100:Order theory
3074:Half-integer
3064:Dedekind cut
3063:
3013:
3000:
2970:
2942:
2936:
2926:
2917:
2708:
2698:
2696:
2688:
2654:. Springer.
2651:
2645:
2636:
2635:. page 203.
2631:
2624:
2609:
2605:
2601:
2597:
2593:
2588:.) Then the
2581:
2577:
2573:
2569:
2565:
2561:
2557:
2549:
2545:
2543:
2538:
2534:
2530:
2526:
2519:
2515:
2507:
2505:
2476:
2466:
2462:
2458:
2456:
2451:
2447:
2301:
2297:
2295:
2234:
1995:
1956:
1953:
1516:works, thus
1185:
1111:
895:
879:
875:
869:
864:
860:
857:greater than
856:
852:
848:
844:
840:
836:
832:
828:
824:
820:
816:
812:
808:
804:
800:
798:
784:
780:
752:
748:
716:
712:
710:
705:
701:
697:
693:
685:
683:
678:
674:
670:
667:
663:
659:
655:
647:
643:
639:
635:
631:
627:
623:
621:
616:
612:
608:
604:
600:
598:
586:
297:is nonempty.
211:
193:where every
186:
183:
174:
170:
166:
162:
158:
154:
150:
146:
142:
141:, such that
138:
134:
128:
122:
118:
114:
110:
102:
98:
93:contains no
90:
86:
82:
78:
74:
47:
41:
36:real numbers
2600:for which (
2486: [
2457:If neither
843:) is again
835:, the cut (
654:(−â,
195:real number
191:number line
44:mathematics
3094:Categories
2964:References
2699:irrational
2576:, and let
2516:completion
2336:such that
2304:is a pair
1226:, that is
890:See also:
638:, the set
537:such that
274:such that
208:Definition
97:. The set
32:irrational
3007:EMS Press
2983:available
2855:∈
2720:≥
2554:inclusion
2405:∈
2379:∈
2347:∪
2197:∈
2171:∈
1874:ϵ
1869:≤
1856:−
1827:ϵ
1811:−
1740:∈
1688:≥
1682:×
1650:×
1624:≥
1606:∈
1594:∀
1585:≤
1579:×
1553:≤
1547:×
1300:∈
1237:×
1088:≥
1074:≥
1053:∈
976:∈
845:less than
809:less than
673:the cut (
522:∈
496:∈
447:∈
421:∈
367:∈
310:≠
202:continuum
113:and
68:partition
62:from the
2931:account.
2522:means a
1891:for the
865:equal to
652:interval
199:complete
3059:Integer
3009:, 2001
2981:. Also
1845:, then
658:), and
650:is the
436:, then
2977:
2949:
2666:
2568:, let
2420:imply
1294:
1286:
410:, and
165:is in
89:, and
2693:(PDF)
2616:Notes
2533:into
2510:is a
2490:]
2186:and
2160:with
1405:with
1352:with
953:with
876:least
807:) as
634:) of
2975:ISBN
2947:ISBN
2890:>
2876:>
2807:and
2740:>
2664:ISBN
2638:....
2604:) =
2514:, a
2450:and
2431:<
2368:and
2300:, a
2259:and
1830:>
1799:>
1773:>
1714:<
1449:<
1431:and
1416:<
1370:<
1314:<
1011:<
997:<
783:and
715:and
700:and
611:and
548:>
395:<
254:and
137:and
77:and
72:sets
2785:in
2656:doi
2592:of
2564:of
2518:of
2467:gap
2302:cut
686:not
681:).
563:. (
485:If
462:. (
350:If
149:in
42:In
3096::
3005:,
2999:,
2678:^
2662:.
2492:.
2488:es
2394:,
2283:.
2232:.
1988:.
1755:,
863:,
859:,
851:,
839:,
815:,
803:,
791:.
677:,
630:,
603:,
591:.
384:,
181:.
157:â€
153:,
46:,
34:,
3040:e
3033:t
3026:v
2955:.
2899:.
2896:}
2893:0
2887:b
2879:2
2871:2
2867:b
2863::
2859:Q
2852:b
2849:{
2846:=
2843:B
2821:0
2818:=
2815:b
2794:Q
2773:2
2770:=
2765:2
2761:x
2672:.
2658::
2610:S
2606:A
2602:A
2598:A
2594:S
2582:A
2578:A
2574:A
2570:A
2566:S
2562:A
2558:S
2546:S
2535:L
2531:S
2527:L
2520:S
2508:S
2463:B
2459:A
2452:B
2448:A
2434:b
2428:a
2408:B
2402:b
2382:A
2376:a
2356:X
2353:=
2350:B
2344:A
2324:)
2321:B
2318:,
2315:A
2312:(
2298:X
2267:B
2247:A
2220:r
2200:B
2194:b
2174:A
2168:a
2148:)
2145:b
2142:,
2139:a
2136:(
2116:r
2096:B
2076:r
2056:A
2036:)
2033:B
2030:,
2027:A
2024:(
2004:r
1972:2
1957:b
1939:2
1919:A
1899:y
1877:2
1864:2
1860:y
1853:2
1833:0
1824:=
1819:2
1815:x
1808:2
1805:,
1802:0
1796:x
1776:r
1768:2
1764:x
1743:A
1737:x
1717:2
1711:r
1691:2
1685:A
1679:A
1659:2
1656:=
1653:A
1647:A
1627:0
1621:y
1618:,
1615:x
1612:,
1609:A
1603:y
1600:,
1597:x
1591:,
1588:2
1582:y
1576:x
1556:2
1550:A
1544:A
1524:A
1501:2
1498:+
1495:x
1490:2
1487:+
1484:x
1481:2
1475:=
1472:y
1452:2
1444:2
1440:y
1419:y
1413:x
1393:y
1373:2
1365:2
1361:x
1340:x
1320:}
1317:2
1311:x
1308:,
1304:Q
1297:x
1290:|
1283:x
1280:{
1260:2
1240:A
1234:A
1214:A
1194:A
1169:2
1147:A
1125:2
1097:.
1094:}
1091:0
1085:b
1077:2
1069:2
1065:b
1061::
1057:Q
1050:b
1047:{
1044:=
1041:B
1020:,
1017:}
1014:0
1008:a
1000:2
992:2
988:a
984::
980:Q
973:a
970:{
967:=
964:A
941:)
938:B
935:,
932:A
929:(
908:Q
880:S
853:D
849:C
847:(
841:B
837:A
833:B
829:D
825:C
821:A
817:D
813:C
805:B
801:A
785:B
781:A
765:2
753:B
749:A
733:2
717:B
713:A
706:b
702:B
698:A
694:b
679:B
675:A
668:b
664:b
660:B
656:b
648:A
644:b
640:B
636:S
632:B
628:A
624:S
617:A
613:B
609:A
605:B
601:A
571:A
551:x
545:y
525:A
519:y
499:A
493:x
470:A
450:A
444:x
424:A
418:y
398:y
392:x
371:Q
364:y
361:,
358:x
335:B
314:Q
307:A
285:A
262:B
242:A
221:Q
187:B
175:A
171:B
167:A
163:x
159:a
155:x
151:A
147:a
143:A
139:B
135:A
123:B
119:A
115:B
111:A
103:B
99:B
91:A
87:B
83:A
79:B
75:A
38:.
23:.
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