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is aleph-null. And so is the cardinality of the obvious co-final subset which is the natural numbers (which are well-ordered). But one could also choose the cofinal subset to be the integers themselves (after all, B is cofinal in itself, right?), since they also have cardinality aleph-null which is the cofinality. And the integers are not well-ordered, contrary to the sentence. Also the integers are not order-isomorhpic to the natural numbers, again contrary to the sentence.
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Which is true? If they are always the same due to properties of cardinalities, it is better to use "min" than "inf", since "inf" implies that a limit must be taken whereas "min" just picks the smallest element, which is then assumed to exist. If they are not the same, then I believe the textual definition should be corrected to use the infimum.
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your point is taken, when we use the word "cofinality" we are usually talking about only a least cardinality (and only apropos a cardinal number), where as with the word "cofinal" we may talk about any set inside any poset. The latter usage is different. I still say merge though. But let's hear what others think as well. -
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However, if the right argument is zero or a successor (such as the "3" in your example), then this does not work because continuous functions are λ-continuous for any infinite regular ordinal λ, but neither 0-continuous nor 1-continuous in general. (But multiplication is 0-continuous; and addition is
1234:
again, "Trichotomy: If two sets are given, then they either have the same cardinality, or one has a smaller cardinality than the other." is one of the theorems which are equivalent to the axiom. In other words, without the axiom there are pairs of cardinal numbers which are incomparable -- neither is
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OK. I see that I was failing to read the second sentence in the context of the first sentence, "If A admits a totally ordered cofinal subset, then we can find a subset B which is well-ordered and cofinal in A.", which says that that B is well-ordered. So the integers would not be a counter-example,
903:
I've reorganized the article a bit. I moved the definition of regular and singular into the lead. I added some content to the "Examples" section (please double check my additions), and moved that section to follow immediately after the lead (I think it helps to have examples as soon as possible). And
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Yeah, I was thinking of cardinals as some least ordinal, and well ordering of ordinals is built into their definitions. But of course, that definition of cardinals relies on AC. And the proof of well ordering in the general case also requires AC. I knew that stuff, but the reason I was momentarily
1599:
In the section "Cofinality of cardinals", the text says "cf(κ) = the cardinality of the smallest collection of sets of strictly smaller cardinals whose sum is κ". But the equation directly following it defines it as the infimum, i.e. a greatest lower bound rather than the least element from the set.
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It is written : "This definition of cofinality relies on the axiom of choice, as it uses the fact that every non-empty set of cardinal numbers has a least member" Isn't it always true by the well ordering of the ordinals that every set of ordinals has a minimum ? Why do one need the axiom of choice
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fit them logically into separate articles is not as relevant to me as whether a single merged article would be too long. I think the answer to the latter question is "no", and hence I support the merger. I also note that having separate articles for the noun and the adjective is a bad policy. But
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The second sentence, "Moreover, any cofinal subset of B whose cardinality is equal to the cofinality of B is well-ordered, and these sets are all order isomorphic.", in the section "Properties" seems (to me) to be incorrect. Let us take B to be the
Integers with the usual order. Their cardinality
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It's not that your browser is defective, but rather that your system doesn't have a unicode enabled font with that symbol. As far as using math tags, I never use math tags for inline math text. I'm fighting hard not to revert your changes, I've seen others do the same revert. But I try to remind
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confused was because well-ordering really means two things: least element, and totalness of ordering. I somehow thought that sentence meant that you need AC to prove every cardinal has a least element, which is certainly not true. But I was just being dumb. Anyway, thanks for the clarification. -
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The second paragraph of the lead includes "... the axiom of choice is assumed ... in the rest of this article ...". Consequently, the cardinal numbers are a subclass of the ordinal numbers and thus well-ordered. The infimum of any nonempty subclass of a well-ordered class exists and is a member of
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For certain infinite sets X, it is also possible to avoid the axiom of choice. For example, suppose that the elements of X are sets of natural numbers. Every nonempty set of natural numbers has a least element, so to specify our choice function we can simply say that it takes each set to the least
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I guess I was wrong about this being a unicode font issue. I guess the real issue is that
Internet Explorer is broken. I don't like making Knowledge (or the web in general) cater to the bugs of Microsoft, but it's unrealistic to expect to be able to not support IE when it constitutes 90% of the
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It has a codepoint of x2135, part of the block of "letterlike symbols". This is a different glyph from the Hebrew letter. But anyway, if you don't have it, you don't have it. I didn't revert you, but I don't agree with your argument. If we insist on everything being readable in every single
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For what it's worth, I use IE 6 SP 2, and
Firefox 1.5.0.1, and I see the "you don't have this character" box in IE, but I see the aleph-lookalike glyph in Firefox. I hypothesize that it's so much a font thing, since both programs surely access the same pool of fonts, as a browser
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element of that set. This gives us a definite choice of an element from each set and we can write down an explicit expression that tells us what value our choice function takes. Any time it is possible to specify such an explicit choice, the axiom of choice is unnecessary.
232:, so in particular the cardinality of the continuum has uncountable cofinality. I don't see how the countable union of countable sets is relevant here, and I suspect this is a mistake based on assuming CH (as I've seen mistakes of this nature on Knowledge before). --
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How does the countable union of countable sets help in establishing that cf(card(R)) is uncountable? Without CH we don't know that sets smaller than R are countable - so maybe we can get R with a countable union of uncountable sets which still are smaller than R.
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Even if I can find a better unicode font than I have or get a Hebrew language package, that would only solve the problem for ME. Surely there are many other readers out there who only see square boxes when you-all use those characters.
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Yes, I feel that we must support IE by avoiding the use of those small alephs. Now that I have down-loaded
Firefox, I checked that the characters which I replaced were in fact alephs, which they were as I had inferred from the context.
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can be well-ordered. This is true of all cardinals, of course, but the proof requires AC. (As with many articles, this one is a mess as regards AC - sometimes mentioning when it is needed, and at other times silently assuming it.)
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This depends on your definition of a cardinal. You can define cardinals such that cardinals are well-ordered independent of choice, but the drawback is that it is then possible to have sets with undefined cardinality.
1188:, as it relies on the fact that every non-empty set of cardinal numbers has a least number." - The first emplies it uses the axiom of choice, but actualy it avoids using it because you are making an explicit choice. -
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Thanks for pointing out the proof to me, I agree with your suspicion - assuming CH clearly rules out card(R) = aleph_omega, which was the question I had when I found this page ;). I edited the article accordingly.
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Like the user at 151.200.whatever (who is probably my nephew Edward), I could not read the small alephs in
Internet Explorer, but now that I am using Firefox (without any change in fonts), they are clear.
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No, it relies on the Axiom of Choice. The explicit choice is the choice of the least member of a non-empty set of cardinals. You are right that we don't need the Axiom of Choice to pick this least member
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Thank you for clarifying that! Suggest, then, to change "inf" to "min", since "inf" implies a more complicated operation (which in this case ends up just yielding the minimum).
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Hmm... maybe what is meant is not that cardinals need not be well-founded, but rather that they need not be totally ordered. If we don't assume AC, that is. -
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995:. Now I am not sure about the merge anymore. I think that enough could be said about the properties of cofinal subsets to justify separate articles. --
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Without the axiom of choice there are non-wellorderable sets. Their cardinal numbers do not have initial ordinals. So you cannot sort them that way,
164:"Moreover, any cofinal subset of B whose cardinality is equal to the cofinality of B is well-ordered and order isomorphic to its own cardinality."
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Is this correct? I don't think so... The last part "order isomorphic to its own cardinality." would imply that B is a cardinal?! Anyone?
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Countable union of countable sets is countable - this requires at least the
Countable Axiom of Choice, as far as I know. Correct?
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Are there any rules to compute the cofinality of the sum, product or power of ordinals? For example, what is the cofinality of
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since they are not well-ordered. Still, the second sentence is hard to understand. Could it be expanded to make it clearer?
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market or whatever. It's unreasonable to tell everyone who wants to view wikipedia to get a different web browser, right? -
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cf(κ) = the cardinality of the smallest collection of sets of strictly smaller cardinals such that their sum is κ
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should redirect there. There's a lot to be said about regular cardinals that goes far beyond their cofinality. --
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I'm not necessarily against a merge, but if they're merged I feel strongly that the article should be at
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myself that not everyone has a capable computer. Please do consider getting some unicode fonts though. -
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Right now there is a certain asymmetry between the content on regular cardinals and regular ordinals:
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762:. Since I'm a newbie to the topic, maybe you can just glance it over? Thanks again for your help. -
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If we do not merge, then I think that "Cofinal (mathematics)" should be renamed "Cofinal subset".
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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Wow. Easy enough to see, but still surprising. I added a section about aleph fixed points to
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Isn't every cardinal well-ordered? In particular, isn't 0 a least element of every cardinal? -
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536:= λ if λ is a limit ordinal? Maybe this equality will hold for the inaccessible cardinals? -
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What is your method of making an "explicit choice" and so avoiding the axiom of choice?
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Hmm.. let's go through that a little more carefully. Let δ be a limit ordinal. Then ℵ
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Yup, that sentence didn't make sense. I think I have it stated correctly now.--
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Agreed, "sets of" seems superfluous. Also, the '=' should be replaced by 'is'.
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IMHO it should be changed to: "This definition of cofinality avoids using the
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union {κ: κ<λ}. But that operation probably isn't valid. For instance,
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the subclass. So "infimum" and "minimum" mean the same thing in this case.
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This section was copied with minor changes from the
Knowledge article on
384:{\displaystyle \mathrm {cf} (\aleph _{\delta })=\mathrm {cf} (\delta )}
1214:. The trouble is that without the Axiom of Choice it may not exist. --
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OK, I think that might be a problem. I use the fact that if ℵ
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is a limit cardinal. Then, according to the article, cf(ℵ
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exists. Somethings not right. Or am I mucking it up? -
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I'm a bit of a mergist by heart. So whether or not we
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I titled the remaining content "Properties". Comments?
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be any cardinal number, and for each positive integer
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987:I have removed the overlap between
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109:and see a list of open tasks.
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742:{\displaystyle \kappa _{n}}
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450:{all ords of cardinality ℵ
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1493:{\displaystyle \omega \,}
1471:{\displaystyle \omega \,}
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966:about merging
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807:to create it.
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192:Luke Gustafson
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175:Luke Gustafson
168:82.157.131.133
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328:limit ordinal
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137:High-priority
132:
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62:High‑priority
60:
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45:
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35:
27:
23:
18:
17:
1598:
1575:
1572:
1569:
1566:
1527:— Preceding
1525:for that ?
1523:
1270:
1243:
1229:
1212:if it exists
1211:
1142:
1129:
1126:
1104:No, I think
1073:
1006:
986:
975:
965:
938:
920:
902:
863:
780:
760:aleph number
523:
519:
515:
496:
488:
447:
425:) ≤ δ. If ℵ
325:
296:
292:
267:
229:
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203:
189:
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172:
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161:
136:
96:
40:WikiProjects
1247:—Preceding
1144:—Preceding
1040:I agree. —
906:Paul August
702:, then let
112:Mathematics
103:mathematics
59:Mathematics
1655:Categories
1092:cofinality
1080:cofinality
1057:cofinality
989:cofinality
968:cofinality
569:for which
248:SirJective
207:SirJective
1618:JRSpriggs
1549:JRSpriggs
1506:JRSpriggs
1237:JRSpriggs
1169:JRSpriggs
1114:Trovatore
1061:Trovatore
1025:JRSpriggs
953:JRSpriggs
942:JRSpriggs
927:JRSpriggs
888:JRSpriggs
867:JRSpriggs
830:JRSpriggs
809:JRSpriggs
472:λ ==: -->
1529:unsigned
1249:unsigned
1158:contribs
1146:unsigned
518:+1 : -->
1563:Wording
1500:itself.
1333:Albmont
1216:Zundark
976:support
751:Zundark
302:Zundark
234:Zundark
139:on the
30:B-class
1082:while
848:thing.
36:scale.
1190:Drysh
1150:Drysh
1130:From
1012:lethe
1007:could
980:lethe
970:with
878:lethe
840:lethe
820:lethe
764:lethe
538:lethe
508:: -->
497:union
489:union
477:: -->
471:: -->
458:: -->
454:} ≥ ℵ
448:union
405:lethe
309:lethe
280:lethe
270:lethe
224:that
1640:talk
1622:talk
1606:talk
1581:talk
1553:talk
1537:talk
1510:talk
1337:talk
1257:talk
1154:talk
1094:. —
991:and
749:. --
649:let
478:λ+1.
131:High
978:. -
475:λ+1
469:λ+1
465:λ+1
461:λ+1
444:λ+1
1657::
1642:)
1624:)
1608:)
1583:)
1555:)
1539:)
1512:)
1486:ω
1464:ω
1442:ω
1436:ω
1430:ω
1421:⋅
1418:ω
1393:⋅
1390:ω
1378:ω
1369:42
1363:ω
1359:ω
1339:)
1331:?
1314:⋅
1311:ω
1299:ω
1290:42
1284:ω
1280:ω
1259:)
1160:)
1156:•
1134::
925:.
793:ℵ
731:κ
710:δ
683:−
676:κ
671:ℵ
658:κ
611:κ
588:δ
584:ℵ
577:δ
557:δ
527:ω.
491:{ℵ
487:=
446:=
376:δ
354:δ
350:ℵ
330:δ
300:--
254:)
246:--
213:)
205:--
1638:(
1620:(
1604:(
1579:(
1551:(
1535:(
1508:(
1439:+
1433:+
1427:=
1424:3
1396:3
1387:+
1382:2
1374:+
1366:+
1335:(
1317:3
1308:+
1303:2
1295:+
1287:+
1255:(
1152:(
909:☎
783:δ
735:n
686:1
680:n
667:=
662:n
637:n
615:0
580:=
534:λ
524:n
520:n
516:n
511:κ
506:κ
493:κ
485:λ
473:ℵ
456:λ
452:λ
435:λ
431:δ
427:δ
423:δ
419:δ
415:δ
397:δ
393:δ
379:)
373:(
369:f
366:c
362:=
359:)
346:(
342:f
339:c
297:k
293:k
250:(
230:k
226:k
209:(
143:.
42::
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