1077:. The pyramid's topmost row is a single room: room 1; its second row is rooms 2 and 3; and so on. The column formed by the set of rightmost rooms will correspond to the triangular numbers. Once they are filled (by the hotel's redistributed occupants), the remaining empty rooms form the shape of a pyramid exactly identical to the original shape. Thus, the process can be repeated for each infinite set. Doing this one at a time for each coach would require an infinite number of steps, but by using the prior formulas, a guest can determine what their room "will be" once their coach has been reached in the process, and can simply go there immediately.
25:
2400:
181:
2390:
1476:
As an added step in this process, one zero can be removed from each section of the number; in this example, the guest's new room is 101000011001 (decimal 2585). This ensures that every room could be filled by a hypothetical guest. If no infinite sets of guests arrive, then only rooms that are a power
142:
number of rooms. Initially every room is occupied, and yet new visitors arrive, each expecting their own room. A normal, finite hotel could not accommodate new guests once every room is full. However, it can be shown that the existing guests and newcomers — even an infinite number of them — can each
1459:
of prime numbers, resulting in very large room numbers even given small inputs. For example, the passenger in the second seat of the third bus on the second ferry (address 2-3-2) would raise the 2nd odd prime (5) to 49, which is the result of the 3rd odd prime (7) being raised to the power of his
849:
Unlike the prime powers solution, this one fills the hotel completely, and we can reconstruct a guest's original coach and seat by reversing the interleaving process. First add a leading zero if the room has an odd number of digits. Then de-interleave the number into two numbers: the coach number
1561:
function which maps the countably infinite set to the set of natural numbers, even if the countably infinite set contains the natural numbers. For example, the set of rational numbers—those numbers which can be written as a quotient of integers—contains the natural numbers as a subset, but is no
1509:. Thus, in an ordinary (finite) hotel with more than one room, the number of odd-numbered rooms is obviously smaller than the total number of rooms. However, in Hilbert's Grand Hotel, the quantity of odd-numbered rooms is not smaller than the total "number" of rooms. In mathematical terms, the
1472:
in which ones are used as separators at the start of each layer, while a number within a given layer (such as a guest's coach number) is represented with that many zeroes. Thus, a guest with the prior address 2-5-1-3-1 (five infinite layers) would go to room 10010000010100010 (decimal 295458).
151:
With one additional guest, the hotel can accommodate him or her and the existing guests if infinitely many guests simultaneously move rooms. The guest currently in room 1 moves to room 2, the guest currently in room 2 to room 3, and so on, moving every guest from their current room
1504:
Initially, this state of affairs might seem to be counter-intuitive. The properties of infinite collections of things are quite different from those of finite collections of things. The paradox of
Hilbert's Grand Hotel can be understood by using Cantor's theory of
1463:
The interleaving method can be used with three interleaved "strands" instead of two. The passenger with the address 2-3-2 would go to room 232, while the one with the address 4935-198-82217 would go to room #008,402,912,391,587 (the leading zeroes can be removed).
160:+1. The infinite hotel has no final room, so every guest has a room to go to. After this, room 1 is empty and the new guest can be moved into that room. By repeating this procedure, it is possible to make room for any finite number of new guests. In general, when
1467:
Anticipating the possibility of any number of layers of infinite guests, the hotel may wish to assign rooms such that no guest will need to move, no matter how many guests arrive afterward. One solution is to convert each arrival's address into a
850:
consists of the odd-numbered digits and the seat number is the even-numbered ones. Of course, the original encoding is arbitrary, and the roles of the two numbers can be reversed (seat-odd and coach-even), so long as it is applied consistently.
845:
the digits to produce a room number: its digits will be ----etc. The hotel (coach #0) guest in room number 1729 moves to room 01070209 (i.e., room 1,070,209). The passenger on seat 1234 of coach 789 goes to room 01728394 (i.e., room 1,728,394).
108:. It is demonstrated that a fully occupied hotel with infinitely many rooms may still accommodate additional guests, even infinitely many of them, and this process may be repeated infinitely often. The idea was introduced by
1147:
1010:
779:
1235:
1430:
1189:
1551:
1521:
of all rooms. Indeed, infinite sets are characterized as sets that have proper subsets of the same cardinality. For countable sets (sets with the same cardinality as the
1280:
903:
710:
665:
473:
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of countably infinite passengers each, by several different methods. Most methods depend on the seats in the coaches being already numbered (or use the
192:
number of new guests: just move the person occupying room 1 to room 2, the guest occupying room 2 to room 4, and, in general, the guest occupying room
1374:
arrive, each bearing an infinite number of coaches, each with an infinite number of passengers. This is a situation involving three "levels" of
675:, it is easy to see all people will have a room, while no two people will end up in the same room. For example, the person in room 2592 (
1836:
184:
By moving each guest to a room number which is twice that of their previous room, an infinite number of new guests can be accommodated
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bigger than the set of natural numbers since the rationals are countable: there is a bijection from the naturals to the rationals.
526:. This solution leaves certain rooms empty (which may or may not be useful to the hotel); specifically, all numbers that are not
1362:
th coach). Thus we have a function assigning each person to a room; furthermore, this assignment does not skip over any rooms.
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712:) was sitting in on the 4th coach, on the 5th seat. Like the prime powers method, this solution leaves certain rooms empty.
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2045:
2014:
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68:
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the number of vacancies created. It is easier to show, by an independent means, that the number of arrivals is also
39:
2429:
2270:
530:, such as 15 or 847, will no longer be occupied. (So, strictly speaking, this shows that the number of arrivals is
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Hilbert imagines a hypothetical hotel with rooms numbered 1, 2, 3, and so on with no upper limit. This is called a
230:
can be used to solve this problem. For each of these methods, consider a passenger's seat number on a coach to be
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1073:
This pairing function can be demonstrated visually by structuring the hotel as a one-room-deep, infinitely tall
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932:
718:
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671:=0 for the people already in the hotel, 1 for the first coach, etc.). Because every number has a unique
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2039:
1979:
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true. The statements "there is a guest to every room" and "no more guests can be accommodated" are not
542:, than to modify the algorithm to an exact fit.) (The algorithm works equally well if one interchanges
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126:
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204:), and all the odd-numbered rooms (which are countably infinite) will be free for the new guests.
33:
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861:
1597: – If there are more items than boxes holding them, one box must contain at least two items
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method can be applied by adding a new prime number for every additional layer of infinity (
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guests seek a room, the hotel can apply the same procedure and move every guest from room
8:
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This method can also easily be expanded for infinite nights, infinite entrances, etc. (
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837:. (Treat each hotel resident as being in coach #0.) If either number is shorter, add
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2122:
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David
Hilbert's Lectures on the Foundations of Arithmetics and Logic 1917-1933
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582:, but whichever choice is made, it must be applied uniformly throughout.)
2137:
1654:
1510:
1460:
seat number (2). This room number would have over thirty decimal digits.
219:
1517:
containing the odd-numbered rooms is the same as the cardinality of the
1070:. In this way all the rooms will be filled by one, and only one, guest.
1614:
Kragh, Helge (2014). "The True (?) Story of
Hilbert's Infinite Hotel".
1526:
1378:, and it can be solved by extensions of any of the previous solutions.
85:
1558:
1705:
180:
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th room (consider the guests already in the hotel as guests of the
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1737:
1659:
Infinity and the Mind. The
Science and Philosophy of the Infinite
1370:
Suppose the hotel is next to an ocean, and an infinite number of
1074:
834:
1574: – List of statements that appear to contradict themselves
1514:
1680:
Hilbert, David (2013), Ewald, William; Sieg, Wilfried (eds.),
1637:
One Two Three... Infinity: Facts and
Speculations of Science
1801:
1557:
Rephrased, for any countably infinite set, there exists a
841:
to it until both values have the same number of digits.
218:
It is possible to accommodate countably infinitely many
208:
Infinitely many coaches with infinitely many guests each
1532:
1455:
The prime power solution can be applied with further
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236:
1142:{\displaystyle S:=\{(a,b)\mid a,b\in \mathbb {N} \}}
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1444:
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1334:
1314:
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1191:is countable, hence we may enumerate its elements
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858:Those already in the hotel will be moved to room
2416:
538:the number of vacancies, and thus that they are
1080:
1721:
1365:
585:
175:
1136:
1098:
853:
789:For each passenger, compare the lengths of
369:, then put the first coach's load in rooms
310:are then fed into the two arguments of the
146:
143:have their own room in the infinite hotel.
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1728:
1714:
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1132:
69:Learn how and when to remove this message
179:
32:This article includes a list of general
1679:
1005:{\displaystyle ((c+n-1)^{2}+c+n-1)/2+n}
774:{\displaystyle 2^{s}3^{c}5^{n}7^{e}...}
117:
2417:
1653:
1501:when there are infinitely many rooms.
784:
317:
120:, p.730), and was popularized through
1735:
1709:
1647:
1634:
1613:
188:It is also possible to accommodate a
929:. Those in a coach will be in room
82:Hilbert's paradox of the Grand Hotel
18:
396:, the second coach's load in rooms
16:Thought experiment of infinite sets
13:
2046:What the Tortoise Said to Achilles
1534:
1230:{\displaystyle s_{1},s_{2},\dots }
38:it lacks sufficient corresponding
14:
2481:
2399:
2398:
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23:
1684:, Heidelberg: Springer-Verlag,
1425:{\displaystyle 2^{s}3^{c}5^{f}}
250:, and their coach number to be
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1037:
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939:
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865:
590:Each person of a certain seat
423:; in general for coach number
133:
1:
1601:
1586: – Paradox in set theory
829:as written in any positional
1184:{\displaystyle \mathbb {N} }
1081:Arbitrary enumeration method
7:
1661:. Paladin. pp. 73–78.
1565:
1546:{\displaystyle \aleph _{0}}
1480:
1275:{\displaystyle s_{n}=(a,b)}
898:{\displaystyle (n^{2}+n)/2}
10:
2486:
1366:Further layers of infinity
705:{\displaystyle 2^{5}3^{4}}
660:{\displaystyle 2^{s}3^{c}}
586:Prime factorization method
211:
176:Infinitely many new guests
2384:
2236:
2065:
1865:
1744:
1690:10.1007/978-3-540-69444-1
1580: – Geometric theorem
1477:of two will be occupied.
468:{\displaystyle p_{c}^{n}}
224:axiom of countable choice
127:One Two Three... Infinity
854:Triangular number method
536:greater than or equal to
147:Finitely many new guests
2430:Paradoxes of set theory
1965:Paradoxes of set theory
1590:Paradoxes of set theory
1485:Hilbert's paradox is a
1050:triangular number plus
1043:{\displaystyle (c+n-1)}
322:Send the guest in room
53:more precise citations.
2440:Mathematical paradoxes
1635:Gamow, George (1947).
1547:
1525:) this cardinality is
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90:Infinite Hotel Paradox
2455:Paradoxes of infinity
1578:Banach–Tarski paradox
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630:can be put into room
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605:
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532:less than or equal to
517:
497:
495:{\displaystyle p_{c}}
470:
438:
418:
416:{\displaystyle 5^{n}}
391:
389:{\displaystyle 3^{n}}
364:
362:{\displaystyle 2^{i}}
337:
305:
285:
265:
245:
212:Further information:
183:
2331:Kavka's toxin puzzle
2103:Income and fertility
1595:Pigeonhole principle
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100:which illustrates a
2425:Eponymous paradoxes
1990:Temperature paradox
1913:Free choice paradox
1777:Fitch's knowability
1507:transfinite numbers
1383:prime factorization
1169:is countable since
785:Interleaving method
673:prime factorization
464:
318:Prime powers method
114:Ăśber das Unendliche
112:in a 1925 lecture "
2460:1925 introductions
2366:Prisoner's dilemma
2052:Heat death paradox
2040:Unexpected hanging
2005:Chicken or the egg
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270:, and the numbers
260:
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226:). In general any
190:countably infinite
186:
140:countably infinite
98:thought experiment
2465:Logical paradoxes
2412:
2411:
2083:Arrow information
1699:978-3-540-20578-4
1584:Galileo's paradox
1572:List of paradoxes
1491:counter-intuitive
1487:veridical paradox
1445:{\displaystyle f}
1355:{\displaystyle 0}
1335:{\displaystyle n}
1315:{\displaystyle a}
1295:{\displaystyle b}
1162:{\displaystyle S}
1063:{\displaystyle n}
927:triangular number
918:{\displaystyle n}
822:{\displaystyle c}
802:{\displaystyle n}
623:{\displaystyle c}
603:{\displaystyle s}
575:{\displaystyle c}
555:{\displaystyle n}
515:{\displaystyle c}
443:we use the rooms
436:{\displaystyle c}
335:{\displaystyle i}
303:{\displaystyle c}
283:{\displaystyle n}
263:{\displaystyle c}
243:{\displaystyle n}
116:", reprinted in (
79:
78:
71:
2477:
2450:Fictional hotels
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2401:
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2203:Service recovery
2057:Olbers's paradox
1757:Buridan's bridge
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1489:: it leads to a
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1322:th coach to the
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1302:th guest of the
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228:pairing function
214:Pairing function
102:counterintuitive
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49:this article by
40:inline citations
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2291:Decision-making
2237:Decision theory
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1985:Hilbert's Hotel
1918:Grelling–Nelson
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1523:natural numbers
1493:result that is
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398:
397:
380:
376:
374:
371:
370:
353:
349:
347:
344:
343:
327:
324:
323:
320:
295:
292:
291:
275:
272:
271:
255:
252:
251:
235:
232:
231:
216:
210:
178:
149:
136:
94:Hilbert's Hotel
75:
64:
58:
55:
45:Please help to
44:
28:
24:
17:
12:
11:
5:
2483:
2473:
2472:
2467:
2462:
2457:
2452:
2447:
2442:
2437:
2432:
2427:
2410:
2409:
2407:
2406:
2396:
2385:
2382:
2381:
2379:
2378:
2373:
2368:
2363:
2358:
2353:
2348:
2343:
2338:
2333:
2328:
2323:
2318:
2313:
2308:
2303:
2298:
2293:
2288:
2283:
2278:
2273:
2268:
2267:
2266:
2261:
2256:
2246:
2240:
2238:
2234:
2233:
2231:
2230:
2225:
2220:
2215:
2210:
2208:St. Petersburg
2205:
2200:
2195:
2190:
2185:
2180:
2175:
2170:
2165:
2160:
2155:
2150:
2145:
2140:
2135:
2130:
2125:
2120:
2115:
2110:
2105:
2100:
2095:
2090:
2085:
2080:
2075:
2069:
2067:
2063:
2062:
2060:
2059:
2054:
2049:
2042:
2037:
2032:
2027:
2022:
2017:
2012:
2007:
2002:
1997:
1992:
1987:
1982:
1977:
1972:
1967:
1962:
1957:
1956:
1955:
1950:
1945:
1940:
1935:
1925:
1920:
1915:
1910:
1905:
1900:
1895:
1890:
1885:
1880:
1875:
1869:
1867:
1863:
1862:
1860:
1859:
1854:
1849:
1844:
1839:
1837:Rule-following
1834:
1829:
1824:
1819:
1814:
1809:
1804:
1799:
1794:
1789:
1784:
1779:
1774:
1769:
1764:
1762:Dream argument
1759:
1754:
1748:
1746:
1742:
1741:
1733:
1732:
1725:
1718:
1710:
1704:
1703:
1698:
1675:
1674:
1667:
1646:
1627:
1605:
1603:
1600:
1599:
1598:
1592:
1587:
1581:
1575:
1567:
1564:
1540:
1536:
1482:
1479:
1457:exponentiation
1441:
1419:
1415:
1409:
1405:
1399:
1395:
1367:
1364:
1351:
1331:
1311:
1291:
1271:
1268:
1265:
1262:
1259:
1256:
1251:
1247:
1226:
1223:
1218:
1214:
1210:
1205:
1201:
1179:
1158:
1138:
1134:
1130:
1127:
1124:
1121:
1118:
1115:
1112:
1109:
1106:
1103:
1100:
1097:
1094:
1082:
1079:
1059:
1039:
1036:
1033:
1030:
1027:
1024:
1021:
1001:
998:
995:
991:
987:
984:
981:
978:
975:
972:
969:
964:
960:
956:
953:
950:
947:
944:
941:
938:
914:
894:
890:
886:
883:
880:
875:
871:
867:
855:
852:
839:leading zeroes
831:numeral system
818:
798:
786:
783:
770:
767:
764:
759:
755:
749:
745:
739:
735:
729:
725:
699:
695:
689:
685:
654:
650:
644:
640:
619:
599:
587:
584:
571:
551:
511:
489:
485:
462:
457:
453:
432:
410:
406:
383:
379:
356:
352:
331:
319:
316:
299:
279:
259:
239:
209:
206:
177:
174:
148:
145:
135:
132:
77:
76:
31:
29:
22:
15:
9:
6:
4:
3:
2:
2482:
2471:
2470:David Hilbert
2468:
2466:
2463:
2461:
2458:
2456:
2453:
2451:
2448:
2446:
2443:
2441:
2438:
2436:
2433:
2431:
2428:
2426:
2423:
2422:
2420:
2405:
2397:
2395:
2387:
2386:
2383:
2377:
2374:
2372:
2369:
2367:
2364:
2362:
2359:
2357:
2354:
2352:
2349:
2347:
2344:
2342:
2339:
2337:
2336:Morton's fork
2334:
2332:
2329:
2327:
2324:
2322:
2319:
2317:
2314:
2312:
2309:
2307:
2304:
2302:
2299:
2297:
2294:
2292:
2289:
2287:
2284:
2282:
2279:
2277:
2276:Buridan's ass
2274:
2272:
2269:
2265:
2262:
2260:
2257:
2255:
2252:
2251:
2250:
2249:Apportionment
2247:
2245:
2242:
2241:
2239:
2235:
2229:
2226:
2224:
2221:
2219:
2216:
2214:
2211:
2209:
2206:
2204:
2201:
2199:
2196:
2194:
2191:
2189:
2186:
2184:
2181:
2179:
2176:
2174:
2171:
2169:
2166:
2164:
2161:
2159:
2156:
2154:
2151:
2149:
2146:
2144:
2141:
2139:
2136:
2134:
2131:
2129:
2126:
2124:
2121:
2119:
2116:
2114:
2111:
2109:
2108:Downs–Thomson
2106:
2104:
2101:
2099:
2096:
2094:
2091:
2089:
2086:
2084:
2081:
2079:
2076:
2074:
2071:
2070:
2068:
2064:
2058:
2055:
2053:
2050:
2047:
2043:
2041:
2038:
2036:
2033:
2031:
2028:
2026:
2025:Plato's beard
2023:
2021:
2018:
2016:
2013:
2011:
2008:
2006:
2003:
2001:
1998:
1996:
1993:
1991:
1988:
1986:
1983:
1981:
1978:
1976:
1973:
1971:
1968:
1966:
1963:
1961:
1958:
1954:
1951:
1949:
1946:
1944:
1941:
1939:
1936:
1934:
1931:
1930:
1929:
1926:
1924:
1923:Kleene–Rosser
1921:
1919:
1916:
1914:
1911:
1909:
1906:
1904:
1901:
1899:
1896:
1894:
1891:
1889:
1886:
1884:
1881:
1879:
1876:
1874:
1871:
1870:
1868:
1864:
1858:
1855:
1853:
1850:
1848:
1847:Theseus' ship
1845:
1843:
1840:
1838:
1835:
1833:
1830:
1828:
1825:
1823:
1820:
1818:
1815:
1813:
1810:
1808:
1807:Mere addition
1805:
1803:
1800:
1798:
1795:
1793:
1790:
1788:
1785:
1783:
1780:
1778:
1775:
1773:
1770:
1768:
1765:
1763:
1760:
1758:
1755:
1753:
1750:
1749:
1747:
1745:Philosophical
1743:
1739:
1731:
1726:
1724:
1719:
1717:
1712:
1711:
1708:
1701:
1695:
1691:
1687:
1683:
1678:
1677:
1670:
1668:0-586-08465-7
1664:
1660:
1656:
1650:
1643:. p. 17.
1642:
1638:
1631:
1622:
1617:
1610:
1606:
1596:
1593:
1591:
1588:
1585:
1582:
1579:
1576:
1573:
1570:
1569:
1563:
1560:
1555:
1553:
1538:
1524:
1520:
1516:
1512:
1508:
1502:
1500:
1496:
1492:
1488:
1478:
1474:
1471:
1470:binary number
1465:
1461:
1458:
1453:
1439:
1417:
1413:
1407:
1403:
1397:
1393:
1384:
1379:
1377:
1373:
1363:
1349:
1329:
1309:
1289:
1282:, assign the
1266:
1263:
1260:
1254:
1249:
1245:
1224:
1221:
1216:
1212:
1208:
1203:
1199:
1156:
1128:
1125:
1122:
1119:
1116:
1110:
1107:
1104:
1095:
1092:
1078:
1076:
1071:
1057:
1034:
1031:
1028:
1025:
1022:
999:
996:
993:
989:
982:
979:
976:
973:
970:
967:
962:
954:
951:
948:
945:
942:
928:
912:
892:
888:
881:
878:
873:
869:
851:
847:
844:
840:
836:
832:
816:
796:
782:
768:
765:
762:
757:
753:
747:
743:
737:
733:
727:
723:
713:
697:
693:
687:
683:
674:
670:
652:
648:
642:
638:
617:
597:
583:
569:
549:
541:
537:
533:
529:
525:
509:
487:
483:
460:
455:
451:
430:
408:
404:
381:
377:
354:
350:
329:
315:
313:
297:
277:
257:
237:
229:
225:
221:
215:
205:
203:
199:
195:
191:
182:
173:
171:
167:
163:
159:
155:
144:
141:
131:
129:
128:
124:'s 1947 book
123:
119:
115:
111:
110:David Hilbert
107:
106:infinite sets
103:
99:
95:
91:
87:
83:
73:
70:
62:
52:
48:
42:
41:
35:
30:
21:
20:
2356:Preparedness
2188:Productivity
2168:Mandeville's
1984:
1960:Opposite Day
1888:Burali-Forti
1883:Bhartrhari's
1681:
1658:
1655:Rucker, Rudy
1649:
1641:Viking Press
1639:. New York:
1636:
1630:
1609:
1556:
1503:
1484:
1475:
1466:
1462:
1454:
1452:the ferry).
1380:
1369:
1084:
1072:
857:
848:
788:
714:
668:
589:
539:
535:
531:
528:prime powers
524:prime number
321:
217:
201:
197:
193:
189:
187:
169:
165:
161:
157:
153:
150:
137:
125:
122:George Gamow
118:Hilbert 2013
113:
104:property of
93:
89:
81:
80:
65:
59:January 2024
56:
37:
2286:Condorcet's
2138:Giffen good
2098:Competition
1852:White horse
1827:Omnipotence
1511:cardinality
1372:car ferries
667:(presuming
134:The paradox
51:introducing
2435:Supertasks
2419:Categories
2361:Prevention
2351:Parrondo's
2341:Navigation
2326:Inventor's
2321:Hedgehog's
2281:Chainstore
2264:Population
2259:New states
2193:Prosperity
2173:Mayfield's
2015:Entailment
1995:Barbershop
1908:Epimenides
1602:References
1499:equivalent
843:Interleave
833:, such as
610:and coach
220:coachloads
86:colloquial
34:references
2376:Willpower
2371:Tolerance
2346:Newcomb's
2311:Fredkin's
2198:Scitovsky
2118:Edgeworth
2113:Easterlin
2078:Antitrust
1975:Russell's
1970:Richard's
1943:Pinocchio
1898:Crocodile
1817:Newcomb's
1787:Goodman's
1782:Free will
1767:Epicurean
1738:paradoxes
1657:(1984) .
1621:1403.0059
1559:bijective
1535:ℵ
1237:. Now if
1225:…
1129:∈
1117:∣
1032:−
1012:, or the
980:−
952:−
905:, or the
200:(2 times
196:to room 2
2445:Infinity
2404:Category
2301:Ellsberg
2153:Leontief
2133:Gibson's
2128:European
2123:Ellsberg
2093:Braess's
2088:Bertrand
2066:Economic
2000:Catch-22
1980:Socratic
1822:Nihilism
1792:Hedonism
1752:Analysis
1736:Notable
1566:See also
1495:provably
1481:Analysis
1376:infinity
342:to room
168:to room
156:to room
2306:Fenno's
2271:Arrow's
2254:Alabama
2244:Abilene
2223:Tullock
2178:Metzler
2020:Lottery
2010:Drinker
1953:Yablo's
1948:Quine's
1903:Curry's
1866:Logical
1842:Sorites
1832:Preface
1812:Moore's
1797:Liberal
1772:Fiction
1513:of the
1432:, with
1075:pyramid
835:decimal
522:th odd
502:is the
96:) is a
47:improve
2213:Thrift
2183:Plenty
2158:Lerner
2148:Jevons
2143:Icarus
2073:Allais
2035:Ross's
1873:Barber
1857:Zeno's
1802:Meno's
1696:
1665:
1515:subset
475:where
36:, but
2316:Green
2296:Downs
2228:Value
2163:Lucas
2030:Raven
1938:No-no
1893:Court
1878:Berry
1616:arXiv
540:equal
170:n + k
2394:List
2218:Toil
1933:Card
1928:Liar
1694:ISBN
1663:ISBN
1381:The
1085:Let
809:and
562:and
290:and
1686:doi
1519:set
925:th
92:or
2421::
1692:,
1554:.
1149:.
1096::=
781:)
314:.
172:.
130:.
88::
2048:"
2044:"
1729:e
1722:t
1715:v
1688::
1671:.
1624:.
1618::
1539:0
1440:f
1418:f
1414:5
1408:c
1404:3
1398:s
1394:2
1350:0
1330:n
1310:a
1290:b
1270:)
1267:b
1264:,
1261:a
1258:(
1255:=
1250:n
1246:s
1222:,
1217:2
1213:s
1209:,
1204:1
1200:s
1178:N
1157:S
1137:}
1133:N
1126:b
1123:,
1120:a
1114:)
1111:b
1108:,
1105:a
1102:(
1099:{
1093:S
1058:n
1038:)
1035:1
1029:n
1026:+
1023:c
1020:(
1000:n
997:+
994:2
990:/
986:)
983:1
977:n
974:+
971:c
968:+
963:2
959:)
955:1
949:n
946:+
943:c
940:(
937:(
913:n
893:2
889:/
885:)
882:n
879:+
874:2
870:n
866:(
817:c
797:n
769:.
766:.
763:.
758:e
754:7
748:n
744:5
738:c
734:3
728:s
724:2
698:4
694:3
688:5
684:2
669:c
653:c
649:3
643:s
639:2
618:c
598:s
570:c
550:n
510:c
488:c
484:p
461:n
456:c
452:p
431:c
409:n
405:5
382:n
378:3
355:i
351:2
330:i
298:c
278:n
258:c
238:n
202:n
198:n
194:n
166:n
162:k
158:n
154:n
84:(
72:)
66:(
61:)
57:(
43:.
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