3637:
2449:
We shall not prove that any models of the axioms are isomorphic. Such a proof can be found in any number of modern analysis or set theory textbooks. We will sketch the basic definitions and properties of a number of constructions, however, because each of these is important for both mathematical and
6687:
is finite. This defines an equivalence relation on the set of almost homomorphisms. Real numbers are defined as the equivalence classes of this relation. Alternatively, the almost homomorphisms taking only finitely many values form a subgroup, and the underlying additive group of the real number is
1300:
4850:
43:
between them. This results from the above definition and is independent of particular constructions. These isomorphisms allow identifying the results of the constructions, and, in practice, to forget which construction has been chosen.
6850:
write: "Few mathematical structures have undergone as many revisions or have been presented in as many guises as the real numbers. Every generation reexamines the reals in the light of its values and mathematical objectives."
5177:
4467:
4685:
4993:
5551:
6688:
the quotient group. To add real numbers defined this way we add the almost homomorphisms that represent them. Multiplication of real numbers corresponds to functional composition of almost homomorphisms. If
1394:
6519:
4599:
2550:
Normally, metrics are defined with real numbers as values, but this does not make the construction/definition circular, since all numbers that are implied (even implicitly) are rational numbers.
1202:
4226:
6685:
5326:
4378:
6428:
6563:
4731:
4300:
5089:
28:
that does not contain any smaller complete ordered field. Such a definition does not prove that such a complete ordered field exists, and the existence proof consists of constructing a
1431:
4497:
6591:
5836:
3935:
3878:
7519:
3066:
5783:
6833:
6307:
6169:
6107:
5990:
5699:
5868:
6064:
5995:
An advantage of this construction is that each real number corresponds to a unique cut. Furthermore, by relaxing the first two requirements of the definition of a cut, the
5604:
5444:
5420:
4545:
4184:
4140:
3823:
1676:
1599:
1105:
7035:
6353:
6269:
3626:
3604:
2906:
2884:
2534:
2512:
2431:
2402:
2371:
2343:
2217:
2085:
2059:
1954:
1914:
1828:
1796:
1762:
1736:
1072:
1035:
988:
931:
869:
808:
743:
698:
645:
611:
585:
544:
518:
481:
440:
379:
306:
221:
180:
80:
2470:
all occurred within a few years of each other. Each has advantages and disadvantages. A major motivation in all three cases was the instruction of mathematics students.
5467:
4726:
6020:
5916:
5747:
6040:
4112:
3739:
6764:
4086:
4060:
4013:
3987:
3961:
131:
669:
6907:
6617:
6244:
5937:
5889:
5720:
5666:
5625:
5368:
5347:
5221:
5200:
5035:
5014:
4894:
4873:
3706:
6804:
6784:
6732:
6212:
6189:
6138:
5957:
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5571:
5396:
4521:
4252:
4160:
4034:
3899:
3846:
3799:
3779:
3759:
6712:
3781:. By doing this we may think intuitively of a real number as being represented by the set of all smaller rational numbers. In more detail, a real number
35:
The article presents several such constructions. They are equivalent in the sense that, given the result of any two such constructions, there is a unique
6986:
6325:(meaning that no real number is infinitely large or infinitely small). This embedding is not unique, though it can be chosen in a canonical way.
6271:
of real numbers. This construction uses a non-principal ultrafilter over the set of natural numbers, the existence of which is guaranteed by the
1161:
satisfies the first three axioms, but not the fourth. In other words, models of the rational numbers are also models of the first three axioms.
6309:. Hence the resulting field is an ordered field. Completeness can be proved in a similar way to the construction from the Cauchy sequences.
5094:
4388:
56:
of the real numbers consists of defining them as the elements of a complete ordered field. This means the following: The real numbers form a
4609:
4899:
3012:
is represented by a Cauchy sequence of rational numbers. This representation is far from unique; every rational sequence that converges to
5484:
6898:
1168:, as it expresses a statement about collections of reals and not just individual such numbers. As such, the reals are not given by a
1305:
1687:
6433:
7645:
7597:
7506:
7456:
7398:
4550:
1295:{\displaystyle (\mathbb {R} ,0_{\mathbb {R} },1_{\mathbb {R} },+_{\mathbb {R} },\times _{\mathbb {R} },\leq _{\mathbb {R} })}
4193:
152:
an isomorphism, and thus, the real numbers can be used and manipulated, without referring to the method of construction.
6999:
4845:{\displaystyle A\times B:=\{a\times b:a\geq 0\land a\in A\land b\geq 0\land b\in B\}\cup \{x\in \mathrm {Q} :x<0\}}
2190:"If a set of reals precedes another set of reals, then there exists at least one real number separating the two sets."
7165:
6971:
6626:
5226:
4315:
6397:
6994:
6889:
As a reviewer of one noted: "The details are all included, but as usual they are tedious and not too instructive."
6524:
4257:
4306:, such a set can have no greatest element and thus fulfills the conditions for being a real number laid out above.
7435:
Knopfmacher, Arnold; Knopfmacher, John (1987). "A new construction of the real numbers (via infinite products)".
5052:
148:, which is proved by constructing such a structure. A consequence of the axioms is that this structure is unique
3024:. This reflects the observation that one can often use different sequences to approximate the same real number.
6947:
1402:
6963:
4472:
6593:.) Almost homomorphisms form an abelian group under pointwise addition. We say that two almost homomorphisms
6568:
6380:
6333:
A relatively less known construction allows to define real numbers using only the additive group of integers
5788:
2933:
Comparison between real numbers is obtained by defining the following comparison between Cauchy sequences:
1169:
3904:
7311:
3853:
5752:
3032:
6809:
6281:
6143:
6081:
5962:
5671:
7124:
6384:
5841:
7133:
6045:
5576:
5425:
5401:
4526:
4165:
4121:
3804:
3044:
3027:
The only real number axiom that does not follow easily from the definitions is the completeness of
1656:
1579:
1085:
7065:
7018:
6336:
6252:
3609:
3587:
2889:
2867:
2517:
2495:
2414:
2385:
2354:
2326:
2200:
2068:
2042:
1937:
1897:
1811:
1779:
1745:
1719:
1055:
1018:
971:
914:
852:
791:
726:
681:
628:
594:
568:
527:
501:
464:
423:
362:
289:
204:
163:
63:
7362:
4500:
2487:
5449:
5043:
4699:
2405:
1185:
1165:
770:
141:
29:
25:
7333:
6002:
5894:
5725:
7640:
7498:
7422:
6025:
4091:
3711:
2797:
if and only if the difference between them tends to zero; that is, for every rational number
341:
7077:
6737:
4065:
4039:
3992:
3966:
3940:
113:
7175:
5996:
4603:
2849:
1147:
654:
7251:
8:
6596:
6247:
6221:
5921:
5873:
5704:
5650:
5609:
5352:
5331:
5205:
5184:
5019:
4998:
4878:
4857:
3690:
1853:
835:
648:
183:
82:, containing two distinguished elements denoted 0 and 1, and on which are defined two
7606:
7557:
7475:
7294:
7277:
7273:
7152:. Graduate Texts in Mathematics. Vol. 188. New York: Springer-Verlag. p. 54.
6789:
6769:
6717:
6197:
6174:
6123:
6118:
5942:
5630:
5556:
5381:
4506:
4237:
4145:
4019:
3884:
3831:
3784:
3764:
3744:
3657:
709:
6691:
7561:
7502:
7491:
7479:
7394:
7376:
7357:
7327:
7161:
6368:
6322:
5474:
3641:
3636:
2857:
2558:
2434:
2087:. We now define two common English verbs in a particular way that suits our purpose:
1923:
1709:
1143:
1042:
57:
1153:
The axiom is crucial in the characterization of the reals. For example, the totally
7616:
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7465:
7371:
7323:
7153:
6360:
6075:
5478:
3681:
3544:
2467:
2459:
1773:
1705:
451:
83:
53:
3547:
can be translated to Cauchy sequences in a natural way. For example, the notation
2455:
7388:
7248:
7171:
6939:
6272:
2479:
2463:
2438:
1739:
555:
87:
7572:
7038:
6318:
4690:
4187:
1769:
1693:
1615:
410:
7621:
7592:
7470:
7451:
7157:
7634:
6215:
6192:
2408:
1697:
1154:
272:
40:
6114:
3653:
2483:
2451:
896:
Addition and multiplication are compatible with the order. In other words,
6901: β Mathematical viewpoint that existence proofs must be constructive
6836:
6110:
4382:
4303:
3645:
2486:
to converge is adding new points to the metric space in a process called
1891:
1079:
1038:
888:
36:
21:
17:
5172:{\displaystyle A/B:=\{a/b:a\in A\land b\in ({\textbf {Q}}\setminus B)\}}
4462:{\displaystyle A-B:=\{a-b:a\in A\land b\in ({\textbf {Q}}\setminus B)\}}
3684:. Real numbers can be constructed as Dedekind cuts of rational numbers.
3315:
This defines two Cauchy sequences of rationals, and so the real numbers
1199:
Saying that any two models are isomorphic means that for any two models
7553:
7520:"A new approach to the real numbers (motivated by continued fractions)"
6387:
to give a
Dedekind-complete ordered field by the IsarMathLib project.
4680:{\displaystyle -B:=\{a-b:a<0\land b\in ({\textbf {Q}}\setminus B)\}}
4234:
the rational numbers into the reals by identifying the rational number
1447:
7341:
de Bruijn, N.G. (1977). "Construction of the system of real numbers".
7299:
7282:
4988:{\displaystyle A\times B=-(A\times -B)=-(-A\times B)=(-A\times -B)\,}
4231:
3628:
is that this construction can be used for every other metric spaces.
1443:
1397:
7230:
6383:
became the basis for this construction. This construction has been
1433:
that preserves both the field operations and the order. Explicitly,
7611:
7527:
7150:
Lectures on the
Hyperreals: An introduction to nonstandard analysis
5375:
4309:
3564:
5546:{\displaystyle A=\{x\in {\textbf {Q}}:x<0\lor x\times x<2\}}
3413:. To see that it is a least upper bound, notice that the limit of
2852:
that is compatible with the operations defined above, and the set
1192:. Any two models are isomorphic; so, the real numbers are unique
145:
6367:, naming them after ancient Greek astronomer and mathematician
7125:"Hyperreals and a Brief Introduction to Non-Standard Analysis"
6714:
denotes the real number represented by an almost homomorphism
6328:
5647:
has no greatest element, i.e. that for any positive rational
1701:
1193:
149:
134:
3825:
of rational numbers that fulfills the following conditions:
2561:
of Cauchy sequences of rational numbers. That is, sequences
1389:{\displaystyle (S,0_{S},1_{S},+_{S},\times _{S},\leq _{S}),}
7411:
7334:
http://alexandria.tue.nl/repository/freearticles/597556.pdf
4036:
contains no greatest element. In other words, there is no
7356:
Faltin, F.; Metropolis, M.; Ross, B.; Rota, G.-C. (1975).
6839:
relation on the set of real numbers constructed this way.
6359:, who attributes this construction to unpublished work by
6278:
It turns out that the maximal ideal respects the order on
3080:. Substituting a larger value if necessary, we may assume
1188:
that satisfies the above axioms. Several models are given
5370:
to a positive number and then apply the definition above.
7355:
7236:
6908:
Decidability of first-order theories of the real numbers
6847:
6514:{\displaystyle \{f(n+m)-f(m)-f(n):n,m\in \mathbb {Z} \}}
6379:, Eudoxus's treatment of quantity using the behavior of
6113:. Here a hyperrational is by definition a ratio of two
5037:
to positive numbers and then apply the definition above.
7540:
Shenitzer, A (1987). "A topics course in mathematics".
7276:(2003). "A natural construction for the real numbers".
7194:
6903:
Pages displaying short descriptions of redirect targets
6899:
Constructivism (mathematics)#Example from real analysis
6069:
4594:{\displaystyle \{x:x\in {\textbf {Q}}\land x\notin B\}}
1802:, denoted by the infix operator +, and the constant 1.
1681:
7449:
7434:
6876:
6872:
5066:
1696:
of the real numbers and their arithmetic was given by
7393:. New York: Oxford University Press US. p. 274.
7021:
6854:
A number of other constructions have been given, by:
6812:
6792:
6772:
6740:
6720:
6694:
6629:
6599:
6571:
6527:
6436:
6400:
6339:
6284:
6255:
6224:
6200:
6177:
6146:
6126:
6084:
6048:
6028:
6005:
5965:
5945:
5924:
5897:
5876:
5844:
5791:
5755:
5728:
5707:
5674:
5653:
5633:
5612:
5579:
5559:
5487:
5452:
5428:
5404:
5384:
5355:
5334:
5229:
5208:
5187:
5097:
5055:
5022:
5001:
4902:
4881:
4860:
4734:
4702:
4612:
4553:
4529:
4509:
4475:
4391:
4318:
4260:
4240:
4196:
4168:
4148:
4124:
4094:
4068:
4042:
4022:
3995:
3969:
3943:
3907:
3887:
3856:
3834:
3807:
3787:
3767:
3747:
3714:
3693:
3612:
3590:
3047:
2892:
2870:
2520:
2498:
2473:
2417:
2388:
2357:
2329:
2203:
2071:
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1940:
1900:
1814:
1782:
1748:
1722:
1659:
1582:
1405:
1308:
1205:
1189:
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1058:
1021:
974:
917:
855:
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729:
684:
657:
631:
597:
571:
530:
504:
467:
426:
365:
292:
207:
166:
116:
66:
7452:"Two concrete new constructions of the real numbers"
6312:
5606:. However, neither claim is immediate. Showing that
3577:
are equivalent, i.e., their difference converges to
2650:. Here the vertical bars denote the absolute value.
1137:
20:, there are several equivalent ways of defining the
7218:
6321:. The real numbers form a maximal subfield that is
4221:{\displaystyle x\leq y\Leftrightarrow x\subseteq y}
2536:of the rational numbers with respect to the metric
2444:
186:under addition and multiplication. In other words,
7490:
7206:
7182:
7094:
7092:
7090:
7088:
7086:
7029:
6827:
6798:
6778:
6758:
6726:
6706:
6679:
6611:
6585:
6557:
6513:
6422:
6347:
6301:
6263:
6238:
6206:
6183:
6163:
6132:
6101:
6058:
6034:
6014:
5984:
5951:
5931:
5910:
5883:
5862:
5830:
5777:
5741:
5714:
5693:
5660:
5639:
5619:
5598:
5565:
5545:
5461:
5438:
5414:
5390:
5362:
5341:
5320:
5215:
5194:
5171:
5083:
5029:
5008:
4987:
4888:
4867:
4844:
4720:
4679:
4593:
4539:
4515:
4491:
4461:
4372:
4294:
4246:
4220:
4178:
4154:
4134:
4106:
4080:
4054:
4028:
4007:
3981:
3955:
3929:
3893:
3872:
3840:
3817:
3793:
3773:
3753:
3733:
3700:
3620:
3598:
3060:
2914:with the equivalence class of the Cauchy sequence
2900:
2878:
2528:
2506:
2425:
2396:
2365:
2337:
2211:
2079:
2053:
1948:
1908:
1822:
1790:
1756:
1730:
1670:
1593:
1425:
1388:
1294:
1099:
1066:
1029:
1015:in the following sense: every non-empty subset of
982:
925:
863:
802:
737:
692:
663:
639:
605:
579:
538:
512:
475:
434:
373:
300:
215:
174:
125:
74:
7293:Arthan, R.D. (2004). "The Eudoxus Real Numbers".
7104:
5553:. It can be seen from the definitions above that
2598:of rational numbers such that for every rational
7632:
6920:
5473:As an example of a Dedekind cut representing an
4142:of real numbers as the set of all Dedekind cuts
3708:as the representative of any given Dedekind cut
3631:
3558:is the equivalence class of the Cauchy sequence
7450:Knopfmacher, Arnold; Knopfmacher, John (1988).
7083:
7047:
6806:takes an infinite number of positive values on
6680:{\displaystyle \{f(n)-g(n):n\in \mathbb {Z} \}}
5321:{\displaystyle A/B=-(A/{-B})=-(-A/B)=-A/{-B}\,}
4373:{\displaystyle A+B:=\{a+b:a\in A\land b\in B\}}
3901:is closed downwards. In other words, for all
3482:is monotonic increasing it is easy to see that
7148:Goldblatt, Robert (1998). "Exercise 5.7 (4)".
6423:{\displaystyle f:\mathbb {Z} \to \mathbb {Z} }
5870:but to show equality requires showing that if
3092:is non-empty, we can choose a rational number
98:of real numbers and denoted respectively with
7312:"Defining reals without the use of rationals"
6558:{\displaystyle f(n)=\lfloor \alpha n\rfloor }
4295:{\displaystyle \{x\in {\textbf {Q}}:x<q\}}
4254:with the set of all smaller rational numbers
2411:under addition with distinguished element 1.
2035:To clarify the above statement somewhat, let
7593:"The real numbers-a survey of constructions"
7343:Nederl. Akad. Wetensch. Verslag Afd. Natuurk
6674:
6630:
6552:
6543:
6508:
6437:
5540:
5494:
5166:
5112:
4839:
4813:
4807:
4747:
4674:
4622:
4588:
4554:
4456:
4404:
4367:
4331:
4289:
4261:
2450:historical reasons. The first three, due to
1006:(preservation of order under multiplication)
7272:
7200:
6317:Every ordered field can be embedded in the
5084:{\displaystyle A\geq 0{\mbox{ and }}B>0}
621:= 1. (existence of multiplicative inverses)
6329:Construction from integers (Eudoxus reals)
6218:hyperrational numbers. The quotient ring
3687:For convenience we may take the lower set
2861:
133:Moreover, the following properties called
7620:
7610:
7539:
7469:
7375:
7340:
7309:
7298:
7281:
7147:
7023:
6862:
6858:
6815:
6670:
6579:
6504:
6416:
6408:
6372:
6341:
6295:
6257:
6157:
6140:of all limited (i.e. finite) elements in
6109:from the rational numbers by means of an
6095:
5981:
5928:
5907:
5880:
5827:
5771:
5738:
5711:
5690:
5657:
5616:
5595:
5359:
5338:
5317:
5212:
5191:
5026:
5005:
4984:
4885:
4864:
3730:
3697:
3614:
3592:
3050:
2894:
2872:
2522:
2500:
2419:
2390:
2359:
2331:
2205:
2073:
2047:
1942:
1902:
1816:
1784:
1750:
1724:
1661:
1584:
1426:{\displaystyle f\colon \mathbb {R} \to S}
1413:
1283:
1268:
1253:
1238:
1223:
1210:
1090:
1060:
1023:
976:
919:
857:
796:
731:
686:
633:
599:
573:
532:
506:
469:
428:
367:
294:
209:
168:
68:
6961:
4492:{\displaystyle {\textbf {Q}}\setminus B}
3635:
2679:can be added and multiplied as follows:
2514:is defined as the completion of the set
1142:Axiom 4, which requires the order to be
957:. (preservation of order under addition)
491:. (existence of multiplicative identity)
47:
7409:
7316:Indagationes Mathematicae (Proceedings)
7224:
7122:
6586:{\displaystyle \alpha \in \mathbb {R} }
5831:{\displaystyle y={\frac {2x+2}{x+2}}\,}
5398:of real numbers has any upper bound in
3640:Dedekind used his cut to construct the
3347:. It is easy to prove, by induction on
7633:
7570:
7517:
7358:"The real numbers as a wreath product"
7292:
7255:
7212:
7188:
6867:
6842:
6376:
6356:
5999:system may be obtained by associating
3560:(3, 3.1, 3.14, 3.141, 3.1415, ...)
7598:Rocky Mountain Journal of Mathematics
7590:
7457:Rocky Mountain Journal of Mathematics
7386:
7110:
6964:"Interactive Notes for Real Analysis"
6926:
6883:
6363:, refers to this construction as the
5422:, then it has a least upper bound in
7488:
7420:
7098:
7053:
6962:Saunders, Bonnie (August 21, 2015).
6877:Knopfmacher & Knopfmacher (1988)
6873:Knopfmacher & Knopfmacher (1987)
6565:is an almost homomorphism for every
6078:, one constructs the hyperrationals
6070:Construction using hyperreal numbers
3930:{\displaystyle x,y\in {\textbf {Q}}}
3676:is nonempty and closed upwards, and
3120:. Now define sequences of rationals
2379:. 1 < 1 + 1.
1688:Tarski's axiomatization of the reals
1682:Tarski's axiomatization of the reals
7037:with respect to other metrics, see
6051:
5505:
5431:
5407:
5223:is negative, we use the identities
5152:
4896:is negative, we use the identities
4660:
4571:
4532:
4478:
4442:
4272:
4171:
4127:
3922:
3873:{\displaystyle r\neq {\textbf {Q}}}
3865:
3810:
3035:. It can be proved as follows: Let
3006:By construction, every real number
13:
6987:"Axioms of the Real Number System"
6029:
6009:
5481:. This can be defined by the set
4823:
4606:is a special case of subtraction:
3672:is nonempty and closed downwards,
3018:is a Cauchy sequence representing
2810:such that for all natural numbers
2611:such that for all natural numbers
2478:A standard procedure to force all
2474:Construction from Cauchy sequences
1115:, such that for every upper bound
24:. One of them is that they form a
14:
7657:
6972:University of Illinois at Chicago
6313:Construction from surreal numbers
5778:{\displaystyle y\times y<2\,.}
5349:to a non-negative number and/or
5157:
4665:
4483:
4447:
4302:. Since the rational numbers are
2908:by identifying a rational number
2886:can be considered as a subset of
1138:On the least upper bound property
520:, there exists an element −
457:0 is not equal to 1, and for all
7123:Krakoff, Gianni (June 8, 2015).
6995:University of California, Irvine
6828:{\displaystyle \mathbb {Z} ^{+}}
6302:{\displaystyle ^{*}\mathbb {Q} }
6164:{\displaystyle ^{*}\mathbb {Q} }
6102:{\displaystyle ^{*}\mathbb {Q} }
5985:{\displaystyle r<x\times x\,}
5694:{\displaystyle x\times x<2\,}
4190:on the real numbers as follows:
2445:Explicit constructions of models
413:of multiplication over addition)
7573:"Update on the efficient reals"
7571:Street, Ross (September 2003).
7497:. New York: Springer. pp.
7265:
7242:
7141:
7116:
5863:{\displaystyle A\times A\leq 2}
2186:Axiom 3 can then be stated as:
344:of addition and multiplication)
275:of addition and multiplication)
32:that satisfies the definition.
7542:The Mathematical Intelligencer
7489:Pugh, Charles Chapman (2002).
7071:
7059:
7009:
6979:
6955:
6948:University of Colorado Boulder
6932:
6753:
6747:
6701:
6695:
6657:
6651:
6642:
6636:
6537:
6531:
6485:
6479:
6470:
6464:
6455:
6443:
6412:
5627:is real requires showing that
5292:
5275:
5266:
5247:
5163:
5147:
4981:
4963:
4957:
4942:
4933:
4918:
4671:
4655:
4453:
4437:
4206:
3727:
3715:
2862:all axioms of the real numbers
1417:
1380:
1309:
1289:
1206:
554:) = 0. (existence of additive
1:
7518:Rieger, Georg Johann (1982).
6913:
6059:{\displaystyle {\textbf {Q}}}
5599:{\displaystyle A\times A=2\,}
5439:{\displaystyle {\textbf {R}}}
5415:{\displaystyle {\textbf {R}}}
4540:{\displaystyle {\textbf {Q}}}
4179:{\displaystyle {\textbf {Q}}}
4135:{\displaystyle {\textbf {R}}}
3818:{\displaystyle {\textbf {Q}}}
3632:Construction by Dedekind cuts
3584:An advantage of constructing
3463:is a smaller upper bound for
3061:{\displaystyle \mathbb {R} '}
1671:{\displaystyle \mathbb {R} .}
1594:{\displaystyle \mathbb {R} .}
1100:{\displaystyle \mathbb {R} ,}
7646:Constructivism (mathematics)
7423:"Cauchy's construction of R"
7390:What is Mathematics, Really?
7377:10.1016/0001-8708(75)90115-2
7328:10.1016/1385-7258(76)90055-X
7030:{\displaystyle \mathbb {Q} }
6348:{\displaystyle \mathbb {Z} }
6264:{\displaystyle \mathbb {R} }
5891:is any rational number with
3621:{\displaystyle \mathbb {Q} }
3599:{\displaystyle \mathbb {R} }
3571:(0, 0.9, 0.99, 0.999,...)
3387:is never an upper bound for
2901:{\displaystyle \mathbb {R} }
2879:{\displaystyle \mathbb {Q} }
2529:{\displaystyle \mathbb {Q} }
2507:{\displaystyle \mathbb {R} }
2426:{\displaystyle \mathbb {R} }
2397:{\displaystyle \mathbb {R} }
2366:{\displaystyle \mathbb {R} }
2338:{\displaystyle \mathbb {R} }
2212:{\displaystyle \mathbb {R} }
2080:{\displaystyle \mathbb {R} }
2054:{\displaystyle \mathbb {R} }
1949:{\displaystyle \mathbb {R} }
1909:{\displaystyle \mathbb {R} }
1890:. In other words, "<" is
1823:{\displaystyle \mathbb {R} }
1791:{\displaystyle \mathbb {R} }
1757:{\displaystyle \mathbb {R} }
1731:{\displaystyle \mathbb {R} }
1704:shown below and a mere four
1175:
1067:{\displaystyle \mathbb {R} }
1030:{\displaystyle \mathbb {R} }
983:{\displaystyle \mathbb {R} }
926:{\displaystyle \mathbb {R} }
864:{\displaystyle \mathbb {R} }
803:{\displaystyle \mathbb {R} }
738:{\displaystyle \mathbb {R} }
693:{\displaystyle \mathbb {R} }
640:{\displaystyle \mathbb {R} }
606:{\displaystyle \mathbb {R} }
580:{\displaystyle \mathbb {R} }
539:{\displaystyle \mathbb {R} }
513:{\displaystyle \mathbb {R} }
476:{\displaystyle \mathbb {R} }
435:{\displaystyle \mathbb {R} }
374:{\displaystyle \mathbb {R} }
301:{\displaystyle \mathbb {R} }
216:{\displaystyle \mathbb {R} }
175:{\displaystyle \mathbb {R} }
90:; the operations are called
75:{\displaystyle \mathbb {R} }
7:
7132:Department of Mathematics,
6892:
5573:is a real number, and that
3530:is a least upper bound for
2967:or there exists an integer
1700:, consisting of only the 8
10:
7662:
7493:Real Mathematical Analysis
6355:with different versions.
3569:states that the sequences
3514:is not an upper bound for
3033:least upper bound property
2804:, there exists an integer
2605:, there exists an integer
1685:
587:, there exists an element
7622:10.1216/RMJ-2015-45-3-737
7471:10.1216/RMJ-1988-18-4-813
7158:10.1007/978-1-4612-0615-6
5918:, then there is positive
5479:positive square root of 2
5462:{\displaystyle \bigcup S}
4721:{\displaystyle A,B\geq 0}
4693:is less straightforward.
3801:is any subset of the set
3656:in an ordered field is a
3041:be a non-empty subset of
2134:if and only if for every
2094:if and only if for every
1926:. More formally, for all
1692:An alternative synthetic
1052:is a non-empty subset of
450:. (existence of additive
155:
106:; the binary relation is
7310:de Bruijn, N.G. (1976).
7134:University of Washington
6015:{\displaystyle -\infty }
5911:{\displaystyle r<2\,}
5742:{\displaystyle x<y\,}
2860:can be shown to satisfy
2382:These axioms imply that
1852:. That is, "<" is an
1170:first-order logic theory
1157:of the rational numbers
1111:has a least upper bound
140:The existence of such a
7363:Advances in Mathematics
6035:{\displaystyle \infty }
6022:with the empty set and
4107:{\displaystyle y\leq x}
3734:{\displaystyle (A,B)\,}
1164:Note that the axiom is
7387:Hersh, Reuben (1997).
7254:(84j:26002) review of
7031:
6829:
6800:
6780:
6760:
6759:{\displaystyle 0\leq }
6728:
6708:
6681:
6613:
6587:
6559:
6521:is finite. (Note that
6515:
6424:
6349:
6303:
6265:
6240:
6208:
6185:
6165:
6134:
6103:
6060:
6036:
6016:
5986:
5953:
5933:
5912:
5885:
5864:
5832:
5779:
5743:
5716:
5701:, there is a rational
5695:
5662:
5641:
5621:
5600:
5567:
5547:
5463:
5440:
5416:
5392:
5364:
5343:
5322:
5217:
5196:
5173:
5085:
5031:
5010:
4989:
4890:
4869:
4846:
4722:
4681:
4595:
4541:
4517:
4493:
4463:
4374:
4296:
4248:
4222:
4180:
4156:
4136:
4108:
4082:
4081:{\displaystyle y\in r}
4056:
4055:{\displaystyle x\in r}
4030:
4009:
4008:{\displaystyle x\in r}
3983:
3982:{\displaystyle y\in r}
3957:
3956:{\displaystyle x<y}
3931:
3895:
3874:
3842:
3819:
3795:
3775:
3761:completely determines
3755:
3735:
3702:
3649:
3622:
3600:
3407:is an upper bound for
3364:is an upper bound for
3222:is an upper bound for
3074:be an upper bound for
3062:
2902:
2880:
2530:
2508:
2427:
2398:
2367:
2339:
2213:
2081:
2055:
1980:, then there exists a
1950:
1910:
1824:
1792:
1758:
1732:
1672:
1595:
1427:
1390:
1296:
1186:mathematical structure
1101:
1068:
1031:
984:
927:
865:
804:
739:
694:
665:
641:
607:
581:
540:
514:
477:
436:
375:
302:
217:
176:
127:
126:{\displaystyle \leq .}
76:
30:mathematical structure
26:complete ordered field
7591:Weiss, Ittay (2015).
7032:
7005:on December 26, 2010.
6882:For an overview, see
6830:
6801:
6781:
6761:
6729:
6709:
6682:
6614:
6588:
6560:
6516:
6425:
6350:
6304:
6266:
6241:
6209:
6186:
6166:
6135:
6104:
6061:
6037:
6017:
5987:
5954:
5934:
5913:
5886:
5865:
5833:
5780:
5744:
5717:
5696:
5663:
5642:
5622:
5601:
5568:
5548:
5464:
5441:
5417:
5393:
5378:. If a nonempty set
5365:
5344:
5323:
5218:
5197:
5174:
5086:
5046:in a similar manner:
5032:
5011:
4990:
4891:
4870:
4847:
4723:
4682:
4596:
4542:
4518:
4494:
4464:
4375:
4297:
4249:
4223:
4181:
4157:
4137:
4109:
4083:
4057:
4031:
4010:
3984:
3958:
3932:
3896:
3875:
3843:
3820:
3796:
3776:
3756:
3736:
3703:
3639:
3623:
3606:as the completion of
3601:
3063:
2903:
2881:
2767:Two Cauchy sequences
2531:
2509:
2428:
2399:
2368:
2340:
2214:
2082:
2056:
1951:
1911:
1825:
1793:
1759:
1733:
1673:
1596:
1428:
1391:
1297:
1182:model of real numbers
1102:
1069:
1032:
985:
928:
866:
805:
740:
695:
666:
664:{\displaystyle \leq }
642:
608:
582:
541:
515:
478:
437:
376:
303:
218:
177:
128:
77:
48:Axiomatic definitions
7410:IsarMathLib (2022).
7019:
6848:Faltin et al. (1975)
6810:
6790:
6770:
6738:
6718:
6692:
6627:
6597:
6569:
6525:
6434:
6398:
6337:
6282:
6253:
6222:
6198:
6175:
6144:
6124:
6082:
6046:
6026:
6003:
5997:extended real number
5963:
5943:
5922:
5895:
5874:
5842:
5789:
5753:
5726:
5705:
5672:
5651:
5631:
5610:
5577:
5557:
5485:
5450:
5426:
5402:
5382:
5353:
5332:
5227:
5206:
5185:
5095:
5053:
5020:
4999:
4900:
4879:
4858:
4732:
4700:
4610:
4551:
4527:
4507:
4473:
4389:
4316:
4258:
4238:
4194:
4166:
4146:
4122:
4092:
4066:
4040:
4020:
3993:
3967:
3941:
3905:
3885:
3854:
3832:
3805:
3785:
3765:
3745:
3712:
3691:
3610:
3588:
3181:consider the number
3045:
2890:
2868:
2850:equivalence relation
2518:
2496:
2415:
2386:
2355:
2327:
2201:
2069:
2043:
1938:
1898:
1812:
1780:
1746:
1720:
1657:
1580:
1403:
1306:
1203:
1148:Archimedean property
1086:
1056:
1019:
972:
915:
853:
792:
727:
682:
655:
629:
595:
569:
528:
502:
465:
424:
363:
290:
205:
164:
114:
64:
54:axiomatic definition
7421:Kemp, Todd (2016).
7015:For completions of
6843:Other constructions
6835:. This defines the
6612:{\displaystyle f,g}
6392:almost homomorphism
6239:{\displaystyle B/I}
5932:{\displaystyle x\,}
5884:{\displaystyle r\,}
5715:{\displaystyle y\,}
5661:{\displaystyle x\,}
5620:{\displaystyle A\,}
5363:{\displaystyle B\,}
5342:{\displaystyle A\,}
5216:{\displaystyle B\,}
5195:{\displaystyle A\,}
5030:{\displaystyle B\,}
5009:{\displaystyle A\,}
4889:{\displaystyle B\,}
4868:{\displaystyle A\,}
4501:relative complement
3701:{\displaystyle A\,}
3086:is rational. Since
2858:equivalence classes
2162: ≠
2154: ∈
2146: ≠
2138: ∈
2106: ∈
2098: ∈
1854:asymmetric relation
1166:nonfirstorderizable
137:must be satisfied.
60:, commonly denoted
7554:10.1007/bf03023955
7237:Faltin et al. 1975
7027:
6825:
6796:
6776:
6756:
6724:
6704:
6677:
6609:
6583:
6555:
6511:
6430:such that the set
6420:
6345:
6299:
6261:
6236:
6204:
6181:
6161:
6130:
6099:
6056:
6032:
6012:
5982:
5949:
5929:
5908:
5881:
5860:
5828:
5775:
5739:
5712:
5691:
5658:
5637:
5617:
5596:
5563:
5543:
5477:, we may take the
5459:
5436:
5412:
5388:
5360:
5339:
5318:
5213:
5192:
5169:
5081:
5070:
5027:
5006:
4985:
4886:
4865:
4842:
4718:
4677:
4591:
4537:
4513:
4489:
4459:
4370:
4292:
4244:
4218:
4176:
4152:
4132:
4104:
4078:
4062:such that for all
4052:
4026:
4005:
3979:
3953:
3927:
3891:
3870:
3838:
3815:
3791:
3771:
3751:
3731:
3698:
3650:
3618:
3596:
3518:and so neither is
3058:
2955:if and only if
2898:
2876:
2526:
2504:
2423:
2394:
2363:
2335:
2209:
2195:Axioms of addition
2077:
2051:
1984:such that for all
1946:
1906:
1820:
1788:
1754:
1728:
1668:
1591:
1423:
1386:
1292:
1097:
1064:
1045:. In other words,
1027:
980:
923:
861:
800:
735:
690:
671:. In other words,
661:
637:
603:
577:
536:
510:
473:
432:
371:
298:
213:
172:
123:
72:
7508:978-0-387-95297-0
7400:978-0-19-513087-4
7066:Math 25 Exercises
6799:{\displaystyle f}
6779:{\displaystyle f}
6727:{\displaystyle f}
6385:formally verified
6369:Eudoxus of Cnidus
6207:{\displaystyle I}
6184:{\displaystyle B}
6133:{\displaystyle B}
6076:hyperreal numbers
6053:
5952:{\displaystyle A}
5825:
5640:{\displaystyle A}
5566:{\displaystyle A}
5507:
5475:irrational number
5446:that is equal to
5433:
5409:
5391:{\displaystyle S}
5154:
5069:
4662:
4573:
4534:
4516:{\displaystyle B}
4480:
4444:
4274:
4247:{\displaystyle q}
4173:
4155:{\displaystyle A}
4129:
4029:{\displaystyle r}
3924:
3894:{\displaystyle r}
3867:
3841:{\displaystyle r}
3812:
3794:{\displaystyle r}
3774:{\displaystyle B}
3754:{\displaystyle A}
2961:is equivalent to
2653:Cauchy sequences
2435:Dedekind-complete
2263:, there exists a
1924:Dedekind-complete
1870:, there exists a
1768:, denoted by the
1706:primitive notions
1144:Dedekind-complete
1043:least upper bound
84:binary operations
7653:
7626:
7624:
7614:
7585:
7583:
7582:
7577:
7565:
7534:
7524:
7512:
7496:
7483:
7473:
7444:
7437:Nieuw Arch. Wisk
7429:
7427:
7415:
7404:
7381:
7379:
7350:
7331:
7304:
7302:
7287:
7285:
7274:A'Campo, Norbert
7259:
7246:
7240:
7234:
7228:
7222:
7216:
7210:
7204:
7198:
7192:
7186:
7180:
7179:
7145:
7139:
7138:
7129:
7120:
7114:
7108:
7102:
7096:
7081:
7075:
7069:
7063:
7057:
7051:
7045:
7036:
7034:
7033:
7028:
7026:
7013:
7007:
7006:
7004:
6998:. Archived from
6991:
6983:
6977:
6976:
6968:
6959:
6953:
6952:
6944:
6936:
6930:
6924:
6904:
6863:de Bruijn (1977)
6859:de Bruijn (1976)
6834:
6832:
6831:
6826:
6824:
6823:
6818:
6805:
6803:
6802:
6797:
6785:
6783:
6782:
6777:
6765:
6763:
6762:
6757:
6733:
6731:
6730:
6725:
6713:
6711:
6710:
6707:{\displaystyle }
6705:
6686:
6684:
6683:
6678:
6673:
6618:
6616:
6615:
6610:
6592:
6590:
6589:
6584:
6582:
6564:
6562:
6561:
6556:
6520:
6518:
6517:
6512:
6507:
6429:
6427:
6426:
6421:
6419:
6411:
6373:Shenitzer (1987)
6361:Stephen Schanuel
6354:
6352:
6351:
6346:
6344:
6308:
6306:
6305:
6300:
6298:
6293:
6292:
6270:
6268:
6267:
6262:
6260:
6245:
6243:
6242:
6237:
6232:
6213:
6211:
6210:
6205:
6190:
6188:
6187:
6182:
6170:
6168:
6167:
6162:
6160:
6155:
6154:
6139:
6137:
6136:
6131:
6117:. Consider the
6108:
6106:
6105:
6100:
6098:
6093:
6092:
6065:
6063:
6062:
6057:
6055:
6054:
6041:
6039:
6038:
6033:
6021:
6019:
6018:
6013:
5991:
5989:
5988:
5983:
5958:
5956:
5955:
5950:
5938:
5936:
5935:
5930:
5917:
5915:
5914:
5909:
5890:
5888:
5887:
5882:
5869:
5867:
5866:
5861:
5837:
5835:
5834:
5829:
5826:
5824:
5813:
5799:
5784:
5782:
5781:
5776:
5748:
5746:
5745:
5740:
5721:
5719:
5718:
5713:
5700:
5698:
5697:
5692:
5667:
5665:
5664:
5659:
5646:
5644:
5643:
5638:
5626:
5624:
5623:
5618:
5605:
5603:
5602:
5597:
5572:
5570:
5569:
5564:
5552:
5550:
5549:
5544:
5509:
5508:
5468:
5466:
5465:
5460:
5445:
5443:
5442:
5437:
5435:
5434:
5421:
5419:
5418:
5413:
5411:
5410:
5397:
5395:
5394:
5389:
5369:
5367:
5366:
5361:
5348:
5346:
5345:
5340:
5327:
5325:
5324:
5319:
5316:
5308:
5288:
5265:
5257:
5237:
5222:
5220:
5219:
5214:
5201:
5199:
5198:
5193:
5178:
5176:
5175:
5170:
5156:
5155:
5122:
5105:
5090:
5088:
5087:
5082:
5071:
5067:
5036:
5034:
5033:
5028:
5015:
5013:
5012:
5007:
4994:
4992:
4991:
4986:
4895:
4893:
4892:
4887:
4874:
4872:
4871:
4866:
4851:
4849:
4848:
4843:
4826:
4727:
4725:
4724:
4719:
4686:
4684:
4683:
4678:
4664:
4663:
4600:
4598:
4597:
4592:
4575:
4574:
4546:
4544:
4543:
4538:
4536:
4535:
4522:
4520:
4519:
4514:
4498:
4496:
4495:
4490:
4482:
4481:
4468:
4466:
4465:
4460:
4446:
4445:
4379:
4377:
4376:
4371:
4301:
4299:
4298:
4293:
4276:
4275:
4253:
4251:
4250:
4245:
4227:
4225:
4224:
4219:
4185:
4183:
4182:
4177:
4175:
4174:
4161:
4159:
4158:
4153:
4141:
4139:
4138:
4133:
4131:
4130:
4118:We form the set
4113:
4111:
4110:
4105:
4087:
4085:
4084:
4079:
4061:
4059:
4058:
4053:
4035:
4033:
4032:
4027:
4014:
4012:
4011:
4006:
3988:
3986:
3985:
3980:
3962:
3960:
3959:
3954:
3936:
3934:
3933:
3928:
3926:
3925:
3900:
3898:
3897:
3892:
3879:
3877:
3876:
3871:
3869:
3868:
3847:
3845:
3844:
3839:
3824:
3822:
3821:
3816:
3814:
3813:
3800:
3798:
3797:
3792:
3780:
3778:
3777:
3772:
3760:
3758:
3757:
3752:
3740:
3738:
3737:
3732:
3707:
3705:
3704:
3699:
3682:greatest element
3627:
3625:
3624:
3619:
3617:
3605:
3603:
3602:
3597:
3595:
3580:
3576:
3575:(1, 1, 1, 1,...)
3572:
3568:
3561:
3557:
3553:
3545:decimal notation
3539:
3535:
3529:
3523:
3517:
3513:
3502:
3496:
3481:
3468:
3462:
3448:
3438:
3434:
3412:
3406:
3398:
3392:
3386:
3375:
3369:
3363:
3352:
3346:
3330:
3311:
3290:
3270:. Otherwise set
3269:
3248:
3227:
3221:
3210:
3180:
3174:
3161:
3145:
3132:
3119:
3113:
3107:
3097:
3091:
3085:
3079:
3073:
3067:
3065:
3064:
3059:
3057:
3053:
3040:
3030:
3023:
3017:
3011:
3002:
2992:
2972:
2966:
2960:
2954:
2929:
2913:
2907:
2905:
2904:
2899:
2897:
2885:
2883:
2882:
2877:
2875:
2848:This defines an
2844:
2819:
2809:
2803:
2792:
2779:
2762:
2721:
2678:
2665:
2649:
2624:
2610:
2604:
2594:
2549:
2547:
2535:
2533:
2532:
2527:
2525:
2513:
2511:
2510:
2505:
2503:
2480:Cauchy sequences
2468:Karl Weierstrass
2460:Richard Dedekind
2432:
2430:
2429:
2424:
2422:
2406:linearly ordered
2403:
2401:
2400:
2395:
2393:
2372:
2370:
2369:
2364:
2362:
2351:. 1 β
2344:
2342:
2341:
2336:
2334:
2313: <
2305: <
2293: <
2218:
2216:
2215:
2210:
2208:
2178: <
2170: <
2123:The real number
2114: <
2086:
2084:
2083:
2078:
2076:
2060:
2058:
2057:
2052:
2050:
2028: <
2020: <
1976: <
1955:
1953:
1952:
1947:
1945:
1915:
1913:
1912:
1907:
1905:
1886: <
1878: <
1866: <
1829:
1827:
1826:
1821:
1819:
1797:
1795:
1794:
1789:
1787:
1774:binary operation
1763:
1761:
1760:
1755:
1753:
1737:
1735:
1734:
1729:
1727:
1714:the real numbers
1677:
1675:
1674:
1669:
1664:
1652:
1646:
1640:
1614:
1600:
1598:
1597:
1592:
1587:
1575:
1569:
1563:
1524:
1483:
1467:
1441:
1432:
1430:
1429:
1424:
1416:
1395:
1393:
1392:
1387:
1379:
1378:
1366:
1365:
1353:
1352:
1340:
1339:
1327:
1326:
1301:
1299:
1298:
1293:
1288:
1287:
1286:
1273:
1272:
1271:
1258:
1257:
1256:
1243:
1242:
1241:
1228:
1227:
1226:
1213:
1106:
1104:
1103:
1098:
1093:
1073:
1071:
1070:
1065:
1063:
1036:
1034:
1033:
1028:
1026:
989:
987:
986:
981:
979:
932:
930:
929:
924:
922:
870:
868:
867:
862:
860:
809:
807:
806:
801:
799:
744:
742:
741:
736:
734:
699:
697:
696:
691:
689:
670:
668:
667:
662:
646:
644:
643:
638:
636:
612:
610:
609:
604:
602:
586:
584:
583:
578:
576:
545:
543:
542:
537:
535:
519:
517:
516:
511:
509:
482:
480:
479:
474:
472:
441:
439:
438:
433:
431:
380:
378:
377:
372:
370:
307:
305:
304:
299:
297:
222:
220:
219:
214:
212:
181:
179:
178:
173:
171:
132:
130:
129:
124:
105:
101:
81:
79:
78:
73:
71:
7661:
7660:
7656:
7655:
7654:
7652:
7651:
7650:
7631:
7630:
7629:
7580:
7578:
7575:
7522:
7509:
7425:
7401:
7268:
7263:
7262:
7247:
7243:
7235:
7231:
7223:
7219:
7211:
7207:
7199:
7195:
7187:
7183:
7168:
7146:
7142:
7127:
7121:
7117:
7109:
7105:
7097:
7084:
7076:
7072:
7064:
7060:
7052:
7048:
7022:
7020:
7017:
7016:
7014:
7010:
7002:
6989:
6985:
6984:
6980:
6966:
6960:
6956:
6942:
6938:
6937:
6933:
6925:
6921:
6916:
6902:
6895:
6845:
6819:
6814:
6813:
6811:
6808:
6807:
6791:
6788:
6787:
6771:
6768:
6767:
6739:
6736:
6735:
6719:
6716:
6715:
6693:
6690:
6689:
6669:
6628:
6625:
6624:
6598:
6595:
6594:
6578:
6570:
6567:
6566:
6526:
6523:
6522:
6503:
6435:
6432:
6431:
6415:
6407:
6399:
6396:
6395:
6371:. As noted by
6340:
6338:
6335:
6334:
6331:
6319:surreal numbers
6315:
6294:
6288:
6285:
6283:
6280:
6279:
6273:axiom of choice
6256:
6254:
6251:
6250:
6228:
6223:
6220:
6219:
6199:
6196:
6195:
6176:
6173:
6172:
6156:
6150:
6147:
6145:
6142:
6141:
6125:
6122:
6121:
6094:
6088:
6085:
6083:
6080:
6079:
6072:
6050:
6049:
6047:
6044:
6043:
6027:
6024:
6023:
6004:
6001:
6000:
5964:
5961:
5960:
5944:
5941:
5940:
5923:
5920:
5919:
5896:
5893:
5892:
5875:
5872:
5871:
5843:
5840:
5839:
5814:
5800:
5798:
5790:
5787:
5786:
5754:
5751:
5750:
5727:
5724:
5723:
5706:
5703:
5702:
5673:
5670:
5669:
5652:
5649:
5648:
5632:
5629:
5628:
5611:
5608:
5607:
5578:
5575:
5574:
5558:
5555:
5554:
5504:
5503:
5486:
5483:
5482:
5451:
5448:
5447:
5430:
5429:
5427:
5424:
5423:
5406:
5405:
5403:
5400:
5399:
5383:
5380:
5379:
5354:
5351:
5350:
5333:
5330:
5329:
5309:
5304:
5284:
5258:
5253:
5233:
5228:
5225:
5224:
5207:
5204:
5203:
5186:
5183:
5182:
5151:
5150:
5118:
5101:
5096:
5093:
5092:
5068: and
5065:
5054:
5051:
5050:
5021:
5018:
5017:
5000:
4997:
4996:
4901:
4898:
4897:
4880:
4877:
4876:
4859:
4856:
4855:
4822:
4733:
4730:
4729:
4701:
4698:
4697:
4659:
4658:
4611:
4608:
4607:
4570:
4569:
4552:
4549:
4548:
4531:
4530:
4528:
4525:
4524:
4508:
4505:
4504:
4477:
4476:
4474:
4471:
4470:
4441:
4440:
4390:
4387:
4386:
4317:
4314:
4313:
4271:
4270:
4259:
4256:
4255:
4239:
4236:
4235:
4195:
4192:
4191:
4186:, and define a
4170:
4169:
4167:
4164:
4163:
4147:
4144:
4143:
4126:
4125:
4123:
4120:
4119:
4093:
4090:
4089:
4067:
4064:
4063:
4041:
4038:
4037:
4021:
4018:
4017:
3994:
3991:
3990:
3968:
3965:
3964:
3942:
3939:
3938:
3921:
3920:
3906:
3903:
3902:
3886:
3883:
3882:
3864:
3863:
3855:
3852:
3851:
3833:
3830:
3829:
3809:
3808:
3806:
3803:
3802:
3786:
3783:
3782:
3766:
3763:
3762:
3746:
3743:
3742:
3713:
3710:
3709:
3692:
3689:
3688:
3634:
3613:
3611:
3608:
3607:
3591:
3589:
3586:
3585:
3578:
3574:
3570:
3563:
3562:. The equation
3559:
3555:
3548:
3537:
3531:
3525:
3519:
3515:
3512:
3504:
3498:
3495:
3483:
3479:
3470:
3464:
3450:
3440:
3436:
3432:
3423:
3414:
3408:
3402:
3394:
3388:
3385:
3377:
3371:
3365:
3362:
3354:
3348:
3344:
3332:
3328:
3316:
3310:
3301:
3292:
3289:
3280:
3271:
3268:
3259:
3250:
3247:
3238:
3229:
3223:
3220:
3212:
3208:
3199:
3190:
3182:
3176:
3169:
3163:
3156:
3150:
3143:
3134:
3130:
3121:
3115:
3109:
3099:
3093:
3087:
3081:
3075:
3069:
3049:
3048:
3046:
3043:
3042:
3036:
3028:
3019:
3013:
3007:
2994:
2991:
2982:
2974:
2968:
2962:
2956:
2952:
2943:
2934:
2915:
2909:
2893:
2891:
2888:
2887:
2871:
2869:
2866:
2865:
2839:
2830:
2821:
2811:
2805:
2798:
2790:
2781:
2777:
2768:
2760:
2751:
2742:
2733:
2724:
2719:
2710:
2701:
2692:
2683:
2676:
2667:
2663:
2654:
2644:
2635:
2626:
2612:
2606:
2599:
2592:
2583:
2574:
2565:
2539:
2537:
2521:
2519:
2516:
2515:
2499:
2497:
2494:
2493:
2476:
2464:Joseph Bertrand
2447:
2418:
2416:
2413:
2412:
2389:
2387:
2384:
2383:
2358:
2356:
2353:
2352:
2345:, <, +, 1):
2330:
2328:
2325:
2324:
2237:) = (
2204:
2202:
2199:
2198:
2072:
2070:
2067:
2066:
2046:
2044:
2041:
2040:
1941:
1939:
1936:
1935:
1901:
1899:
1896:
1895:
1815:
1813:
1810:
1809:
1806:Axioms of order
1783:
1781:
1778:
1777:
1749:
1747:
1744:
1743:
1740:binary relation
1723:
1721:
1718:
1717:
1690:
1684:
1660:
1658:
1655:
1654:
1648:
1642:
1631:
1618:
1610:
1603:
1583:
1581:
1578:
1577:
1571:
1565:
1554:
1537:
1526:
1515:
1498:
1487:
1482:
1476:
1469:
1466:
1460:
1453:
1437:
1412:
1404:
1401:
1400:
1374:
1370:
1361:
1357:
1348:
1344:
1335:
1331:
1322:
1318:
1307:
1304:
1303:
1282:
1281:
1277:
1267:
1266:
1262:
1252:
1251:
1247:
1237:
1236:
1232:
1222:
1221:
1217:
1209:
1204:
1201:
1200:
1178:
1140:
1089:
1087:
1084:
1083:
1059:
1057:
1054:
1053:
1022:
1020:
1017:
1016:
1011:The order β€ is
975:
973:
970:
969:
918:
916:
913:
912:
856:
854:
851:
850:
795:
793:
790:
789:
730:
728:
725:
724:
685:
683:
680:
679:
656:
653:
652:
649:totally ordered
632:
630:
627:
626:
598:
596:
593:
592:
572:
570:
567:
566:
531:
529:
526:
525:
505:
503:
500:
499:
468:
466:
463:
462:
427:
425:
422:
421:
366:
364:
361:
360:
293:
291:
288:
287:
208:
206:
203:
202:
167:
165:
162:
161:
158:
115:
112:
111:
103:
99:
88:binary relation
67:
65:
62:
61:
50:
12:
11:
5:
7659:
7649:
7648:
7643:
7628:
7627:
7605:(3): 737β762.
7587:
7586:
7567:
7566:
7536:
7535:
7514:
7513:
7507:
7485:
7484:
7464:(4): 813β824.
7446:
7445:
7431:
7430:
7417:
7416:
7406:
7405:
7399:
7383:
7382:
7370:(3): 278β304.
7352:
7351:
7337:
7336:
7322:(2): 100β108.
7306:
7305:
7289:
7288:
7269:
7267:
7264:
7261:
7260:
7241:
7229:
7217:
7205:
7193:
7181:
7166:
7140:
7115:
7103:
7082:
7070:
7058:
7046:
7025:
7008:
6978:
6954:
6940:"Real Numbers"
6931:
6918:
6917:
6915:
6912:
6911:
6910:
6905:
6894:
6891:
6880:
6879:
6870:
6865:
6844:
6841:
6822:
6817:
6795:
6786:is bounded or
6775:
6755:
6752:
6749:
6746:
6743:
6723:
6703:
6700:
6697:
6676:
6672:
6668:
6665:
6662:
6659:
6656:
6653:
6650:
6647:
6644:
6641:
6638:
6635:
6632:
6608:
6605:
6602:
6581:
6577:
6574:
6554:
6551:
6548:
6545:
6542:
6539:
6536:
6533:
6530:
6510:
6506:
6502:
6499:
6496:
6493:
6490:
6487:
6484:
6481:
6478:
6475:
6472:
6469:
6466:
6463:
6460:
6457:
6454:
6451:
6448:
6445:
6442:
6439:
6418:
6414:
6410:
6406:
6403:
6343:
6330:
6327:
6314:
6311:
6297:
6291:
6287:
6259:
6235:
6231:
6227:
6203:
6180:
6159:
6153:
6149:
6129:
6097:
6091:
6087:
6071:
6068:
6031:
6011:
6008:
5980:
5977:
5974:
5971:
5968:
5948:
5927:
5906:
5903:
5900:
5879:
5859:
5856:
5853:
5850:
5847:
5838:works. Then
5823:
5820:
5817:
5812:
5809:
5806:
5803:
5797:
5794:
5774:
5770:
5767:
5764:
5761:
5758:
5737:
5734:
5731:
5710:
5689:
5686:
5683:
5680:
5677:
5656:
5636:
5615:
5594:
5591:
5588:
5585:
5582:
5562:
5542:
5539:
5536:
5533:
5530:
5527:
5524:
5521:
5518:
5515:
5512:
5502:
5499:
5496:
5493:
5490:
5471:
5470:
5458:
5455:
5387:
5373:
5372:
5371:
5358:
5337:
5315:
5312:
5307:
5303:
5300:
5297:
5294:
5291:
5287:
5283:
5280:
5277:
5274:
5271:
5268:
5264:
5261:
5256:
5252:
5249:
5246:
5243:
5240:
5236:
5232:
5211:
5190:
5179:
5168:
5165:
5162:
5159:
5149:
5146:
5143:
5140:
5137:
5134:
5131:
5128:
5125:
5121:
5117:
5114:
5111:
5108:
5104:
5100:
5080:
5077:
5074:
5064:
5061:
5058:
5040:
5039:
5038:
5025:
5004:
4983:
4980:
4977:
4974:
4971:
4968:
4965:
4962:
4959:
4956:
4953:
4950:
4947:
4944:
4941:
4938:
4935:
4932:
4929:
4926:
4923:
4920:
4917:
4914:
4911:
4908:
4905:
4884:
4863:
4852:
4841:
4838:
4835:
4832:
4829:
4825:
4821:
4818:
4815:
4812:
4809:
4806:
4803:
4800:
4797:
4794:
4791:
4788:
4785:
4782:
4779:
4776:
4773:
4770:
4767:
4764:
4761:
4758:
4755:
4752:
4749:
4746:
4743:
4740:
4737:
4717:
4714:
4711:
4708:
4705:
4691:multiplication
4687:
4676:
4673:
4670:
4667:
4657:
4654:
4651:
4648:
4645:
4642:
4639:
4636:
4633:
4630:
4627:
4624:
4621:
4618:
4615:
4601:
4590:
4587:
4584:
4581:
4578:
4568:
4565:
4562:
4559:
4556:
4512:
4488:
4485:
4458:
4455:
4452:
4449:
4439:
4436:
4433:
4430:
4427:
4424:
4421:
4418:
4415:
4412:
4409:
4406:
4403:
4400:
4397:
4394:
4380:
4369:
4366:
4363:
4360:
4357:
4354:
4351:
4348:
4345:
4342:
4339:
4336:
4333:
4330:
4327:
4324:
4321:
4307:
4291:
4288:
4285:
4282:
4279:
4269:
4266:
4263:
4243:
4228:
4217:
4214:
4211:
4208:
4205:
4202:
4199:
4188:total ordering
4151:
4115:
4114:
4103:
4100:
4097:
4077:
4074:
4071:
4051:
4048:
4045:
4025:
4015:
4004:
4001:
3998:
3978:
3975:
3972:
3952:
3949:
3946:
3919:
3916:
3913:
3910:
3890:
3880:
3862:
3859:
3849:
3837:
3790:
3770:
3750:
3729:
3726:
3723:
3720:
3717:
3696:
3633:
3630:
3616:
3594:
3508:
3491:
3475:
3449:. Now suppose
3428:
3419:
3381:
3358:
3340:
3324:
3306:
3296:
3285:
3275:
3264:
3254:
3243:
3233:
3216:
3204:
3195:
3186:
3167:
3154:
3139:
3126:
3056:
3052:
2987:
2978:
2948:
2939:
2896:
2874:
2835:
2826:
2786:
2773:
2765:
2764:
2756:
2747:
2738:
2729:
2722:
2715:
2706:
2697:
2688:
2672:
2659:
2640:
2631:
2596:
2595:
2588:
2579:
2570:
2524:
2502:
2475:
2472:
2446:
2443:
2421:
2392:
2361:
2333:
2321:Axioms for one
2245:) +
2229: + (
2207:
2192:
2191:
2184:
2183:
2120:
2119:
2075:
2049:
1944:
1904:
1818:
1786:
1770:infix operator
1752:
1726:
1694:axiomatization
1686:Main article:
1683:
1680:
1679:
1678:
1667:
1663:
1627:
1616:if and only if
1608:
1601:
1590:
1586:
1550:
1535:
1511:
1496:
1485:
1478:
1474:
1462:
1458:
1451:
1422:
1419:
1415:
1411:
1408:
1385:
1382:
1377:
1373:
1369:
1364:
1360:
1356:
1351:
1347:
1343:
1338:
1334:
1330:
1325:
1321:
1317:
1314:
1311:
1291:
1285:
1280:
1276:
1270:
1265:
1261:
1255:
1250:
1246:
1240:
1235:
1231:
1225:
1220:
1216:
1212:
1208:
1196:isomorphisms.
1177:
1174:
1146:, implies the
1139:
1136:
1135:
1134:
1133:
1132:
1096:
1092:
1062:
1025:
1009:
1008:
1007:
978:
958:
921:
894:
893:
892:
859:
839:
798:
774:
733:
713:
688:
660:
635:
624:
623:
622:
601:
575:
559:
534:
508:
492:
471:
455:
430:
414:
411:distributivity
369:
345:
296:
276:
211:
170:
157:
154:
122:
119:
96:multiplication
70:
49:
46:
9:
6:
4:
3:
2:
7658:
7647:
7644:
7642:
7639:
7638:
7636:
7623:
7618:
7613:
7608:
7604:
7600:
7599:
7594:
7589:
7588:
7574:
7569:
7568:
7563:
7559:
7555:
7551:
7547:
7543:
7538:
7537:
7532:
7528:
7521:
7516:
7515:
7510:
7504:
7500:
7495:
7494:
7487:
7486:
7481:
7477:
7472:
7467:
7463:
7459:
7458:
7453:
7448:
7447:
7442:
7438:
7433:
7432:
7424:
7419:
7418:
7413:
7412:"IsarMathLib"
7408:
7407:
7402:
7396:
7392:
7391:
7385:
7384:
7378:
7373:
7369:
7365:
7364:
7359:
7354:
7353:
7349:(9): 121β125.
7348:
7344:
7339:
7338:
7335:
7329:
7325:
7321:
7317:
7313:
7308:
7307:
7301:
7296:
7291:
7290:
7284:
7279:
7275:
7271:
7270:
7257:
7253:
7250:
7245:
7238:
7233:
7226:
7221:
7214:
7209:
7202:
7197:
7190:
7185:
7177:
7173:
7169:
7167:0-387-98464-X
7163:
7159:
7155:
7151:
7144:
7136:
7135:
7126:
7119:
7112:
7107:
7100:
7095:
7093:
7091:
7089:
7087:
7079:
7074:
7067:
7062:
7055:
7050:
7043:
7042:-adic numbers
7041:
7012:
7001:
6997:
6996:
6988:
6982:
6974:
6973:
6965:
6958:
6950:
6949:
6941:
6935:
6928:
6923:
6919:
6909:
6906:
6900:
6897:
6896:
6890:
6887:
6885:
6878:
6874:
6871:
6869:
6868:Rieger (1982)
6866:
6864:
6860:
6857:
6856:
6855:
6852:
6849:
6840:
6838:
6820:
6793:
6773:
6750:
6744:
6741:
6721:
6698:
6666:
6663:
6660:
6654:
6648:
6645:
6639:
6633:
6622:
6606:
6603:
6600:
6575:
6572:
6549:
6546:
6540:
6534:
6528:
6500:
6497:
6494:
6491:
6488:
6482:
6476:
6473:
6467:
6461:
6458:
6452:
6449:
6446:
6440:
6404:
6401:
6393:
6388:
6386:
6382:
6378:
6377:Arthan (2004)
6374:
6370:
6366:
6365:Eudoxus reals
6362:
6358:
6357:Arthan (2004)
6326:
6324:
6320:
6310:
6289:
6286:
6276:
6274:
6249:
6233:
6229:
6225:
6217:
6216:infinitesimal
6201:
6194:
6193:maximal ideal
6191:has a unique
6178:
6151:
6148:
6127:
6120:
6116:
6115:hyperintegers
6112:
6089:
6086:
6077:
6067:
6006:
5998:
5993:
5978:
5975:
5972:
5969:
5966:
5946:
5925:
5904:
5901:
5898:
5877:
5857:
5854:
5851:
5848:
5845:
5821:
5818:
5815:
5810:
5807:
5804:
5801:
5795:
5792:
5772:
5768:
5765:
5762:
5759:
5756:
5735:
5732:
5729:
5708:
5687:
5684:
5681:
5678:
5675:
5654:
5634:
5613:
5592:
5589:
5586:
5583:
5580:
5560:
5537:
5534:
5531:
5528:
5525:
5522:
5519:
5516:
5513:
5510:
5500:
5497:
5491:
5488:
5480:
5476:
5456:
5453:
5385:
5377:
5374:
5356:
5335:
5313:
5310:
5305:
5301:
5298:
5295:
5289:
5285:
5281:
5278:
5272:
5269:
5262:
5259:
5254:
5250:
5244:
5241:
5238:
5234:
5230:
5209:
5188:
5180:
5160:
5144:
5141:
5138:
5135:
5132:
5129:
5126:
5123:
5119:
5115:
5109:
5106:
5102:
5098:
5078:
5075:
5072:
5062:
5059:
5056:
5048:
5047:
5045:
5041:
5023:
5002:
4978:
4975:
4972:
4969:
4966:
4960:
4954:
4951:
4948:
4945:
4939:
4936:
4930:
4927:
4924:
4921:
4915:
4912:
4909:
4906:
4903:
4882:
4861:
4853:
4836:
4833:
4830:
4827:
4819:
4816:
4810:
4804:
4801:
4798:
4795:
4792:
4789:
4786:
4783:
4780:
4777:
4774:
4771:
4768:
4765:
4762:
4759:
4756:
4753:
4750:
4744:
4741:
4738:
4735:
4715:
4712:
4709:
4706:
4703:
4695:
4694:
4692:
4688:
4668:
4652:
4649:
4646:
4643:
4640:
4637:
4634:
4631:
4628:
4625:
4619:
4616:
4613:
4605:
4602:
4585:
4582:
4579:
4576:
4566:
4563:
4560:
4557:
4510:
4502:
4486:
4450:
4434:
4431:
4428:
4425:
4422:
4419:
4416:
4413:
4410:
4407:
4401:
4398:
4395:
4392:
4384:
4381:
4364:
4361:
4358:
4355:
4352:
4349:
4346:
4343:
4340:
4337:
4334:
4328:
4325:
4322:
4319:
4311:
4308:
4305:
4286:
4283:
4280:
4277:
4267:
4264:
4241:
4233:
4229:
4215:
4212:
4209:
4203:
4200:
4197:
4189:
4149:
4117:
4116:
4101:
4098:
4095:
4075:
4072:
4069:
4049:
4046:
4043:
4023:
4016:
4002:
3999:
3996:
3976:
3973:
3970:
3950:
3947:
3944:
3917:
3914:
3911:
3908:
3888:
3881:
3860:
3857:
3850:
3835:
3828:
3827:
3826:
3788:
3768:
3748:
3724:
3721:
3718:
3694:
3685:
3683:
3679:
3675:
3671:
3668:), such that
3667:
3663:
3659:
3655:
3647:
3643:
3638:
3629:
3582:
3566:
3551:
3546:
3541:
3540:is complete.
3534:
3528:
3522:
3511:
3507:
3501:
3494:
3490:
3486:
3478:
3474:
3467:
3461:
3457:
3453:
3447:
3443:
3431:
3427:
3422:
3418:
3411:
3405:
3399:
3397:
3391:
3384:
3380:
3374:
3368:
3361:
3357:
3351:
3343:
3339:
3335:
3327:
3323:
3319:
3313:
3309:
3305:
3299:
3295:
3288:
3284:
3278:
3274:
3267:
3263:
3257:
3253:
3246:
3242:
3236:
3232:
3226:
3219:
3215:
3207:
3203:
3198:
3194:
3189:
3185:
3179:
3173:
3166:
3160:
3153:
3147:
3142:
3138:
3129:
3125:
3118:
3112:
3106:
3102:
3096:
3090:
3084:
3078:
3072:
3054:
3039:
3034:
3025:
3022:
3016:
3010:
3004:
3001:
2997:
2990:
2986:
2981:
2977:
2971:
2965:
2959:
2951:
2947:
2942:
2938:
2931:
2927:
2923:
2919:
2912:
2863:
2859:
2855:
2851:
2846:
2843:
2838:
2834:
2829:
2825:
2818:
2814:
2808:
2801:
2796:
2789:
2785:
2776:
2772:
2759:
2755:
2750:
2746:
2741:
2737:
2732:
2728:
2723:
2718:
2714:
2709:
2705:
2700:
2696:
2691:
2687:
2682:
2681:
2680:
2675:
2671:
2662:
2658:
2651:
2648:
2643:
2639:
2634:
2630:
2623:
2619:
2615:
2609:
2602:
2591:
2587:
2582:
2578:
2573:
2569:
2564:
2563:
2562:
2560:
2556:
2551:
2546:
2542:
2491:
2489:
2485:
2481:
2471:
2469:
2465:
2461:
2457:
2456:Charles MΓ©ray
2453:
2442:
2440:
2436:
2410:
2409:abelian group
2407:
2380:
2378:
2374:
2350:
2346:
2323:(primitives:
2322:
2318:
2316:
2312:
2308:
2304:
2300:
2297: +
2296:
2292:
2289: +
2288:
2284:
2280:
2278:
2275: =
2274:
2271: +
2270:
2266:
2262:
2258:
2254:
2250:
2248:
2244:
2241: +
2240:
2236:
2233: +
2232:
2228:
2224:
2220:
2197:(primitives:
2196:
2189:
2188:
2187:
2181:
2177:
2173:
2169:
2165:
2161:
2157:
2153:
2149:
2145:
2141:
2137:
2133:
2129:
2126:
2122:
2121:
2117:
2113:
2109:
2105:
2101:
2097:
2093:
2090:
2089:
2088:
2065: β
2064:
2039: β
2038:
2033:
2031:
2027:
2023:
2019:
2015:
2012: β
2011:
2007:
2004: β
2003:
1999:
1996: β
1995:
1991:
1988: β
1987:
1983:
1979:
1975:
1971:
1968: β
1967:
1963:
1960: β
1959:
1956:, if for all
1934: β
1933:
1929:
1925:
1921:
1917:
1893:
1889:
1885:
1881:
1877:
1873:
1869:
1865:
1861:
1857:
1855:
1851:
1847:
1843:
1839:
1835:
1831:
1808:(primitives:
1807:
1803:
1801:
1775:
1771:
1767:
1741:
1715:
1711:
1707:
1703:
1699:
1698:Alfred Tarski
1695:
1689:
1665:
1651:
1645:
1638:
1634:
1630:
1625:
1621:
1617:
1613:
1606:
1602:
1588:
1574:
1568:
1561:
1557:
1553:
1548:
1544:
1540:
1533:
1529:
1522:
1518:
1514:
1509:
1505:
1501:
1494:
1490:
1486:
1481:
1472:
1465:
1456:
1452:
1449:
1445:
1440:
1436:
1435:
1434:
1420:
1409:
1406:
1399:
1383:
1375:
1371:
1367:
1362:
1358:
1354:
1349:
1345:
1341:
1336:
1332:
1328:
1323:
1319:
1315:
1312:
1278:
1274:
1263:
1259:
1248:
1244:
1233:
1229:
1218:
1214:
1197:
1195:
1191:
1187:
1183:
1173:
1171:
1167:
1162:
1160:
1156:
1155:ordered field
1151:
1149:
1145:
1130:
1126:
1122:
1118:
1114:
1110:
1094:
1081:
1077:
1051:
1047:
1046:
1044:
1040:
1039:bounded above
1014:
1010:
1005:
1001:
997:
993:
967:
963:
959:
956:
952:
948:
944:
940:
936:
910:
906:
902:
898:
897:
895:
890:
886:
882:
878:
874:
848:
844:
840:
837:
833:
829:
825:
821:
817:
813:
787:
783:
779:
775:
772:
768:
764:
760:
756:
752:
748:
722:
718:
714:
711:
707:
703:
677:
673:
672:
658:
650:
625:
620:
616:
590:
564:
560:
557:
553:
549:
523:
497:
493:
490:
486:
460:
456:
453:
449:
445:
419:
415:
412:
408:
404:
400:
396:
392:
388:
384:
358:
354:
350:
346:
343:
342:commutativity
339:
335:
331:
327:
323:
319:
315:
311:
285:
281:
277:
274:
273:associativity
270:
266:
262:
258:
254:
250:
246:
242:
238:
234:
230:
226:
200:
196:
192:
188:
187:
185:
160:
159:
153:
151:
147:
143:
138:
136:
120:
117:
109:
97:
93:
89:
85:
59:
55:
45:
42:
41:ordered field
38:
33:
31:
27:
23:
19:
7641:Real numbers
7602:
7596:
7579:. Retrieved
7548:(3): 44β52.
7545:
7541:
7530:
7526:
7492:
7461:
7455:
7440:
7436:
7389:
7367:
7361:
7346:
7342:
7319:
7315:
7300:math/0405454
7283:math/0301015
7266:Bibliography
7244:
7232:
7220:
7208:
7201:A'Campo 2003
7196:
7184:
7149:
7143:
7131:
7118:
7106:
7073:
7061:
7049:
7039:
7011:
7000:the original
6993:
6981:
6970:
6957:
6946:
6934:
6922:
6888:
6884:Weiss (2015)
6881:
6853:
6846:
6837:linear order
6734:we say that
6621:almost equal
6620:
6391:
6389:
6364:
6332:
6316:
6277:
6073:
6042:with all of
5994:
5472:
4499:denotes the
3848:is not empty
3686:
3680:contains no
3677:
3673:
3669:
3665:
3661:
3654:Dedekind cut
3651:
3646:real numbers
3583:
3549:
3542:
3532:
3526:
3520:
3509:
3505:
3499:
3492:
3488:
3484:
3476:
3472:
3465:
3459:
3455:
3451:
3445:
3441:
3429:
3425:
3420:
3416:
3409:
3403:
3400:
3395:
3389:
3382:
3378:
3372:
3366:
3359:
3355:
3349:
3341:
3337:
3333:
3325:
3321:
3317:
3314:
3307:
3303:
3297:
3293:
3286:
3282:
3276:
3272:
3265:
3261:
3255:
3251:
3244:
3240:
3234:
3230:
3224:
3217:
3213:
3205:
3201:
3196:
3192:
3187:
3183:
3177:
3171:
3164:
3158:
3151:
3148:
3146:as follows:
3140:
3136:
3127:
3123:
3116:
3110:
3104:
3100:
3094:
3088:
3082:
3076:
3070:
3037:
3026:
3020:
3014:
3008:
3005:
2999:
2995:
2993:for all
2988:
2984:
2979:
2975:
2969:
2963:
2957:
2949:
2945:
2940:
2936:
2932:
2925:
2921:
2917:
2910:
2853:
2847:
2841:
2836:
2832:
2827:
2823:
2816:
2812:
2806:
2799:
2794:
2787:
2783:
2774:
2770:
2766:
2757:
2753:
2748:
2744:
2739:
2735:
2730:
2726:
2716:
2712:
2707:
2703:
2698:
2694:
2689:
2685:
2673:
2669:
2660:
2656:
2652:
2646:
2641:
2637:
2632:
2628:
2621:
2617:
2613:
2607:
2600:
2597:
2589:
2585:
2580:
2576:
2571:
2567:
2554:
2552:
2544:
2540:
2492:
2484:metric space
2477:
2452:Georg Cantor
2448:
2381:
2376:
2375:
2348:
2347:
2320:
2319:
2314:
2310:
2306:
2302:
2298:
2294:
2290:
2286:
2282:
2281:
2276:
2272:
2268:
2264:
2260:
2256:
2252:
2251:
2246:
2242:
2238:
2234:
2230:
2226:
2222:
2221:
2219:, <, +):
2194:
2193:
2185:
2179:
2175:
2171:
2167:
2163:
2159:
2155:
2151:
2147:
2143:
2139:
2135:
2131:
2127:
2124:
2115:
2111:
2107:
2103:
2099:
2095:
2092:X precedes Y
2091:
2062:
2036:
2034:
2029:
2025:
2021:
2017:
2013:
2009:
2005:
2001:
1997:
1993:
1989:
1985:
1981:
1977:
1973:
1969:
1965:
1961:
1957:
1931:
1927:
1922:. "<" is
1919:
1918:
1887:
1883:
1879:
1875:
1871:
1867:
1863:
1859:
1858:
1849:
1845:
1841:
1837:
1833:
1832:
1805:
1804:
1799:
1765:
1713:
1691:
1649:
1643:
1636:
1632:
1628:
1623:
1619:
1611:
1604:
1572:
1566:
1559:
1555:
1551:
1546:
1542:
1538:
1531:
1527:
1520:
1516:
1512:
1507:
1503:
1499:
1492:
1488:
1479:
1470:
1463:
1454:
1438:
1198:
1181:
1179:
1163:
1158:
1152:
1141:
1128:
1124:
1120:
1116:
1112:
1108:
1075:
1049:
1012:
1003:
999:
995:
991:
965:
961:
954:
950:
946:
942:
938:
934:
908:
904:
900:
884:
880:
876:
872:
846:
842:
836:transitivity
831:
827:
823:
819:
815:
811:
785:
781:
777:
771:antisymmetry
766:
762:
758:
754:
750:
746:
720:
716:
705:
701:
675:
618:
614:
613:, such that
588:
562:
551:
547:
546:, such that
521:
495:
488:
484:
458:
447:
443:
417:
406:
402:
398:
394:
390:
386:
382:
356:
352:
348:
337:
333:
329:
325:
321:
317:
313:
309:
283:
279:
268:
264:
260:
256:
252:
248:
244:
240:
236:
232:
228:
224:
198:
194:
190:
139:
107:
95:
91:
51:
34:
22:real numbers
15:
7443:(5): 19β31.
7225:IsarMathLib
7213:Street 2003
7189:Arthan 2004
7068:ucdavis.edu
6623:if the set
6381:proportions
6323:Archimedean
6111:ultrafilter
5785:The choice
5328:to convert
4995:to convert
4383:Subtraction
3554:means that
3552:= 3.1415...
3175:. For each
3031:, i.e. the
2793:are called
2125:z separates
1844:, then not
1641:, for all
1396:there is a
1080:upper bound
998:, then 0 β€
710:reflexivity
37:isomorphism
18:mathematics
7635:Categories
7612:1506.03467
7581:2010-10-23
7533:: 205β217.
7256:Rieger1982
7111:Hersh 1997
7080:furman.edu
6927:Weiss 2015
6914:References
6246:gives the
6074:As in the
5181:if either
5042:We define
4854:if either
3937:such that
3642:irrational
3543:The usual
3098:such that
2973:such that
2820:, one has
2795:equivalent
2734:) × (
2625:, one has
2488:completion
2267:such that
2255:. For all
2150:and every
2102:and every
1874:such that
1716:, denoted
1564:, for all
1448:surjective
561:For every
550:+ (−
494:For every
110:, denoted
108:inequality
7562:122199850
7480:122161507
7099:Pugh 2002
7054:Kemp 2016
6745:≤
6667:∈
6646:−
6576:∈
6573:α
6553:⌋
6547:α
6544:⌊
6501:∈
6474:−
6459:−
6413:→
6394:be a map
6290:∗
6152:∗
6090:∗
6030:∞
6010:∞
6007:−
5976:×
5855:≤
5849:×
5760:×
5679:×
5584:×
5529:×
5523:∨
5501:∈
5454:⋃
5311:−
5299:−
5279:−
5273:−
5260:−
5245:−
5158:∖
5145:∈
5139:∧
5133:∈
5060:≥
4976:−
4973:×
4967:−
4952:×
4946:−
4940:−
4928:−
4925:×
4916:−
4907:×
4820:∈
4811:∪
4802:∈
4796:∧
4790:≥
4784:∧
4778:∈
4772:∧
4766:≥
4754:×
4739:×
4713:≥
4689:Defining
4666:∖
4653:∈
4647:∧
4629:−
4614:−
4583:∉
4577:∧
4567:∈
4484:∖
4448:∖
4435:∈
4429:∧
4423:∈
4411:−
4396:−
4362:∈
4356:∧
4350:∈
4268:∈
4213:⊆
4207:⇔
4201:≤
4099:≤
4073:∈
4047:∈
4000:∈
3974:∈
3918:∈
3861:≠
3658:partition
3497:for some
3439:, and so
3108:for some
2439:divisible
1830:, <):
1444:injective
1418:→
1410::
1398:bijection
1372:≤
1359:×
1279:≤
1264:×
1176:On models
1074:, and if
990:, if 0 β€
659:≤
142:structure
118:≤
7332:also at
7078:1.2βCuts
6893:See also
6171:. Then
5376:Supremum
5044:division
4604:Negation
4310:Addition
3741:, since
3660:of it, (
3565:0.999...
3524:. Hence
3469:. Since
3393:for any
3370:for all
3055:′
2433:is also
1800:addition
1772:<, a
1442:is both
1037:that is
1013:complete
994:and 0 β€
960:For all
899:For all
889:totality
841:For all
776:For all
715:For all
674:For all
556:inverses
452:identity
416:For all
347:For all
278:For all
189:For all
92:addition
86:and one
7176:1643950
6390:Let an
5016:and/or
2856:of all
2840:| <
2752:×
2645:| <
2557:be the
2377:Axiom 8
2349:Axiom 7
2301:, then
2283:Axiom 6
2253:Axiom 5
2223:Axiom 4
2016:, then
1930:,
1920:Axiom 3
1860:Axiom 2
1834:Axiom 1
1798:called
1764:called
1712:called
1078:has an
941:, then
826:, then
761:, then
565:β 0 in
146:theorem
7560:
7505:
7478:
7397:
7252:693180
7174:
7164:
6214:, the
4469:where
3503:. But
3228:, set
2944:) β₯ (
2928:, ...)
2802:> 0
2603:> 0
2548:|
2538:|
1702:axioms
1041:has a
784:, and
487:Γ 1 =
446:+ 0 =
355:, and
197:, and
156:Axioms
135:axioms
7607:arXiv
7576:(PDF)
7558:S2CID
7523:(PDF)
7501:β15.
7476:S2CID
7426:(PDF)
7295:arXiv
7278:arXiv
7128:(PDF)
7003:(PDF)
6990:(PDF)
6967:(PDF)
6943:(PDF)
6248:field
5959:with
5722:with
5668:with
5091:then
4728:then
4304:dense
4232:embed
3989:then
3963:, if
3487:<
3454:<
3401:Thus
3353:that
3211:. If
3103:<
2998:>
2815:>
2743:) = (
2702:) = (
2693:) + (
2620:>
2593:,...)
2482:in a
2404:is a
2285:. If
2158:with
2142:with
2000:, if
1892:dense
1862:. If
1848:<
1840:<
1836:. If
1776:over
1766:order
1742:over
1477:) = 1
1461:) = 0
1194:up to
1190:below
1184:is a
1107:then
933:, if
810:, if
745:, if
401:) + (
393:) = (
259:) = (
235:) = (
184:field
182:is a
150:up to
144:is a
7503:ISBN
7395:ISBN
7162:ISBN
6619:are
6375:and
6119:ring
5970:<
5902:<
5766:<
5749:and
5733:<
5685:<
5535:<
5517:<
5076:>
4834:<
4641:<
4284:<
3948:<
3573:and
3536:and
3376:and
3331:and
3291:and
3249:and
3162:and
3149:Set
3133:and
3068:and
2780:and
2666:and
2553:Let
2466:and
2437:and
2174:and
2130:and
2061:and
2024:and
2008:and
1992:and
1964:and
1882:and
1738:, a
1708:: a
1647:and
1570:and
1541:) =
1525:and
1502:) =
1468:and
1446:and
1302:and
964:and
907:and
845:and
818:and
753:and
719:and
651:for
409:). (
324:and
282:and
267:) Γ
247:and
243:) +
102:and
94:and
7617:doi
7550:doi
7466:doi
7372:doi
7324:doi
7154:doi
6766:if
5939:in
5202:or
5049:if
4875:or
4696:if
4523:in
4503:of
4230:We
4162:of
3567:= 1
3435:is
3336:= (
3320:= (
3209:)/2
3191:= (
3114:in
2983:β₯
2559:set
2309:or
1894:in
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1653:in
1626:) β€
1576:in
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1048:If
968:in
911:in
887:. (
879:or
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834:. (
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769:. (
723:in
708:. (
678:in
647:is
591:in
524:in
498:in
461:in
420:in
385:Γ (
359:in
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286:in
271:. (
251:Γ (
227:+ (
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58:set
52:An
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7637::
7615:.
7603:45
7601:.
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7499:11
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7460:.
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7439:.
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7360:.
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7320:79
7318:.
7314:.
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7085:^
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6992:.
6969:.
6945:.
6886:.
6875:,
6861:,
6275:.
6066:.
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5110::=
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4402::=
4385:.
4329::=
4312:.
4088:,
3664:,
3652:A
3644:,
3581:.
3458:=
3444:=
3424:β
3312:.
3302:=
3300:+1
3281:=
3279:+1
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3258:+1
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3157:=
3003:.
2930:.
2924:,
2920:,
2864:.
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2831:β
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2458:,
2441:.
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1972:,
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883:β€
875:β€
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830:β€
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757:β€
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704:β€
700:,
617:Γ
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405:Γ
397:Γ
389:+
381:,
351:,
336:Γ
332:=
328:Γ
320:+
316:=
312:+
308:,
263:Γ
255:Γ
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223:,
193:,
7625:.
7619::
7609::
7584:.
7564:.
7552::
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6699:f
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6675:}
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5270:=
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5167:}
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4958:)
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4883:B
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