7049:
4195:
29:
4331:). Zorn's lemma can be expressed straightforwardly by observing that the set having no maximal element would be equivalent to stating that the set's ordering relation would be entire, which would allow us to apply the axiom of dependent choice to construct a countable chain. As a result, any partially ordered set with exclusively finite chains must have a maximal element.
184:
and the assumptions of the situation to get a contradiction. Zorn's lemma tidies up the conditions a situation needs to satisfy in order for such an argument to work and enables mathematicians to not have to repeat the transfinite induction argument by hand each time, but just check the conditions of
4334:
More generally, strengthening the axiom of dependent choice to higher ordinals allows us to generalize the statement in the previous paragraph to higher cardinalities. In the limit where we allow arbitrarily large ordinals, we recover the proof of the full Zorn's lemma using the axiom of choice in
655:
The difference may seem subtle, but in many proofs that invoke Zorn's lemma one takes unions of some sort to produce an upper bound, and so the case of the empty chain may be overlooked; that is, the verification that all chains have upper bounds may have to deal with empty and non-empty chains
40:. The set of all sub-graphs that are trees is ordered by inclusion, and the union of a chain is an upper bound. Zorn's lemma says that a maximal tree must exist, which is a spanning tree since the graph is connected. Zorn's lemma is not needed for finite graphs, such as the one pictured here.
4156:
proved in 1922 a version of the lemma close to its modern formulation (it applies to sets ordered by inclusion and closed under unions of well-ordered chains). Essentially the same formulation (weakened by using arbitrary chains, not just well-ordered) was independently given by
2499:.) Note the above is a weak form of Zorn's lemma since Zorn's lemma says in particular that any set of subsets satisfying the above (1) and (2) has a maximal element. The point is that, conversely, Zorn's lemma follows from this weak form. Indeed, let
1016:
189:
If you are building a mathematical object in stages and find that (i) you have not finished even after infinitely many stages, and (ii) there seems to be nothing to stop you continuing to build, then Zorn’s lemma may well be able to help
3172:
4165:
of set theory replacing the well-ordering theorem, exhibited some of its applications in algebra, and promised to show its equivalence with the axiom of choice in another paper, which never appeared.
1902:
4734:
2750:
3376:
2917:
Indeed, trivially, Zorn's lemma implies the above lemma. Conversely, the above lemma implies the aforementioned weak form of Zorn's lemma, since a union gives a least upper bound.
4261:
2805:
3718:
3288:
4251:
3909:
3786:
1956:
915:
3427:
2925:
A proof that Zorn's lemma implies the axiom of choice illustrates a typical application of Zorn's lemma. (The structure of the proof is exactly the same as the one for the
2979:
3460:
3749:
3056:
466:. Depending on the order relation, a partially ordered set may have any number of maximal elements. However, a totally ordered set can have at most one maximal element.
4050:
3018:
3202:
2703:
4010:
3944:
3486:
3082:
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3540:
3513:
4765:
2831:
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4274:, a result from linear algebra (to which it is equivalent). In particular, the real numbers, as a vector space over the rational numbers, possess a Hamel basis.
4136:
4116:
4096:
4076:
4030:
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3308:
3242:
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2911:
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2617:
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2429:
2409:
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2323:
2300:
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168:
which states that every totally ordered subset of a given partially ordered set is contained in a maximal totally ordered subset of that partially ordered set.
2539:. Then it satisfies all of the above properties (it is nonempty since the empty subset is a chain.) Thus, by the above weak form, we find a maximal element
5428:
6103:
6186:
5327:
7736:
5029:
7719:
2025:), more than there are elements in any set (in other words, given any set of ordinals, there exists a larger ordinal), and the set
164:
without the axiom of choice) any one of the three is sufficient to prove the other two. An earlier formulation of Zorn's lemma is
7249:
6500:
4327:
A weakened form of Zorn's lemma can be proven from ZF + DC (Zermelo–Fraenkel set theory with the axiom of choice replaced by the
7085:
3090:
2165:}). Contradiction is reached by noting that we can always find a "next" initial segment either by taking the union of all such
6658:
5251:
5228:
5202:
4965:
4840:
4821:
4802:
4684:
4607:
4576:
4553:
4535:
4257:
Moreover, Zorn's lemma (or one of its equivalent forms) implies some major results in other mathematical areas. For example,
4247:: "The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?"
5446:
1833:
7566:
6513:
5836:
4694:
4649:
4787:
from the weak form of Zorn's lemma. The meaning of passage there was unclear and so here we gave an alternative reasoning.
6518:
6508:
6245:
6098:
5451:
344:. The word "partial" is meant to indicate that not every pair of elements of a partially ordered set is required to be
5442:
7702:
7561:
6654:
5186:
5011:
5996:
7556:
6751:
6495:
5320:
180:
in some way, one can try proving the existence of such an object by assuming there is no maximal element and using
165:
2708:
7192:
6056:
5749:
4219:
161:
5490:
2104:
The above proof can be formulated without explicitly referring to ordinals by considering the initial segments {
7274:
7012:
6714:
6477:
6472:
6297:
5718:
5402:
5103:
2837:
4403:
7774:
7593:
7513:
7007:
6790:
6707:
6420:
6351:
6228:
5470:
5288:
contains a formal proof down to the finest detail of the equivalence of the axiom of choice and Zorn's Lemma.
5278:
7378:
7307:
7187:
6932:
6758:
6444:
6078:
5677:
4382:
4185:
33:
7281:
7269:
7232:
7207:
7182:
7136:
7105:
6810:
6805:
6415:
6154:
6083:
5412:
5313:
5273:
4388:
4226:
4147:
2231:
636:
is assumed to be non-empty and satisfies the hypothesis that every non-empty chain has an upper bound in
2840:. Indeed, the existence of a maximal chain is exactly the assertion of the Hausdorff maximal principle.
2775:
7578:
7212:
7202:
7078:
6739:
6329:
5723:
5691:
5382:
3333:
2252:
The basic idea of the proof is to reduce the proof to proving the following weak form of Zorn's lemma:
1587:
1011:{\displaystyle a_{1}\mathbf {v} _{1}+a_{2}\mathbf {v} _{2}+\cdots +a_{k}\mathbf {v} _{k}=\mathbf {0} .}
3247:
7551:
7217:
7029:
6978:
6875:
6373:
6334:
5811:
5456:
4624:
4530:. Graduate Texts in Mathematics. Vol. 211 (Revised 3rd ed.). Springer-Verlag. p. 880.
4377:
4328:
4322:
75:
5485:
1911:
7769:
7483:
7110:
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6339:
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5897:
5377:
3381:
338:
4508:
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7731:
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6702:
6679:
6640:
6526:
6467:
6113:
6033:
5877:
5821:
5434:
4864:
4265:
4243:
A well-known joke alluding to this equivalency (which may defy human intuition) is attributed to
3692:
2926:
102:
5268:
5101:
Wolk, Elliot S. (1983), "On the principle of dependent choices and some forms of Zorn's lemma",
3432:
176:
To prove the existence of a mathematical object that can be viewed as a maximal element in some
7643:
7259:
6992:
6719:
6697:
6664:
6557:
6403:
6388:
6361:
6312:
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6131:
5956:
5922:
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4288:
4271:
3871:
3754:
3026:
1824:
681:
242:
118:
114:
5181:. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974.
5131:
4035:
2991:
7779:
7621:
7456:
7447:
7316:
7197:
7151:
7115:
7071:
6922:
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6567:
6285:
6021:
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5786:
5771:
5652:
5627:
4236:
4213:
4055:
3181:
2682:
2040:
528:
392:
310:
181:
177:
153:
101:
in 1935. It occurs in the proofs of several theorems of crucial importance, for instance the
56:
3989:
3914:
3465:
3061:
652:
serves as an upper bound for the empty chain (that is, the empty subset viewed as a chain).
7709:
7668:
7658:
7648:
7393:
7356:
7346:
7326:
7311:
6895:
6857:
6734:
6538:
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6302:
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5932:
5796:
5584:
5495:
4359:
4303:
4153:
3585:
3518:
3491:
2058:
contains an upper bound for the empty set and is thus not empty) and for any other ordinal
886:
535:
412:
94:
64:
4741:
4689:
NB: in the reference, this deduction is by noting there is an order-preserving embedding
3723:
8:
7636:
7547:
7493:
7452:
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7331:
7264:
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7024:
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6724:
6674:
6600:
6545:
6482:
6275:
6270:
6218:
5986:
5975:
5647:
5547:
5475:
5466:
5462:
5397:
5392:
4313:
In this sense, Zorn's lemma is a powerful tool, applicable to many areas of mathematics.
2810:
1803:
such that every totally ordered subset has an upper bound, and that for every element in
1267:
712:
377:
276:
142:
106:
86:
1779:
The hypothesis of Zorn's lemma has been checked, and thus there is a maximal element in
1150:
The hypothesis of Zorn's lemma has been checked, and thus there is a maximal element in
7748:
7675:
7528:
7437:
7427:
7368:
7286:
7222:
7053:
6822:
6785:
6770:
6763:
6746:
6550:
6532:
6398:
6324:
6307:
6260:
6073:
5982:
5816:
5801:
5761:
5713:
5698:
5686:
5642:
5617:
5387:
5336:
5243:
5220:
4770:
4345:
4282:
4264:
which is used to prove one of the most fundamental results in functional analysis, the
4200:
4121:
4101:
4081:
4061:
4015:
3969:
3949:
3851:
3831:
3811:
3791:
3672:
3652:
3632:
3612:
3565:
3545:
3313:
3293:
3227:
3207:
2935:
2896:
2876:
2856:
2755:
2662:
2642:
2622:
2602:
2582:
2562:
2542:
2522:
2502:
2482:
2458:
2434:
2414:
2394:
2374:
2351:
2331:
2308:
2285:
2265:
1242:
826:
224:
7588:
6006:
2479:(Note that, strictly speaking, (1) is redundant since (2) implies the empty set is in
1823:
has an upper bound, and that upper bound has a bigger element. To actually define the
7685:
7663:
7523:
7508:
7488:
7291:
7048:
6988:
6795:
6605:
6595:
6487:
6368:
6203:
6179:
5960:
5944:
5849:
5826:
5703:
5672:
5637:
5532:
5367:
5247:
5224:
5198:
5182:
5166:
5149:
5007:
4961:
4855:
4603:
4572:
4549:
4531:
4423:
4394:
4307:
4194:
724:
213:
146:
4918:
4901:
2184:
This proof shows that actually a slightly stronger version of Zorn's lemma is true:
7498:
7351:
7002:
6997:
6890:
6847:
6669:
6630:
6625:
6610:
6436:
6393:
6290:
6088:
6038:
5612:
5574:
5161:
5112:
5065:
5034:
4999:
4953:
4945:
4913:
4873:
4665:
4661:
4177:
2236:
2029:
will be exhausted before long and then we will run into the desired contradiction.
1799:. Suppose the lemma is false. Then there exists a partially ordered set, or poset,
604:), in fact the two formulations are equivalent: To verify this, suppose first that
600:
the additional condition of being non-empty, but obtains the same conclusion about
130:
4856:"Une méthode d'élimination des nombres transfinis des raisonnements mathématiques"
4184:
of 1939 refers to a similar maximal principle as "le théorème de Zorn". The name "
7680:
7463:
7341:
7336:
7321:
7146:
7131:
6983:
6973:
6927:
6910:
6865:
6827:
6729:
6649:
6456:
6383:
6356:
6344:
6250:
6164:
6138:
6093:
6061:
5862:
5664:
5607:
5557:
5522:
5480:
4858:[A method of disposing of transfinite numbers of mathematical reasoning]
4231:
4158:
1796:
752:
555:
431:
220:
157:
98:
90:
79:
5003:
4957:
562:
Variants of this formulation are sometimes used, such as requiring that the set
7598:
7583:
7573:
7432:
7410:
7388:
6968:
6947:
6905:
6885:
6780:
6635:
6233:
6223:
6213:
6208:
6142:
6016:
5892:
5781:
5776:
5754:
5355:
5285:
5038:
4593:
4562:
4504:
2982:
2014:
2010:
1725:
1474:
contains at least one element, and that element contains at least 0, the union
195:
134:
4738:
and that the "passage" allows to deduce the existence of a maximal element of
7763:
7697:
7653:
7631:
7503:
7373:
7361:
7166:
6942:
6620:
6127:
5912:
5902:
5872:
5857:
5527:
4295:
4278:
1253:
621:
345:
138:
126:
37:
2843:
The same proof also shows the following equivalent variant of Zorn's lemma:
1236:
596:
Although this formulation appears to be formally weaker (since it places on
7518:
7400:
7383:
7301:
7141:
7094:
6842:
6689:
6590:
6582:
6462:
6410:
6319:
6255:
6238:
6169:
6028:
5887:
5589:
5372:
5117:
4597:
4566:
4354:
2022:
674:
110:
5299:
5291:
5070:
5053:
4878:
2873:
be a partially ordered set in which each chain has a least upper bound in
1807:
there is another element bigger than it. For every totally ordered subset
656:
separately. So many authors prefer to verify the non-emptiness of the set
7724:
7417:
7296:
7161:
6952:
6832:
6011:
6001:
5948:
5632:
5552:
5537:
5417:
5362:
5212:
5174:
4994:
Blass, Andreas (1984). "Existence of bases implies the Axiom of Choice".
4299:
1249:
1178:
624:; so the hypothesis implies that this subset must have an upper bound in
543:
486:
376:. An ordered set in which every pair of elements is comparable is called
68:
60:
28:
17:
806:
has an upper bound, that is, there exists a linearly independent subset
7692:
7626:
7467:
5882:
5737:
5708:
5514:
4523:
4244:
4169:
2203:
52:
4385:– a partially ordered set in which every chain has a least upper bound
7743:
7616:
7422:
7034:
6937:
5990:
5907:
5867:
5831:
5767:
5579:
5569:
5542:
5305:
4984:. North Holland Publishing Company. Chapter 5, Theorem 4.3, page 103.
4371:
4054:
Essentially the same proof also shows that Zorn's lemma implies the
7538:
7405:
7156:
7019:
6817:
6265:
5970:
5564:
5295:
5240:
Zermelo's axiom of choice: Its origins, development & influence
4470:, ch. 2, §2 Some applications of the Axiom of Choice in mathematics
3946:. (Note this step does not need the axiom of choice.) The function
122:
4316:
1294:
is partially ordered by set inclusion. Finding a maximal ideal in
584:
has the property that every non-empty chain has an upper bound in
6615:
5407:
4602:. Universitext (Second ed.). Springer-Verlag. p. 164.
1760:, then it would contain 1, and that means one of the members of
1129:. This tells us there is a linearly dependent set of vectors in
7063:
4397: – Mathematical set containing a finite number of elements
2125:. Such sets can be easily characterized as well-ordered chains
748:
660:
rather than deal with the empty chain in the general argument.
71:
348:
under the order relation, that is, in a partially ordered set
6159:
5505:
5350:
4571:. Universitext (2nd ed.). Springer-Verlag. p. 162.
4162:
2222:. That is, there is a maximal element which is comparable to
2199:
5132:"Zorn's Lemma | The Simpsons and their Mathematical Secrets"
4599:
An
Invitation to General Algebra and Universal Constructions
4568:
An
Invitation to General Algebra and Universal Constructions
1795:
A sketch of the proof of Zorn's lemma follows, assuming the
1732:
then it contains 1; on the other hand, if it contains 1 and
5054:"The Tychonoff product theorem implies the axiom of choice"
4950:
Handbook of Logic and Proof
Techniques for Computer Science
2920:
2282:
be a set consisting of subsets of some fixed set such that
1349:(otherwise it would not be a proper ideal, so it is not in
1317:
is empty, then the trivial ideal {0} is an upper bound for
4998:. Contemporary Mathematics. Vol. 31. pp. 31–33.
4430:, Springer Monographs in Mathematics, Springer, p. 23
2169:(corresponding to the limit ordinal case) or by appending
23:
Mathematical proposition equivalent to the axiom of choice
4078:
to be the set of all well-ordered subsets of a given set
3167:{\displaystyle P=\{f:X'\to U\mid X'\subset X,f(S)\in S\}}
1830:, we need to employ the axiom of choice (explicitly: let
1748:
is an element of the ideal, and so the ideal is equal to
1237:
Every nontrivial ring with unity contains a maximal ideal
2101:
are totally ordered, this is a well-founded definition.
4399:
Pages displaying short descriptions of redirect targets
1478:
contains at least 0 and is not empty. Every element of
1241:
Zorn's lemma can be used to show that every nontrivial
1154:, in other words a maximal linearly independent subset
860:
is not linearly independent. Then there exists vectors
4503:
2368:, where the ordering is with respect to set inclusion.
1897:{\displaystyle B(T)=\{b\in P:\forall t\in T,b\geq t\}}
32:
Zorn's lemma can be used to show that every connected
4773:
4744:
4729:{\displaystyle s:P\hookrightarrow {\mathfrak {P}}(P)}
4697:
4124:
4104:
4084:
4064:
4038:
4018:
3992:
3972:
3952:
3917:
3874:
3854:
3834:
3814:
3794:
3757:
3726:
3695:
3675:
3655:
3649:. Hence, by Zorn's lemma, there is a maximal element
3635:
3615:
3588:
3568:
3548:
3521:
3494:
3468:
3435:
3384:
3336:
3316:
3296:
3250:
3230:
3210:
3184:
3093:
3064:
3029:
2994:
2958:
2938:
2899:
2879:
2859:
2813:
2778:
2758:
2711:
2685:
2665:
2645:
2625:
2605:
2585:
2565:
2545:
2525:
2505:
2485:
2475:
has a maximal element with respect to set inclusion.
2461:
2437:
2417:
2397:
2377:
2354:
2334:
2311:
2288:
2268:
1914:
1836:
918:
648:
chain has an upper bound, as an arbitrary element of
4190:
755:). Finding a maximal linearly independent subset of
434:(with respect to ≤) if there is no other element of
4207:
1333:has an upper bound, that is, there exists an ideal
1143:is linearly independent (because it is a member of
493:if it is greater than or equal to every element of
137:with identity every proper ideal is contained in a
4779:
4759:
4728:
4130:
4110:
4090:
4070:
4044:
4024:
4004:
3978:
3958:
3938:
3903:
3860:
3840:
3820:
3800:
3780:
3743:
3712:
3681:
3661:
3641:
3621:
3601:
3574:
3554:
3534:
3507:
3480:
3454:
3421:
3370:
3322:
3302:
3282:
3236:
3216:
3196:
3166:
3076:
3050:
3012:
2973:
2944:
2905:
2885:
2865:
2825:
2799:
2764:
2744:
2697:
2671:
2651:
2631:
2611:
2591:
2571:
2551:
2531:
2511:
2491:
2467:
2443:
2423:
2403:
2383:
2360:
2340:
2317:
2294:
2274:
2230:Alternatively, one can use the same proof for the
1950:
1896:
1388:to be an ideal, it must satisfy three conditions:
1010:
419:) if it is totally ordered in the inherited order.
2234:. This is the proof given for example in Halmos'
668:
505:is required to be comparable to every element of
7761:
2017:. In fact, the sequence is too long for the set
4317:Analogs under weakenings of the axiom of choice
4150:is an early statement similar to Zorn's lemma.
2218:has a maximal element greater than or equal to
2181:(corresponding to the successor ordinal case).
4391: – Mathematical result on order relations
1101:must contain the others, so there is some set
7079:
5321:
4906:Bulletin of the American Mathematical Society
4625:"A Proof that every Vector Space has a Basis"
4168:The name "Zorn's lemma" appears to be due to
2328:The union of each totally ordered subsets of
1290:contains the trivial ideal {0}. Furthermore,
1185:. Suppose for the sake of contradiction that
5096:
5094:
4839:harvnb error: no target: CITEREFHalmos1960 (
4820:harvnb error: no target: CITEREFHalmos1960 (
4548:(2nd ed.). Prentice Hall. p. 875.
4544:Dummit, David S.; Foote, Richard M. (1998).
4291:in topology (to which it is also equivalent)
3178:It is partially ordered by extension; i.e.,
3161:
3100:
2745:{\displaystyle {\widetilde {C}}=C\cup \{y\}}
2739:
2733:
1891:
1852:
1298:is the same as finding a maximal element in
759:is the same as finding a maximal element in
673:Zorn's lemma can be used to show that every
608:satisfies the condition that every chain in
352:with order relation ≤ there may be elements
4902:"A remark on method in transfinite algebra"
4543:
3609:'s. Thus, we have shown that each chain in
1764:would contain 1 and would thus be equal to
1329:is non-empty. It is necessary to show that
620:is a chain, as it satisfies the definition
391:can itself be seen as partially ordered by
7737:Positive cone of a partially ordered group
7086:
7072:
5513:
5328:
5314:
5192:
5030:Journal of the London Mathematical Society
4853:
1728:it contains 1. (It is clear that if it is
739:contains the linearly independent subset {
580:Suppose a non-empty partially ordered set
5165:
5116:
5091:
5069:
5027:Hodges, W. (1979). "Krull implies Zorn".
4917:
4877:
3751:. Suppose otherwise; then there is a set
203:
7720:Positive cone of an ordered vector space
5195:Set Theory for the Working Mathematician
5147:
4932:
4801:harvnb error: no target: CITEREFHalmos (
4683:harvnb error: no target: CITEREFHalmos (
4452:
2921:Zorn's lemma implies the axiom of choice
1189:is not spanning. Then there exists some
27:
4592:
4561:
4374: – Subset of incomparable elements
4250:Zorn's lemma is also equivalent to the
4012:, a contradiction to the maximality of
2985:), we want to show there is a function
2836:The proof of the weak form is given in
1904:, that is, the set of upper bounds for
845:. To do this, it suffices to show that
663:
592:contains at least one maximal element.
7762:
5335:
5026:
4944:
4834:
4815:
4457:. New York: Blaisdell. pp. 16–17.
4174:Convergence and Uniformity in Topology
1209:} is a linearly independent subset of
632:is in fact non-empty. Conversely, if
7067:
5309:
5237:
5084:J.L. Bell & A.B. Slomson (1969).
4980:J.L. Bell & A.B. Slomson (1969).
4647:
4491:
4440:
4422:
4338:
4285:, to which Zorn's lemma is equivalent
4281:, a result from ring theory known as
2599:. By the hypothesis of Zorn's lemma,
1360:to be the union of all the ideals in
695:}, then the empty set is a basis for
509:but need not itself be an element of
16:For the film by Hollis Frampton, see
5211:
5100:
4899:
4522:
4479:
4467:
4416:
4277:Every commutative unital ring has a
4098:and then shows a maximal element of
3808:is nonempty, it contains an element
2302:satisfies the following properties:
1811:we may then define a bigger element
1783:, in other words a maximal ideal in
1305:To apply Zorn's lemma, take a chain
1080:is totally ordered, one of the sets
849:is a linearly independent subset of
766:To apply Zorn's lemma, take a chain
517:Zorn's lemma can then be stated as:
5148:Campbell, Paul J. (February 1978).
5088:. North Holland Publishing Company.
4712:
2054:arbitrary (this is possible, since
802:is non-empty. We need to show that
501:is not required to be a chain, and
13:
7247:Properties & Types (
4796:
4678:
4214:Axiom of choice § Equivalents
4161:in 1935, who proposed it as a new
3310:, then we can define the function
2800:{\displaystyle {\widetilde {C}}=C}
2206:subset has an upper bound, and if
1965:, we are going to define elements
1867:
1217:, contradicting the maximality of
644:also satisfies the condition that
628:, and this upper bound shows that
395:the order relation inherited from
152:Zorn's lemma is equivalent to the
14:
7791:
7703:Positive cone of an ordered field
5261:
4654:The American Mathematical Monthly
4622:
4188:" prevails in Poland and Russia.
3582:and is a common extension of all
3371:{\displaystyle X'=\cup _{i}X_{i}}
3084:. For that end, consider the set
2838:Hausdorff maximal principle#Proof
2021:; there are too many ordinals (a
1570:is totally ordered, we know that
727:, there exists a nonzero element
7557:Ordered topological vector space
7093:
7047:
4650:"A simple proof of Zorn's lemma"
4208:Equivalent forms of Zorn's lemma
4193:
3462:. This is well-defined since if
3283:{\displaystyle f_{i}:X_{i}\to U}
1908:. The axiom of choice furnishes
1025:is the union of all the sets in
1001:
987:
956:
931:
129:is compact, and the theorems in
5124:
5078:
5045:
5020:
4987:
4974:
4948:(2002), "The Axiom of Choice",
4938:
4926:
4919:10.1090/S0002-9904-1935-06166-X
4893:
4847:
4828:
4809:
4790:
4672:
4641:
4616:
4406:(sometimes named Tukey's lemma)
4218:Zorn's lemma is equivalent (in
2952:of nonempty sets and its union
2009:: the indices are not just the
2005:. This uncountable sequence is
1790:
160:, in the sense that within ZF (
5197:. Cambridge University Press.
5193:Ciesielski, Krzysztof (1997).
5150:"The Origin of 'Zorn's Lemma'"
5104:Canadian Mathematical Bulletin
4952:, Springer, pp. 121–126,
4754:
4748:
4723:
4717:
4707:
4666:10.1080/00029890.1991.12000768
4586:
4515:
4497:
4485:
4473:
4461:
4446:
4434:
3927:
3921:
3880:
3416:
3410:
3394:
3388:
3274:
3152:
3146:
3117:
3039:
3033:
3004:
2679:is a maximal element since if
1951:{\displaystyle b:b(T)\in B(T)}
1945:
1939:
1930:
1924:
1846:
1840:
1590:, assume the first case. Both
1538:. Then there exist two ideals
1341:containing all the members of
814:containing all the members of
669:Every vector space has a basis
125:stating that every product of
1:
7514:Series-parallel partial order
7008:History of mathematical logic
5141:
3422:{\displaystyle f(x)=f_{i}(x)}
1666:. Then there exists an ideal
1490:only consists of elements in
1263:be the set consisting of all
782:that is totally ordered). If
711:be the set consisting of all
566:and the chains be non-empty.
171:
166:Hausdorff's maximum principle
97:in 1922 and independently by
7193:Cantor's isomorphism theorem
6933:Primitive recursive function
5167:10.1016/0315-0860(78)90136-2
4854:Kuratowski, Casimir (1922).
4383:Chain-complete partial order
4352:The lemma was referenced on
2974:{\displaystyle U:=\bigcup X}
2519:be the set of all chains in
1772:is explicitly excluded from
1721:ideal. An ideal is equal to
534:has the property that every
7:
7233:Szpilrajn extension theorem
7208:Hausdorff maximal principle
7183:Boolean prime ideal theorem
5298:is another formal proof. (
5274:Encyclopedia of Mathematics
5238:Moore, Gregory H. (2013) .
5051:
4958:10.1007/978-1-4612-0115-1_9
4648:Lewin, Jonathan W. (1991).
4389:Szpilrajn extension theorem
4365:
4302:, a result that yields the
4252:strong completeness theorem
4227:Hausdorff maximal principle
4148:Hausdorff maximal principle
3713:{\displaystyle X'\subset X}
2579:; i.e., a maximal chain in
2232:Hausdorff maximal principle
1736:is an arbitrary element of
1197:not covered by the span of
1173:. It suffices to show that
616:. Then the empty subset of
473:of a partially ordered set
426:of a partially ordered set
407:of a partially ordered set
387:of a partially ordered set
198:, "How to use Zorn’s lemma"
162:Zermelo–Fraenkel set theory
10:
7796:
7579:Topological vector lattice
5997:Schröder–Bernstein theorem
5724:Monadic predicate calculus
5383:Foundations of mathematics
5286:Zorn's Lemma at ProvenMath
5189:(Springer-Verlag edition).
4993:
4349:is named after the lemma.
4320:
4262:Banach's extension theorem
4211:
4172:, who used it in his book
4141:
3455:{\displaystyle x\in X_{i}}
2241:
1588:Without loss of generality
1376:. We will first show that
909:, not all zero, such that
442:, that is, if there is no
15:
7609:
7537:
7476:
7246:
7175:
7124:
7101:
7043:
7030:Philosophy of mathematics
6979:Automated theorem proving
6961:
6856:
6688:
6581:
6433:
6150:
6126:
6104:Von Neumann–Bernays–Gödel
6049:
5943:
5847:
5745:
5736:
5663:
5598:
5504:
5426:
5343:
4767:or equivalently, that of
4509:"How to use Zorn's lemma"
4453:Wilansky, Albert (1964).
4329:axiom of dependent choice
4323:Axiom of dependent choice
4270:Every vector space has a
4222:) to three main results:
3904:{\displaystyle g|_{X'}=f}
3781:{\displaystyle S\in X-X'}
3051:{\displaystyle f(S)\in S}
1598:are members of the ideal
430:with order relation ≤ is
313:by ≤. Given two elements
109:, the theorem that every
89:was proved (assuming the
7188:Cantor–Bernstein theorem
5086:Models and Ultraproducts
5052:Kelley, John L. (1950).
5039:10.1112/jlms/s2-19.2.285
4982:Models and Ultraproducts
4410:
4045:{\displaystyle \square }
3689:that is defined on some
3013:{\displaystyle f:X\to U}
2247:
1698:and hence an element of
1462:is a nonempty subset of
1395:is a nonempty subset of
1286:is non-trivial, the set
1274:(that is, all ideals in
856:Suppose otherwise, that
790:} is an upper bound for
786:is the empty set, then {
747:is partially ordered by
339:greater than or equal to
7732:Partially ordered group
7552:Specialization preorder
6680:Self-verifying theories
6501:Tarski's axiomatization
5452:Tarski's undefinability
5447:incompleteness theorems
5058:Fundamenta Mathematicae
5004:10.1090/conm/031/763890
4865:Fundamenta Mathematicae
4404:Teichmüller–Tukey lemma
4335:the preceding section.
3197:{\displaystyle f\leq g}
2913:has a maximal element.
2698:{\displaystyle y\geq x}
1364:. We wish to show that
1345:but still smaller than
1229:, and thus, a basis of
833:. We wish to show that
7218:Kruskal's tree theorem
7213:Knaster–Tarski theorem
7203:Dushnik–Miller theorem
7054:Mathematics portal
6665:Proof of impossibility
6313:propositional variable
5623:Propositional calculus
5118:10.4153/CMB-1983-062-5
5033:. s2-19 (2): 285–287.
4781:
4761:
4730:
4505:William Timothy Gowers
4254:of first-order logic.
4132:
4112:
4092:
4072:
4046:
4026:
4006:
4005:{\displaystyle f<g}
3980:
3960:
3940:
3939:{\displaystyle g(S)=s}
3905:
3862:
3842:
3822:
3802:
3782:
3745:
3714:
3683:
3663:
3643:
3629:has an upper bound in
3623:
3603:
3576:
3562:is also an element of
3556:
3536:
3515:is the restriction of
3509:
3482:
3481:{\displaystyle i<j}
3456:
3423:
3372:
3324:
3304:
3284:
3238:
3224:is the restriction of
3218:
3198:
3168:
3078:
3077:{\displaystyle S\in X}
3052:
3014:
2975:
2946:
2907:
2887:
2867:
2827:
2801:
2766:
2746:
2699:
2673:
2653:
2633:
2613:
2593:
2573:
2553:
2533:
2513:
2493:
2469:
2445:
2425:
2405:
2385:
2362:
2342:
2319:
2296:
2276:
1952:
1898:
1602:, therefore their sum
1368:is an upper bound for
1165:Finally, we show that
1029:, there are some sets
1012:
837:is an upper bound for
612:has an upper bound in
554:contains at least one
204:Statement of the lemma
201:
196:William Timothy Gowers
78:contains at least one
51:, is a proposition of
41:
6923:Kolmogorov complexity
6876:Computably enumerable
6776:Model complete theory
6568:Principia Mathematica
5628:Propositional formula
5457:Banach–Tarski paradox
5302:for recent browsers.)
5219:. Mineola, New York:
5071:10.4064/fm-37-1-75-76
4879:10.4064/fm-3-1-76-108
4782:
4762:
4731:
4378:Bourbaki–Witt theorem
4237:Well-ordering theorem
4186:Kuratowski–Zorn lemma
4182:Théorie des Ensembles
4133:
4113:
4093:
4073:
4056:well-ordering theorem
4047:
4027:
4007:
3981:
3961:
3941:
3906:
3863:
3843:
3828:. We can then extend
3823:
3803:
3783:
3746:
3715:
3684:
3664:
3644:
3624:
3604:
3602:{\displaystyle f_{i}}
3577:
3557:
3537:
3535:{\displaystyle f_{j}}
3510:
3508:{\displaystyle f_{i}}
3483:
3457:
3424:
3373:
3325:
3305:
3285:
3239:
3219:
3199:
3169:
3079:
3053:
3015:
2981:(which exists by the
2976:
2947:
2908:
2888:
2868:
2828:
2802:
2767:
2747:
2700:
2674:
2654:
2634:
2614:
2594:
2574:
2554:
2534:
2514:
2494:
2470:
2446:
2426:
2406:
2386:
2363:
2343:
2320:
2297:
2277:
2041:transfinite recursion
1953:
1899:
1225:is a spanning set of
1169:is indeed a basis of
1136:, contradicting that
1108:that contains all of
1013:
529:partially ordered set
208:Preliminary notions:
187:
182:transfinite induction
178:partially ordered set
154:well-ordering theorem
57:partially ordered set
49:Kuratowski–Zorn lemma
31:
7775:Lemmas in set theory
7710:Ordered vector space
6871:Church–Turing thesis
6858:Computability theory
6067:continuum hypothesis
5585:Square of opposition
5443:Gödel's completeness
5154:Historia Mathematica
4996:Axiomatic Set Theory
4771:
4760:{\displaystyle s(P)}
4742:
4695:
4304:completeness theorem
4154:Kazimierz Kuratowski
4122:
4102:
4082:
4062:
4036:
4016:
3990:
3970:
3950:
3915:
3872:
3852:
3832:
3812:
3792:
3755:
3744:{\displaystyle X'=X}
3724:
3693:
3673:
3653:
3633:
3613:
3586:
3566:
3546:
3519:
3492:
3466:
3433:
3382:
3334:
3314:
3294:
3248:
3228:
3208:
3182:
3091:
3062:
3027:
2992:
2956:
2936:
2897:
2877:
2857:
2811:
2776:
2756:
2709:
2683:
2663:
2643:
2623:
2603:
2583:
2563:
2543:
2523:
2503:
2483:
2459:
2435:
2415:
2395:
2375:
2352:
2332:
2309:
2286:
2266:
1912:
1834:
1213:that is larger than
916:
798:. Suppose then that
713:linearly independent
699:. Now, suppose that
664:Example applications
575:(for non-empty sets)
95:Kazimierz Kuratowski
47:, also known as the
7548:Alexandrov topology
7494:Lexicographic order
7453:Well-quasi-ordering
7025:Mathematical object
6916:P versus NP problem
6881:Computable function
6675:Reverse mathematics
6601:Logical consequence
6478:primitive recursive
6473:elementary function
6246:Free/bound variable
6099:Tarski–Grothendieck
5618:Logical connectives
5548:Logical equivalence
5398:Logical consequence
5217:The Axiom of Choice
4455:Functional Analysis
4298:is contained in an
4289:Tychonoff's theorem
4266:Hahn–Banach theorem
2927:Hahn–Banach theorem
2851: —
2826:{\displaystyle y=x}
2619:has an upper bound
2260: —
2192: —
1961:Using the function
1614:, which shows that
1325:. Assume then that
829:of all the sets in
578: —
525: —
311:(partially) ordered
119:Tychonoff's theorem
107:functional analysis
103:Hahn–Banach theorem
55:. It states that a
7529:Transitive closure
7489:Converse/Transpose
7198:Dilworth's theorem
6823:Transfer principle
6786:Semantics of logic
6771:Categorical theory
6747:Non-standard model
6261:Logical connective
5388:Information theory
5337:Mathematical logic
5244:Dover Publications
5221:Dover Publications
4900:Zorn, Max (1935).
4777:
4757:
4726:
4507:(12 August 2008).
4424:Serre, Jean-Pierre
4339:In popular culture
4201:Mathematics portal
4128:
4108:
4088:
4068:
4042:
4022:
4002:
3976:
3956:
3936:
3901:
3858:
3838:
3818:
3798:
3778:
3741:
3720:. We want to show
3710:
3679:
3659:
3639:
3619:
3599:
3572:
3552:
3532:
3505:
3478:
3452:
3419:
3368:
3320:
3300:
3280:
3234:
3214:
3194:
3164:
3074:
3048:
3010:
2971:
2942:
2903:
2883:
2863:
2849:
2823:
2797:
2762:
2742:
2695:
2669:
2649:
2629:
2609:
2589:
2569:
2549:
2529:
2509:
2489:
2465:
2441:
2421:
2401:
2381:
2358:
2338:
2315:
2292:
2272:
2258:
2210:is any element of
2190:
1948:
1894:
1713:Now, we show that
1008:
572:
523:
42:
7757:
7756:
7715:Partially ordered
7524:Symmetric closure
7509:Reflexive closure
7252:
7061:
7060:
6993:Abstract category
6796:Theories of truth
6606:Rule of inference
6596:Natural deduction
6577:
6576:
6122:
6121:
5827:Cartesian product
5732:
5731:
5638:Many-valued logic
5613:Boolean functions
5496:Russell's paradox
5471:diagonal argument
5368:First-order logic
5253:978-0-486-48841-7
5230:978-0-486-46624-8
5204:978-0-521-59465-3
4967:978-1-4612-6619-8
4946:Krantz, Steven G.
4837:, § 17. Exercise.
4818:, § 16. Exercise.
4799:, § 16. Exercise.
4780:{\displaystyle P}
4609:978-3-319-11477-4
4594:Bergman, George M
4578:978-3-319-11477-4
4563:Bergman, George M
4555:978-0-13-569302-5
4537:978-0-387-95385-4
4395:Tarski finiteness
4360:Bart's New Friend
4308:first-order logic
4131:{\displaystyle X}
4111:{\displaystyle P}
4091:{\displaystyle X}
4071:{\displaystyle P}
4025:{\displaystyle f}
3979:{\displaystyle P}
3959:{\displaystyle g}
3861:{\displaystyle g}
3841:{\displaystyle f}
3821:{\displaystyle s}
3801:{\displaystyle S}
3682:{\displaystyle P}
3662:{\displaystyle f}
3642:{\displaystyle P}
3622:{\displaystyle P}
3575:{\displaystyle P}
3555:{\displaystyle f}
3323:{\displaystyle f}
3303:{\displaystyle P}
3237:{\displaystyle g}
3217:{\displaystyle f}
2945:{\displaystyle X}
2906:{\displaystyle P}
2886:{\displaystyle P}
2866:{\displaystyle P}
2847:
2788:
2765:{\displaystyle C}
2721:
2672:{\displaystyle x}
2652:{\displaystyle P}
2632:{\displaystyle x}
2612:{\displaystyle C}
2592:{\displaystyle P}
2572:{\displaystyle F}
2552:{\displaystyle C}
2532:{\displaystyle P}
2512:{\displaystyle F}
2492:{\displaystyle F}
2468:{\displaystyle F}
2444:{\displaystyle F}
2424:{\displaystyle S}
2411:, each subset of
2404:{\displaystyle F}
2384:{\displaystyle S}
2361:{\displaystyle F}
2341:{\displaystyle F}
2318:{\displaystyle F}
2295:{\displaystyle F}
2275:{\displaystyle F}
2256:
2188:
1694:is an element of
1662:is an element of
1562:is an element of
1554:is an element of
1201:. This says that
725:zero vector space
570:
521:
147:algebraic closure
7787:
7499:Linear extension
7248:
7228:Mirsky's theorem
7088:
7081:
7074:
7065:
7064:
7052:
7051:
7003:History of logic
6998:Category of sets
6891:Decision problem
6670:Ordinal analysis
6611:Sequent calculus
6509:Boolean algebras
6449:
6448:
6423:
6394:logical/constant
6148:
6147:
6134:
6057:Zermelo–Fraenkel
5808:Set operations:
5743:
5742:
5680:
5511:
5510:
5491:Löwenheim–Skolem
5378:Formal semantics
5330:
5323:
5316:
5307:
5306:
5282:
5257:
5234:
5208:
5179:Naive set theory
5171:
5169:
5136:
5135:
5128:
5122:
5121:
5120:
5098:
5089:
5082:
5076:
5075:
5073:
5049:
5043:
5042:
5024:
5018:
5017:
4991:
4985:
4978:
4972:
4970:
4942:
4936:
4930:
4924:
4923:
4921:
4897:
4891:
4890:
4888:
4886:
4881:
4861:
4851:
4845:
4844:
4832:
4826:
4825:
4813:
4807:
4806:
4794:
4788:
4786:
4784:
4783:
4778:
4766:
4764:
4763:
4758:
4735:
4733:
4732:
4727:
4716:
4715:
4688:
4676:
4670:
4669:
4645:
4639:
4638:
4636:
4634:
4629:
4620:
4614:
4613:
4590:
4584:
4582:
4559:
4546:Abstract Algebra
4541:
4519:
4513:
4512:
4501:
4495:
4489:
4483:
4477:
4471:
4465:
4459:
4458:
4450:
4444:
4438:
4432:
4431:
4420:
4400:
4358:in the episode "
4203:
4198:
4197:
4137:
4135:
4134:
4129:
4117:
4115:
4114:
4109:
4097:
4095:
4094:
4089:
4077:
4075:
4074:
4069:
4051:
4049:
4048:
4043:
4031:
4029:
4028:
4023:
4011:
4009:
4008:
4003:
3985:
3983:
3982:
3977:
3965:
3963:
3962:
3957:
3945:
3943:
3942:
3937:
3910:
3908:
3907:
3902:
3894:
3893:
3892:
3883:
3867:
3865:
3864:
3859:
3847:
3845:
3844:
3839:
3827:
3825:
3824:
3819:
3807:
3805:
3804:
3799:
3787:
3785:
3784:
3779:
3777:
3750:
3748:
3747:
3742:
3734:
3719:
3717:
3716:
3711:
3703:
3688:
3686:
3685:
3680:
3668:
3666:
3665:
3660:
3648:
3646:
3645:
3640:
3628:
3626:
3625:
3620:
3608:
3606:
3605:
3600:
3598:
3597:
3581:
3579:
3578:
3573:
3561:
3559:
3558:
3553:
3541:
3539:
3538:
3533:
3531:
3530:
3514:
3512:
3511:
3506:
3504:
3503:
3487:
3485:
3484:
3479:
3461:
3459:
3458:
3453:
3451:
3450:
3428:
3426:
3425:
3420:
3409:
3408:
3377:
3375:
3374:
3369:
3367:
3366:
3357:
3356:
3344:
3329:
3327:
3326:
3321:
3309:
3307:
3306:
3301:
3289:
3287:
3286:
3281:
3273:
3272:
3260:
3259:
3243:
3241:
3240:
3235:
3223:
3221:
3220:
3215:
3203:
3201:
3200:
3195:
3173:
3171:
3170:
3165:
3133:
3116:
3083:
3081:
3080:
3075:
3057:
3055:
3054:
3049:
3019:
3017:
3016:
3011:
2980:
2978:
2977:
2972:
2951:
2949:
2948:
2943:
2912:
2910:
2909:
2904:
2892:
2890:
2889:
2884:
2872:
2870:
2869:
2864:
2852:
2832:
2830:
2829:
2824:
2806:
2804:
2803:
2798:
2790:
2789:
2781:
2771:
2769:
2768:
2763:
2751:
2749:
2748:
2743:
2723:
2722:
2714:
2704:
2702:
2701:
2696:
2678:
2676:
2675:
2670:
2658:
2656:
2655:
2650:
2638:
2636:
2635:
2630:
2618:
2616:
2615:
2610:
2598:
2596:
2595:
2590:
2578:
2576:
2575:
2570:
2558:
2556:
2555:
2550:
2538:
2536:
2535:
2530:
2518:
2516:
2515:
2510:
2498:
2496:
2495:
2490:
2474:
2472:
2471:
2466:
2450:
2448:
2447:
2442:
2430:
2428:
2427:
2422:
2410:
2408:
2407:
2402:
2390:
2388:
2387:
2382:
2367:
2365:
2364:
2359:
2347:
2345:
2344:
2339:
2324:
2322:
2321:
2316:
2301:
2299:
2298:
2293:
2281:
2279:
2278:
2273:
2261:
2237:Naive Set Theory
2193:
2177:) to the "last"
2121:} as subsets of
2092:}). Because the
1957:
1955:
1954:
1949:
1903:
1901:
1900:
1895:
1534:are elements of
1017:
1015:
1014:
1009:
1004:
996:
995:
990:
984:
983:
965:
964:
959:
953:
952:
940:
939:
934:
928:
927:
743:}. Furthermore,
579:
576:
526:
309:) is said to be
308:
298:
288:
274:
264:
254:
236:
219:equipped with a
199:
156:and also to the
131:abstract algebra
67:(that is, every
7795:
7794:
7790:
7789:
7788:
7786:
7785:
7784:
7770:Axiom of choice
7760:
7759:
7758:
7753:
7749:Young's lattice
7605:
7533:
7472:
7322:Heyting algebra
7270:Boolean algebra
7242:
7223:Laver's theorem
7171:
7137:Boolean algebra
7132:Binary relation
7120:
7097:
7092:
7062:
7057:
7046:
7039:
6984:Category theory
6974:Algebraic logic
6957:
6928:Lambda calculus
6866:Church encoding
6852:
6828:Truth predicate
6684:
6650:Complete theory
6573:
6442:
6438:
6434:
6429:
6421:
6141: and
6137:
6132:
6118:
6094:New Foundations
6062:axiom of choice
6045:
6007:Gödel numbering
5947: and
5939:
5843:
5728:
5678:
5659:
5608:Boolean algebra
5594:
5558:Equiconsistency
5523:Classical logic
5500:
5481:Halting problem
5469: and
5445: and
5433: and
5432:
5427:Theorems (
5422:
5339:
5334:
5300:Unicode version
5267:
5264:
5254:
5231:
5205:
5144:
5139:
5130:
5129:
5125:
5099:
5092:
5083:
5079:
5050:
5046:
5025:
5021:
5014:
4992:
4988:
4979:
4975:
4968:
4943:
4939:
4931:
4927:
4912:(10): 667–670.
4898:
4894:
4884:
4882:
4859:
4852:
4848:
4838:
4833:
4829:
4819:
4814:
4810:
4800:
4795:
4791:
4772:
4769:
4768:
4743:
4740:
4739:
4711:
4710:
4696:
4693:
4692:
4682:
4677:
4673:
4646:
4642:
4632:
4630:
4627:
4621:
4617:
4610:
4591:
4587:
4579:
4556:
4538:
4520:
4516:
4502:
4498:
4490:
4486:
4478:
4474:
4466:
4462:
4451:
4447:
4439:
4435:
4421:
4417:
4413:
4398:
4368:
4341:
4325:
4319:
4283:Krull's theorem
4232:Axiom of choice
4216:
4210:
4199:
4192:
4144:
4123:
4120:
4119:
4103:
4100:
4099:
4083:
4080:
4079:
4063:
4060:
4059:
4037:
4034:
4033:
4017:
4014:
4013:
3991:
3988:
3987:
3971:
3968:
3967:
3951:
3948:
3947:
3916:
3913:
3912:
3885:
3884:
3879:
3878:
3873:
3870:
3869:
3853:
3850:
3849:
3833:
3830:
3829:
3813:
3810:
3809:
3793:
3790:
3789:
3770:
3756:
3753:
3752:
3727:
3725:
3722:
3721:
3696:
3694:
3691:
3690:
3674:
3671:
3670:
3654:
3651:
3650:
3634:
3631:
3630:
3614:
3611:
3610:
3593:
3589:
3587:
3584:
3583:
3567:
3564:
3563:
3547:
3544:
3543:
3542:. The function
3526:
3522:
3520:
3517:
3516:
3499:
3495:
3493:
3490:
3489:
3467:
3464:
3463:
3446:
3442:
3434:
3431:
3430:
3404:
3400:
3383:
3380:
3379:
3362:
3358:
3352:
3348:
3337:
3335:
3332:
3331:
3315:
3312:
3311:
3295:
3292:
3291:
3268:
3264:
3255:
3251:
3249:
3246:
3245:
3229:
3226:
3225:
3209:
3206:
3205:
3204:if and only if
3183:
3180:
3179:
3126:
3109:
3092:
3089:
3088:
3063:
3060:
3059:
3028:
3025:
3024:
2993:
2990:
2989:
2957:
2954:
2953:
2937:
2934:
2933:
2923:
2915:
2898:
2895:
2894:
2878:
2875:
2874:
2858:
2855:
2854:
2850:
2812:
2809:
2808:
2780:
2779:
2777:
2774:
2773:
2757:
2754:
2753:
2752:is larger than
2713:
2712:
2710:
2707:
2706:
2684:
2681:
2680:
2664:
2661:
2660:
2644:
2641:
2640:
2624:
2621:
2620:
2604:
2601:
2600:
2584:
2581:
2580:
2564:
2561:
2560:
2544:
2541:
2540:
2524:
2521:
2520:
2504:
2501:
2500:
2484:
2481:
2480:
2477:
2460:
2457:
2456:
2436:
2433:
2432:
2416:
2413:
2412:
2396:
2393:
2392:
2376:
2373:
2372:
2353:
2350:
2349:
2333:
2330:
2329:
2310:
2307:
2306:
2287:
2284:
2283:
2267:
2264:
2263:
2259:
2250:
2228:
2202:in which every
2191:
2112:
2100:
2083:
2070:
2049:
2039:are defined by
2037:
2011:natural numbers
2000:
1996:
1993:< ... < a
1992:
1985:
1978:
1971:
1913:
1910:
1909:
1835:
1832:
1831:
1797:axiom of choice
1793:
1706:is an ideal in
1630:#3 - For every
1622:is a member of
1610:is a member of
1498:#2 - For every
1486:, so the union
1482:is a subset of
1380:is an ideal of
1282:itself). Since
1239:
1142:
1135:
1128:
1121:
1114:
1107:
1100:
1093:
1086:
1067:
1060:
1049:
1042:
1035:
1000:
991:
986:
985:
979:
975:
960:
955:
954:
948:
944:
935:
930:
929:
923:
919:
917:
914:
913:
908:
901:
894:
880:
873:
866:
778:is a subset of
753:inclusion order
671:
666:
594:
588:. Then the set
577:
574:
560:
556:maximal element
550:. Then the set
524:
469:Given a subset
378:totally ordered
300:
290:
280:
266:
256:
246:
228:
221:binary relation
206:
200:
194:
174:
158:axiom of choice
141:and that every
91:axiom of choice
80:maximal element
69:totally ordered
24:
21:
12:
11:
5:
7793:
7783:
7782:
7777:
7772:
7755:
7754:
7752:
7751:
7746:
7741:
7740:
7739:
7729:
7728:
7727:
7722:
7717:
7707:
7706:
7705:
7695:
7690:
7689:
7688:
7683:
7676:Order morphism
7673:
7672:
7671:
7661:
7656:
7651:
7646:
7641:
7640:
7639:
7629:
7624:
7619:
7613:
7611:
7607:
7606:
7604:
7603:
7602:
7601:
7596:
7594:Locally convex
7591:
7586:
7576:
7574:Order topology
7571:
7570:
7569:
7567:Order topology
7564:
7554:
7544:
7542:
7535:
7534:
7532:
7531:
7526:
7521:
7516:
7511:
7506:
7501:
7496:
7491:
7486:
7480:
7478:
7474:
7473:
7471:
7470:
7460:
7450:
7445:
7440:
7435:
7430:
7425:
7420:
7415:
7414:
7413:
7403:
7398:
7397:
7396:
7391:
7386:
7381:
7379:Chain-complete
7371:
7366:
7365:
7364:
7359:
7354:
7349:
7344:
7334:
7329:
7324:
7319:
7314:
7304:
7299:
7294:
7289:
7284:
7279:
7278:
7277:
7267:
7262:
7256:
7254:
7244:
7243:
7241:
7240:
7235:
7230:
7225:
7220:
7215:
7210:
7205:
7200:
7195:
7190:
7185:
7179:
7177:
7173:
7172:
7170:
7169:
7164:
7159:
7154:
7149:
7144:
7139:
7134:
7128:
7126:
7122:
7121:
7119:
7118:
7113:
7108:
7102:
7099:
7098:
7091:
7090:
7083:
7076:
7068:
7059:
7058:
7044:
7041:
7040:
7038:
7037:
7032:
7027:
7022:
7017:
7016:
7015:
7005:
7000:
6995:
6986:
6981:
6976:
6971:
6969:Abstract logic
6965:
6963:
6959:
6958:
6956:
6955:
6950:
6948:Turing machine
6945:
6940:
6935:
6930:
6925:
6920:
6919:
6918:
6913:
6908:
6903:
6898:
6888:
6886:Computable set
6883:
6878:
6873:
6868:
6862:
6860:
6854:
6853:
6851:
6850:
6845:
6840:
6835:
6830:
6825:
6820:
6815:
6814:
6813:
6808:
6803:
6793:
6788:
6783:
6781:Satisfiability
6778:
6773:
6768:
6767:
6766:
6756:
6755:
6754:
6744:
6743:
6742:
6737:
6732:
6727:
6722:
6712:
6711:
6710:
6705:
6698:Interpretation
6694:
6692:
6686:
6685:
6683:
6682:
6677:
6672:
6667:
6662:
6652:
6647:
6646:
6645:
6644:
6643:
6633:
6628:
6618:
6613:
6608:
6603:
6598:
6593:
6587:
6585:
6579:
6578:
6575:
6574:
6572:
6571:
6563:
6562:
6561:
6560:
6555:
6554:
6553:
6548:
6543:
6523:
6522:
6521:
6519:minimal axioms
6516:
6505:
6504:
6503:
6492:
6491:
6490:
6485:
6480:
6475:
6470:
6465:
6452:
6450:
6431:
6430:
6428:
6427:
6426:
6425:
6413:
6408:
6407:
6406:
6401:
6396:
6391:
6381:
6376:
6371:
6366:
6365:
6364:
6359:
6349:
6348:
6347:
6342:
6337:
6332:
6322:
6317:
6316:
6315:
6310:
6305:
6295:
6294:
6293:
6288:
6283:
6278:
6273:
6268:
6258:
6253:
6248:
6243:
6242:
6241:
6236:
6231:
6226:
6216:
6211:
6209:Formation rule
6206:
6201:
6200:
6199:
6194:
6184:
6183:
6182:
6172:
6167:
6162:
6157:
6151:
6145:
6128:Formal systems
6124:
6123:
6120:
6119:
6117:
6116:
6111:
6106:
6101:
6096:
6091:
6086:
6081:
6076:
6071:
6070:
6069:
6064:
6053:
6051:
6047:
6046:
6044:
6043:
6042:
6041:
6031:
6026:
6025:
6024:
6017:Large cardinal
6014:
6009:
6004:
5999:
5994:
5980:
5979:
5978:
5973:
5968:
5953:
5951:
5941:
5940:
5938:
5937:
5936:
5935:
5930:
5925:
5915:
5910:
5905:
5900:
5895:
5890:
5885:
5880:
5875:
5870:
5865:
5860:
5854:
5852:
5845:
5844:
5842:
5841:
5840:
5839:
5834:
5829:
5824:
5819:
5814:
5806:
5805:
5804:
5799:
5789:
5784:
5782:Extensionality
5779:
5777:Ordinal number
5774:
5764:
5759:
5758:
5757:
5746:
5740:
5734:
5733:
5730:
5729:
5727:
5726:
5721:
5716:
5711:
5706:
5701:
5696:
5695:
5694:
5684:
5683:
5682:
5669:
5667:
5661:
5660:
5658:
5657:
5656:
5655:
5650:
5645:
5635:
5630:
5625:
5620:
5615:
5610:
5604:
5602:
5596:
5595:
5593:
5592:
5587:
5582:
5577:
5572:
5567:
5562:
5561:
5560:
5550:
5545:
5540:
5535:
5530:
5525:
5519:
5517:
5508:
5502:
5501:
5499:
5498:
5493:
5488:
5483:
5478:
5473:
5461:Cantor's
5459:
5454:
5449:
5439:
5437:
5424:
5423:
5421:
5420:
5415:
5410:
5405:
5400:
5395:
5390:
5385:
5380:
5375:
5370:
5365:
5360:
5359:
5358:
5347:
5345:
5341:
5340:
5333:
5332:
5325:
5318:
5310:
5304:
5303:
5289:
5283:
5263:
5262:External links
5260:
5259:
5258:
5252:
5235:
5229:
5209:
5203:
5190:
5172:
5143:
5140:
5138:
5137:
5123:
5111:(3): 365–367,
5090:
5077:
5044:
5019:
5012:
4986:
4973:
4966:
4937:
4925:
4892:
4846:
4827:
4808:
4789:
4776:
4756:
4753:
4750:
4747:
4737:
4736:
4725:
4722:
4719:
4714:
4709:
4706:
4703:
4700:
4671:
4660:(4): 353–354.
4640:
4615:
4608:
4585:
4577:
4554:
4536:
4514:
4496:
4484:
4472:
4460:
4445:
4433:
4414:
4412:
4409:
4408:
4407:
4401:
4392:
4386:
4380:
4375:
4367:
4364:
4343:The 1970 film
4340:
4337:
4318:
4315:
4311:
4310:
4292:
4286:
4275:
4268:
4241:
4240:
4234:
4229:
4209:
4206:
4205:
4204:
4143:
4140:
4127:
4107:
4087:
4067:
4041:
4021:
4001:
3998:
3995:
3975:
3955:
3935:
3932:
3929:
3926:
3923:
3920:
3900:
3897:
3891:
3888:
3882:
3877:
3857:
3848:to a function
3837:
3817:
3797:
3776:
3773:
3769:
3766:
3763:
3760:
3740:
3737:
3733:
3730:
3709:
3706:
3702:
3699:
3678:
3658:
3638:
3618:
3596:
3592:
3571:
3551:
3529:
3525:
3502:
3498:
3477:
3474:
3471:
3449:
3445:
3441:
3438:
3418:
3415:
3412:
3407:
3403:
3399:
3396:
3393:
3390:
3387:
3365:
3361:
3355:
3351:
3347:
3343:
3340:
3319:
3299:
3290:is a chain in
3279:
3276:
3271:
3267:
3263:
3258:
3254:
3233:
3213:
3193:
3190:
3187:
3176:
3175:
3163:
3160:
3157:
3154:
3151:
3148:
3145:
3142:
3139:
3136:
3132:
3129:
3125:
3122:
3119:
3115:
3112:
3108:
3105:
3102:
3099:
3096:
3073:
3070:
3067:
3047:
3044:
3041:
3038:
3035:
3032:
3021:
3020:
3009:
3006:
3003:
3000:
2997:
2983:axiom of union
2970:
2967:
2964:
2961:
2941:
2922:
2919:
2902:
2882:
2862:
2845:
2822:
2819:
2816:
2796:
2793:
2787:
2784:
2761:
2741:
2738:
2735:
2732:
2729:
2726:
2720:
2717:
2694:
2691:
2688:
2668:
2648:
2628:
2608:
2588:
2568:
2548:
2528:
2508:
2488:
2464:
2453:
2452:
2440:
2420:
2400:
2380:
2369:
2357:
2337:
2326:
2314:
2291:
2271:
2254:
2249:
2246:
2186:
2108:
2096:
2079:
2066:
2047:
2035:
1998:
1994:
1990:
1983:
1976:
1969:
1947:
1944:
1941:
1938:
1935:
1932:
1929:
1926:
1923:
1920:
1917:
1893:
1890:
1887:
1884:
1881:
1878:
1875:
1872:
1869:
1866:
1863:
1860:
1857:
1854:
1851:
1848:
1845:
1842:
1839:
1792:
1789:
1756:were equal to
1726:if and only if
1646:, the product
1455:
1454:
1445:, the product
1427:
1400:
1238:
1235:
1140:
1133:
1126:
1119:
1112:
1105:
1098:
1091:
1084:
1065:
1058:
1047:
1040:
1033:
1019:
1018:
1007:
1003:
999:
994:
989:
982:
978:
974:
971:
968:
963:
958:
951:
947:
943:
938:
933:
926:
922:
906:
899:
892:
878:
871:
864:
670:
667:
665:
662:
568:
519:
515:
514:
467:
420:
381:
337:is said to be
205:
202:
192:
185:Zorn's lemma.
173:
170:
127:compact spaces
22:
9:
6:
4:
3:
2:
7792:
7781:
7778:
7776:
7773:
7771:
7768:
7767:
7765:
7750:
7747:
7745:
7742:
7738:
7735:
7734:
7733:
7730:
7726:
7723:
7721:
7718:
7716:
7713:
7712:
7711:
7708:
7704:
7701:
7700:
7699:
7698:Ordered field
7696:
7694:
7691:
7687:
7684:
7682:
7679:
7678:
7677:
7674:
7670:
7667:
7666:
7665:
7662:
7660:
7657:
7655:
7654:Hasse diagram
7652:
7650:
7647:
7645:
7642:
7638:
7635:
7634:
7633:
7632:Comparability
7630:
7628:
7625:
7623:
7620:
7618:
7615:
7614:
7612:
7608:
7600:
7597:
7595:
7592:
7590:
7587:
7585:
7582:
7581:
7580:
7577:
7575:
7572:
7568:
7565:
7563:
7560:
7559:
7558:
7555:
7553:
7549:
7546:
7545:
7543:
7540:
7536:
7530:
7527:
7525:
7522:
7520:
7517:
7515:
7512:
7510:
7507:
7505:
7504:Product order
7502:
7500:
7497:
7495:
7492:
7490:
7487:
7485:
7482:
7481:
7479:
7477:Constructions
7475:
7469:
7465:
7461:
7458:
7454:
7451:
7449:
7446:
7444:
7441:
7439:
7436:
7434:
7431:
7429:
7426:
7424:
7421:
7419:
7416:
7412:
7409:
7408:
7407:
7404:
7402:
7399:
7395:
7392:
7390:
7387:
7385:
7382:
7380:
7377:
7376:
7375:
7374:Partial order
7372:
7370:
7367:
7363:
7362:Join and meet
7360:
7358:
7355:
7353:
7350:
7348:
7345:
7343:
7340:
7339:
7338:
7335:
7333:
7330:
7328:
7325:
7323:
7320:
7318:
7315:
7313:
7309:
7305:
7303:
7300:
7298:
7295:
7293:
7290:
7288:
7285:
7283:
7280:
7276:
7273:
7272:
7271:
7268:
7266:
7263:
7261:
7260:Antisymmetric
7258:
7257:
7255:
7251:
7245:
7239:
7236:
7234:
7231:
7229:
7226:
7224:
7221:
7219:
7216:
7214:
7211:
7209:
7206:
7204:
7201:
7199:
7196:
7194:
7191:
7189:
7186:
7184:
7181:
7180:
7178:
7174:
7168:
7167:Weak ordering
7165:
7163:
7160:
7158:
7155:
7153:
7152:Partial order
7150:
7148:
7145:
7143:
7140:
7138:
7135:
7133:
7130:
7129:
7127:
7123:
7117:
7114:
7112:
7109:
7107:
7104:
7103:
7100:
7096:
7089:
7084:
7082:
7077:
7075:
7070:
7069:
7066:
7056:
7055:
7050:
7042:
7036:
7033:
7031:
7028:
7026:
7023:
7021:
7018:
7014:
7011:
7010:
7009:
7006:
7004:
7001:
6999:
6996:
6994:
6990:
6987:
6985:
6982:
6980:
6977:
6975:
6972:
6970:
6967:
6966:
6964:
6960:
6954:
6951:
6949:
6946:
6944:
6943:Recursive set
6941:
6939:
6936:
6934:
6931:
6929:
6926:
6924:
6921:
6917:
6914:
6912:
6909:
6907:
6904:
6902:
6899:
6897:
6894:
6893:
6892:
6889:
6887:
6884:
6882:
6879:
6877:
6874:
6872:
6869:
6867:
6864:
6863:
6861:
6859:
6855:
6849:
6846:
6844:
6841:
6839:
6836:
6834:
6831:
6829:
6826:
6824:
6821:
6819:
6816:
6812:
6809:
6807:
6804:
6802:
6799:
6798:
6797:
6794:
6792:
6789:
6787:
6784:
6782:
6779:
6777:
6774:
6772:
6769:
6765:
6762:
6761:
6760:
6757:
6753:
6752:of arithmetic
6750:
6749:
6748:
6745:
6741:
6738:
6736:
6733:
6731:
6728:
6726:
6723:
6721:
6718:
6717:
6716:
6713:
6709:
6706:
6704:
6701:
6700:
6699:
6696:
6695:
6693:
6691:
6687:
6681:
6678:
6676:
6673:
6671:
6668:
6666:
6663:
6660:
6659:from ZFC
6656:
6653:
6651:
6648:
6642:
6639:
6638:
6637:
6634:
6632:
6629:
6627:
6624:
6623:
6622:
6619:
6617:
6614:
6612:
6609:
6607:
6604:
6602:
6599:
6597:
6594:
6592:
6589:
6588:
6586:
6584:
6580:
6570:
6569:
6565:
6564:
6559:
6558:non-Euclidean
6556:
6552:
6549:
6547:
6544:
6542:
6541:
6537:
6536:
6534:
6531:
6530:
6528:
6524:
6520:
6517:
6515:
6512:
6511:
6510:
6506:
6502:
6499:
6498:
6497:
6493:
6489:
6486:
6484:
6481:
6479:
6476:
6474:
6471:
6469:
6466:
6464:
6461:
6460:
6458:
6454:
6453:
6451:
6446:
6440:
6435:Example
6432:
6424:
6419:
6418:
6417:
6414:
6412:
6409:
6405:
6402:
6400:
6397:
6395:
6392:
6390:
6387:
6386:
6385:
6382:
6380:
6377:
6375:
6372:
6370:
6367:
6363:
6360:
6358:
6355:
6354:
6353:
6350:
6346:
6343:
6341:
6338:
6336:
6333:
6331:
6328:
6327:
6326:
6323:
6321:
6318:
6314:
6311:
6309:
6306:
6304:
6301:
6300:
6299:
6296:
6292:
6289:
6287:
6284:
6282:
6279:
6277:
6274:
6272:
6269:
6267:
6264:
6263:
6262:
6259:
6257:
6254:
6252:
6249:
6247:
6244:
6240:
6237:
6235:
6232:
6230:
6227:
6225:
6222:
6221:
6220:
6217:
6215:
6212:
6210:
6207:
6205:
6202:
6198:
6195:
6193:
6192:by definition
6190:
6189:
6188:
6185:
6181:
6178:
6177:
6176:
6173:
6171:
6168:
6166:
6163:
6161:
6158:
6156:
6153:
6152:
6149:
6146:
6144:
6140:
6135:
6129:
6125:
6115:
6112:
6110:
6107:
6105:
6102:
6100:
6097:
6095:
6092:
6090:
6087:
6085:
6082:
6080:
6079:Kripke–Platek
6077:
6075:
6072:
6068:
6065:
6063:
6060:
6059:
6058:
6055:
6054:
6052:
6048:
6040:
6037:
6036:
6035:
6032:
6030:
6027:
6023:
6020:
6019:
6018:
6015:
6013:
6010:
6008:
6005:
6003:
6000:
5998:
5995:
5992:
5988:
5984:
5981:
5977:
5974:
5972:
5969:
5967:
5964:
5963:
5962:
5958:
5955:
5954:
5952:
5950:
5946:
5942:
5934:
5931:
5929:
5926:
5924:
5923:constructible
5921:
5920:
5919:
5916:
5914:
5911:
5909:
5906:
5904:
5901:
5899:
5896:
5894:
5891:
5889:
5886:
5884:
5881:
5879:
5876:
5874:
5871:
5869:
5866:
5864:
5861:
5859:
5856:
5855:
5853:
5851:
5846:
5838:
5835:
5833:
5830:
5828:
5825:
5823:
5820:
5818:
5815:
5813:
5810:
5809:
5807:
5803:
5800:
5798:
5795:
5794:
5793:
5790:
5788:
5785:
5783:
5780:
5778:
5775:
5773:
5769:
5765:
5763:
5760:
5756:
5753:
5752:
5751:
5748:
5747:
5744:
5741:
5739:
5735:
5725:
5722:
5720:
5717:
5715:
5712:
5710:
5707:
5705:
5702:
5700:
5697:
5693:
5690:
5689:
5688:
5685:
5681:
5676:
5675:
5674:
5671:
5670:
5668:
5666:
5662:
5654:
5651:
5649:
5646:
5644:
5641:
5640:
5639:
5636:
5634:
5631:
5629:
5626:
5624:
5621:
5619:
5616:
5614:
5611:
5609:
5606:
5605:
5603:
5601:
5600:Propositional
5597:
5591:
5588:
5586:
5583:
5581:
5578:
5576:
5573:
5571:
5568:
5566:
5563:
5559:
5556:
5555:
5554:
5551:
5549:
5546:
5544:
5541:
5539:
5536:
5534:
5531:
5529:
5528:Logical truth
5526:
5524:
5521:
5520:
5518:
5516:
5512:
5509:
5507:
5503:
5497:
5494:
5492:
5489:
5487:
5484:
5482:
5479:
5477:
5474:
5472:
5468:
5464:
5460:
5458:
5455:
5453:
5450:
5448:
5444:
5441:
5440:
5438:
5436:
5430:
5425:
5419:
5416:
5414:
5411:
5409:
5406:
5404:
5401:
5399:
5396:
5394:
5391:
5389:
5386:
5384:
5381:
5379:
5376:
5374:
5371:
5369:
5366:
5364:
5361:
5357:
5354:
5353:
5352:
5349:
5348:
5346:
5342:
5338:
5331:
5326:
5324:
5319:
5317:
5312:
5311:
5308:
5301:
5297:
5293:
5290:
5287:
5284:
5280:
5276:
5275:
5270:
5266:
5265:
5255:
5249:
5245:
5241:
5236:
5232:
5226:
5222:
5218:
5214:
5210:
5206:
5200:
5196:
5191:
5188:
5187:0-387-90092-6
5184:
5180:
5176:
5173:
5168:
5163:
5159:
5155:
5151:
5146:
5145:
5133:
5127:
5119:
5114:
5110:
5106:
5105:
5097:
5095:
5087:
5081:
5072:
5067:
5063:
5059:
5055:
5048:
5040:
5036:
5032:
5031:
5023:
5015:
5013:9780821850268
5009:
5005:
5001:
4997:
4990:
4983:
4977:
4969:
4963:
4959:
4955:
4951:
4947:
4941:
4935:, p. 82.
4934:
4933:Campbell 1978
4929:
4920:
4915:
4911:
4907:
4903:
4896:
4880:
4875:
4871:
4868:(in French).
4867:
4866:
4857:
4850:
4842:
4836:
4831:
4823:
4817:
4812:
4804:
4798:
4793:
4774:
4751:
4745:
4720:
4704:
4701:
4698:
4691:
4690:
4686:
4680:
4675:
4667:
4663:
4659:
4655:
4651:
4644:
4626:
4619:
4611:
4605:
4601:
4600:
4595:
4589:
4580:
4574:
4570:
4569:
4564:
4557:
4551:
4547:
4539:
4533:
4529:
4525:
4521:For example,
4518:
4510:
4506:
4500:
4494:, p. 168
4493:
4488:
4481:
4476:
4469:
4464:
4456:
4449:
4443:, p. 168
4442:
4437:
4429:
4425:
4419:
4415:
4405:
4402:
4396:
4393:
4390:
4387:
4384:
4381:
4379:
4376:
4373:
4370:
4369:
4363:
4361:
4357:
4356:
4350:
4348:
4347:
4336:
4332:
4330:
4324:
4314:
4309:
4305:
4301:
4297:
4296:proper filter
4293:
4290:
4287:
4284:
4280:
4279:maximal ideal
4276:
4273:
4269:
4267:
4263:
4260:
4259:
4258:
4255:
4253:
4248:
4246:
4238:
4235:
4233:
4230:
4228:
4225:
4224:
4223:
4221:
4215:
4202:
4196:
4191:
4189:
4187:
4183:
4179:
4175:
4171:
4166:
4164:
4160:
4155:
4151:
4149:
4139:
4125:
4105:
4085:
4065:
4057:
4052:
4039:
4019:
3999:
3996:
3993:
3973:
3953:
3933:
3930:
3924:
3918:
3898:
3895:
3889:
3886:
3875:
3855:
3835:
3815:
3795:
3774:
3771:
3767:
3764:
3761:
3758:
3738:
3735:
3731:
3728:
3707:
3704:
3700:
3697:
3676:
3656:
3636:
3616:
3594:
3590:
3569:
3549:
3527:
3523:
3500:
3496:
3475:
3472:
3469:
3447:
3443:
3439:
3436:
3413:
3405:
3401:
3397:
3391:
3385:
3363:
3359:
3353:
3349:
3345:
3341:
3338:
3330:on the union
3317:
3297:
3277:
3269:
3265:
3261:
3256:
3252:
3231:
3211:
3191:
3188:
3185:
3158:
3155:
3149:
3143:
3140:
3137:
3134:
3130:
3127:
3123:
3120:
3113:
3110:
3106:
3103:
3097:
3094:
3087:
3086:
3085:
3071:
3068:
3065:
3045:
3042:
3036:
3030:
3007:
3001:
2998:
2995:
2988:
2987:
2986:
2984:
2968:
2965:
2962:
2959:
2939:
2930:
2928:
2918:
2914:
2900:
2880:
2860:
2844:
2841:
2839:
2834:
2820:
2817:
2814:
2794:
2791:
2785:
2782:
2759:
2736:
2730:
2727:
2724:
2718:
2715:
2692:
2689:
2686:
2666:
2646:
2626:
2606:
2586:
2566:
2546:
2526:
2506:
2486:
2476:
2462:
2438:
2418:
2398:
2378:
2371:For each set
2370:
2355:
2335:
2327:
2312:
2305:
2304:
2303:
2289:
2269:
2253:
2245:
2243:
2239:
2238:
2233:
2227:
2225:
2221:
2217:
2213:
2209:
2205:
2201:
2197:
2185:
2182:
2180:
2176:
2172:
2168:
2164:
2160:
2156:
2152:
2148:
2144:
2140:
2136:
2132:
2128:
2124:
2120:
2116:
2111:
2107:
2102:
2099:
2095:
2091:
2087:
2082:
2078:
2074:
2069:
2065:
2061:
2057:
2053:
2046:
2042:
2038:
2030:
2028:
2024:
2020:
2016:
2012:
2008:
2004:
1989:
1982:
1975:
1968:
1964:
1959:
1942:
1936:
1933:
1927:
1921:
1918:
1915:
1907:
1888:
1885:
1882:
1879:
1876:
1873:
1870:
1864:
1861:
1858:
1855:
1849:
1843:
1837:
1829:
1826:
1822:
1818:
1814:
1810:
1806:
1802:
1798:
1788:
1786:
1782:
1777:
1775:
1771:
1767:
1763:
1759:
1755:
1751:
1747:
1743:
1739:
1735:
1731:
1727:
1724:
1720:
1716:
1711:
1709:
1705:
1701:
1697:
1693:
1689:
1685:
1681:
1677:
1673:
1669:
1665:
1661:
1656:
1655:
1653:
1649:
1645:
1641:
1637:
1633:
1627:
1625:
1621:
1617:
1613:
1609:
1605:
1601:
1597:
1593:
1589:
1585:
1581:
1577:
1573:
1569:
1565:
1561:
1557:
1553:
1549:
1545:
1541:
1537:
1533:
1529:
1524:
1523:
1521:
1517:
1513:
1509:
1505:
1501:
1495:
1493:
1489:
1485:
1481:
1477:
1473:
1468:
1467:
1465:
1461:
1452:
1448:
1444:
1440:
1436:
1432:
1428:
1425:
1421:
1417:
1413:
1409:
1405:
1401:
1398:
1394:
1391:
1390:
1389:
1387:
1383:
1379:
1375:
1371:
1367:
1363:
1359:
1354:
1352:
1348:
1344:
1340:
1336:
1332:
1328:
1324:
1320:
1316:
1312:
1308:
1303:
1301:
1297:
1293:
1289:
1285:
1281:
1277:
1273:
1269:
1266:
1262:
1257:
1255:
1254:maximal ideal
1251:
1247:
1244:
1234:
1232:
1228:
1224:
1221:. Therefore,
1220:
1216:
1212:
1208:
1204:
1200:
1196:
1192:
1188:
1184:
1180:
1176:
1172:
1168:
1163:
1161:
1157:
1153:
1148:
1146:
1139:
1132:
1125:
1118:
1111:
1104:
1097:
1090:
1083:
1079:
1075:
1072:= 1, 2, ...,
1071:
1064:
1057:
1053:
1046:
1039:
1032:
1028:
1024:
1005:
997:
992:
980:
976:
972:
969:
966:
961:
949:
945:
941:
936:
924:
920:
912:
911:
910:
905:
898:
891:
888:
884:
877:
870:
863:
859:
854:
852:
848:
844:
840:
836:
832:
828:
824:
819:
817:
813:
809:
805:
801:
797:
793:
789:
785:
781:
777:
773:
769:
764:
762:
758:
754:
750:
749:set inclusion
746:
742:
738:
734:
730:
726:
722:
718:
714:
710:
706:
702:
698:
694:
690:
685:
683:
679:
676:
661:
659:
653:
651:
647:
643:
639:
635:
631:
627:
623:
619:
615:
611:
607:
603:
599:
593:
591:
587:
583:
567:
565:
559:
557:
553:
549:
545:
541:
537:
533:
530:
518:
512:
508:
504:
500:
496:
492:
488:
484:
480:
477:, an element
476:
472:
468:
465:
461:
457:
453:
449:
445:
441:
438:greater than
437:
433:
429:
425:
421:
418:
414:
410:
406:
402:
398:
394:
390:
386:
383:Every subset
382:
379:
375:
371:
367:
363:
360:with neither
359:
355:
351:
347:
343:
340:
336:
332:
328:
324:
320:
316:
312:
307:
303:
297:
293:
287:
283:
278:
273:
269:
263:
259:
253:
249:
244:
243:antisymmetric
240:
235:
231:
226:
222:
218:
215:
211:
210:
209:
197:
191:
186:
183:
179:
169:
167:
163:
159:
155:
150:
148:
144:
140:
139:maximal ideal
136:
132:
128:
124:
120:
116:
112:
108:
104:
100:
96:
92:
88:
83:
81:
77:
73:
70:
66:
62:
58:
54:
50:
46:
39:
38:spanning tree
35:
30:
26:
19:
7780:Order theory
7541:& Orders
7519:Star product
7448:Well-founded
7401:Prefix order
7357:Distributive
7347:Complemented
7317:Foundational
7282:Completeness
7238:Zorn's lemma
7237:
7142:Cyclic order
7125:Key concepts
7095:Order theory
7045:
6843:Ultraproduct
6690:Model theory
6655:Independence
6591:Formal proof
6583:Proof theory
6566:
6539:
6496:real numbers
6468:second-order
6379:Substitution
6256:Metalanguage
6197:conservative
6170:Axiom schema
6114:Constructive
6084:Morse–Kelley
6050:Set theories
6029:Aleph number
6022:inaccessible
5928:Grothendieck
5812:intersection
5699:Higher-order
5687:Second-order
5633:Truth tables
5590:Venn diagram
5373:Formal proof
5292:Zorn's Lemma
5272:
5269:"Zorn lemma"
5239:
5216:
5213:Jech, Thomas
5194:
5178:
5160:(1): 77–89.
5157:
5153:
5126:
5108:
5102:
5085:
5080:
5061:
5057:
5047:
5028:
5022:
4995:
4989:
4981:
4976:
4949:
4940:
4928:
4909:
4905:
4895:
4883:. Retrieved
4869:
4863:
4849:
4830:
4811:
4792:
4674:
4657:
4653:
4643:
4631:. Retrieved
4623:Smits, Tim.
4618:
4598:
4588:
4567:
4545:
4527:
4517:
4499:
4487:
4475:
4463:
4454:
4448:
4436:
4427:
4418:
4355:The Simpsons
4353:
4351:
4344:
4342:
4333:
4326:
4312:
4256:
4249:
4242:
4217:
4181:
4173:
4167:
4152:
4145:
4053:
3177:
3022:
2932:Given a set
2931:
2924:
2916:
2846:
2842:
2835:
2659:. Then this
2478:
2454:
2325:is nonempty.
2255:
2251:
2242:§ Proof
2235:
2229:
2223:
2219:
2215:
2211:
2207:
2204:well-ordered
2195:
2187:
2183:
2178:
2174:
2170:
2166:
2162:
2158:
2154:
2150:
2146:
2142:
2138:
2134:
2130:
2126:
2122:
2118:
2114:
2109:
2105:
2103:
2097:
2093:
2089:
2085:
2080:
2076:
2072:
2067:
2063:
2059:
2055:
2051:
2044:
2033:
2031:
2026:
2023:proper class
2018:
2006:
2002:
1987:
1980:
1973:
1966:
1962:
1960:
1905:
1827:
1820:
1816:
1812:
1808:
1804:
1800:
1794:
1791:Proof sketch
1784:
1780:
1778:
1773:
1769:
1765:
1761:
1757:
1753:
1749:
1745:
1741:
1737:
1733:
1729:
1722:
1718:
1714:
1712:
1707:
1703:
1699:
1695:
1691:
1687:
1683:
1679:
1675:
1671:
1667:
1663:
1659:
1657:
1651:
1647:
1643:
1639:
1635:
1631:
1629:
1628:
1623:
1619:
1615:
1611:
1607:
1603:
1599:
1595:
1591:
1583:
1579:
1575:
1571:
1567:
1563:
1559:
1555:
1551:
1547:
1543:
1539:
1535:
1531:
1527:
1525:
1519:
1515:
1511:
1507:
1503:
1499:
1497:
1496:
1491:
1487:
1483:
1479:
1475:
1471:
1469:
1463:
1459:
1457:
1456:
1450:
1446:
1442:
1438:
1434:
1430:
1423:
1419:
1415:
1411:
1407:
1403:
1396:
1392:
1385:
1381:
1377:
1373:
1369:
1365:
1361:
1357:
1355:
1350:
1346:
1342:
1338:
1334:
1330:
1326:
1322:
1318:
1314:
1310:
1306:
1304:
1299:
1295:
1291:
1287:
1283:
1279:
1275:
1271:
1264:
1260:
1258:
1245:
1240:
1230:
1226:
1222:
1218:
1214:
1210:
1206:
1202:
1198:
1194:
1190:
1186:
1182:
1179:spanning set
1174:
1170:
1166:
1164:
1159:
1155:
1151:
1149:
1144:
1137:
1130:
1123:
1116:
1109:
1102:
1095:
1088:
1081:
1077:
1073:
1069:
1062:
1055:
1051:
1044:
1037:
1030:
1026:
1022:
1020:
903:
896:
889:
882:
875:
868:
861:
857:
855:
850:
846:
842:
838:
834:
830:
822:
820:
815:
811:
807:
803:
799:
795:
791:
787:
783:
779:
775:
771:
767:
765:
760:
756:
744:
740:
736:
732:
728:
720:
716:
708:
704:
700:
696:
692:
688:
686:
677:
675:vector space
672:
657:
654:
649:
645:
641:
637:
633:
629:
625:
617:
613:
609:
605:
601:
597:
595:
589:
585:
581:
571:Zorn's lemma
569:
563:
561:
551:
547:
539:
531:
522:Zorn's lemma
520:
516:
510:
506:
502:
498:
494:
490:
482:
478:
474:
470:
463:
459:
455:
451:
447:
443:
439:
435:
427:
423:
416:
411:is called a
408:
404:
400:
396:
388:
384:
373:
369:
365:
361:
357:
353:
349:
341:
334:
330:
326:
322:
318:
314:
305:
301:
295:
291:
285:
281:
271:
267:
261:
257:
251:
247:
238:
233:
229:
216:
207:
188:
175:
151:
111:vector space
84:
61:upper bounds
48:
45:Zorn's lemma
44:
43:
25:
7725:Riesz space
7686:Isomorphism
7562:Normal cone
7484:Composition
7418:Semilattice
7327:Homogeneous
7312:Equivalence
7162:Total order
6953:Type theory
6901:undecidable
6833:Truth value
6720:equivalence
6399:non-logical
6012:Enumeration
6002:Isomorphism
5949:cardinality
5933:Von Neumann
5898:Ultrafilter
5863:Uncountable
5797:equivalence
5714:Quantifiers
5704:Fixed-point
5673:First-order
5553:Consistency
5538:Proposition
5515:Traditional
5486:Lindström's
5476:Compactness
5418:Type theory
5363:Cardinality
5175:Paul Halmos
4835:Halmos 1960
4816:Halmos 1960
4524:Lang, Serge
4482:, p. 9
4346:Zorns Lemma
4300:ultrafilter
3868:by setting
3378:by setting
2133:where each
2007:really long
1819:), because
1252:contains a
723:is not the
715:subsets of
544:upper bound
487:upper bound
422:An element
403:. A subset
393:restricting
265:hold, then
76:necessarily
59:containing
18:Zorns Lemma
7764:Categories
7693:Order type
7627:Cofinality
7468:Well-order
7443:Transitive
7332:Idempotent
7265:Asymmetric
6764:elementary
6457:arithmetic
6325:Quantifier
6303:functional
6175:Expression
5893:Transitive
5837:identities
5822:complement
5755:hereditary
5738:Set theory
5142:References
4872:: 76–108.
4492:Moore 2013
4441:Moore 2013
4321:See also:
4245:Jerry Bona
4212:See also:
4170:John Tukey
3023:such that
2141:satisfies
2043:: we pick
2013:, but all
2001:<…, in
1752:.) So, if
1674:such that
1638:and every
1550:such that
1510:, the sum
1437:and every
1429:For every
1414:, the sum
1402:For every
1068:for every
1054:such that
825:to be the
774:(that is,
527:Suppose a
346:comparable
277:transitive
237:for every
223:≤ that is
172:Motivation
133:that in a
63:for every
53:set theory
7744:Upper set
7681:Embedding
7617:Antichain
7438:Tolerance
7428:Symmetric
7423:Semiorder
7369:Reflexive
7287:Connected
7035:Supertask
6938:Recursion
6896:decidable
6730:saturated
6708:of models
6631:deductive
6626:axiomatic
6546:Hilbert's
6533:Euclidean
6514:canonical
6437:axiomatic
6369:Signature
6298:Predicate
6187:Extension
6109:Ackermann
6034:Operation
5913:Universal
5903:Recursive
5878:Singleton
5873:Inhabited
5858:Countable
5848:Types of
5832:power set
5802:partition
5719:Predicate
5665:Predicate
5580:Syllogism
5570:Soundness
5543:Inference
5533:Tautology
5435:paradoxes
5279:EMS Press
5215:(2008) .
5064:: 75–76.
4708:↪
4633:14 August
4480:Jech 2008
4468:Jech 2008
4372:Antichain
4176:in 1940.
4040:◻
3768:−
3762:∈
3705:⊂
3440:∈
3350:∪
3275:→
3189:≤
3156:∈
3135:⊂
3124:∣
3118:→
3069:∈
3058:for each
3043:∈
3005:→
2966:⋃
2786:~
2731:∪
2719:~
2690:≥
1934:∈
1886:≥
1874:∈
1868:∀
1859:∈
970:⋯
622:vacuously
245:(if both
225:reflexive
7539:Topology
7406:Preorder
7389:Eulerian
7352:Complete
7302:Directed
7292:Covering
7157:Preorder
7116:Category
7111:Glossary
7020:Logicism
7013:timeline
6989:Concrete
6848:Validity
6818:T-schema
6811:Kripke's
6806:Tarski's
6801:semantic
6791:Strength
6740:submodel
6735:spectrum
6703:function
6551:Tarski's
6540:Elements
6527:geometry
6483:Robinson
6404:variable
6389:function
6362:spectrum
6352:Sentence
6308:variable
6251:Language
6204:Relation
6165:Automata
6155:Alphabet
6139:language
5993:-jection
5971:codomain
5957:Function
5918:Universe
5888:Infinite
5792:Relation
5575:Validity
5565:Argument
5463:theorem,
5296:Metamath
4885:24 April
4596:(2015).
4565:(2015).
4526:(2002).
4426:(2003),
4366:See also
4178:Bourbaki
4159:Max Zorn
3890:′
3775:′
3732:′
3701:′
3342:′
3131:′
3114:′
2893:. Then
2807:. Thus,
2157: :
2113: :
2084: :
2015:ordinals
1825:function
1702:. Thus,
1658:Suppose
1566:. Since
1526:Suppose
1470:Because
719:. Since
497:. Here,
193:—
123:topology
99:Max Zorn
7644:Duality
7622:Cofinal
7610:Related
7589:Fréchet
7466:)
7342:Bounded
7337:Lattice
7310:)
7308:Partial
7176:Results
7147:Lattice
6962:Related
6759:Diagram
6657: (
6636:Hilbert
6621:Systems
6616:Theorem
6494:of the
6439:systems
6219:Formula
6214:Grammar
6130: (
6074:General
5787:Forcing
5772:Element
5692:Monadic
5467:paradox
5408:Theorem
5344:General
5281:, 2001
4681:, § 16.
4528:Algebra
4142:History
4058:: take
3488:, then
2772:and so
2705:, then
2244:below.
2214:, then
2062:we set
1740:, then
1690:, then
1278:except
1122:, ...,
1094:, ...,
1043:, ...,
902:, ...,
887:scalars
874:, ...,
707:}. Let
640:, then
542:has an
432:maximal
275:), and
145:has an
7669:Subnet
7649:Filter
7599:Normed
7584:Banach
7550:&
7457:Better
7394:Strict
7384:Graded
7275:topics
7106:Topics
6725:finite
6488:Skolem
6441:
6416:Theory
6384:Symbol
6374:String
6357:atomic
6234:ground
6229:closed
6224:atomic
6180:ground
6143:syntax
6039:binary
5966:domain
5883:Finite
5648:finite
5506:Logics
5465:
5413:Theory
5250:
5227:
5201:
5185:
5010:
4964:
4797:Halmos
4679:Halmos
4606:
4575:
4560:, and
4552:
4534:
4294:Every
3966:is in
2431:is in
2348:is in
2240:or in
1997:< a
1768:– but
1719:proper
1678:is in
1650:is in
1518:is in
1449:is in
1422:is in
1384:. For
1268:ideals
1265:proper
1021:Since
680:has a
573:
485:is an
113:has a
72:subset
36:has a
7659:Ideal
7637:Graph
7433:Total
7411:Total
7297:Dense
6715:Model
6463:Peano
6320:Proof
6160:Arity
6089:Naive
5976:image
5908:Fuzzy
5868:Empty
5817:union
5762:Class
5403:Model
5393:Lemma
5351:Axiom
4860:(PDF)
4628:(PDF)
4428:Trees
4411:Notes
4272:basis
4163:axiom
3788:. As
3429:when
3244:. If
2848:Lemma
2455:Then
2257:Lemma
2248:Proof
2200:poset
2198:is a
2189:Lemma
2161:<
2117:<
2088:<
1986:<
1979:<
1972:<
1717:is a
1682:. If
1458:#1 -
1356:Take
1313:. If
1250:unity
1248:with
1177:is a
1076:. As
827:union
821:Take
751:(see
735:, so
682:basis
646:every
536:chain
450:with
413:chain
325:with
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