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It might be nice if the ones that are considered more central were clearly identified. Any working set theorist needs to know about say a weak compact or measurable cardinal, but nobody talkes about say ineffable cardinals.
It might make sense to move this article to something like List of large cardinals or List of large cardinal properties. Just a thought; I don't feel very strongly about it. (On the other hand, I might feel more strongly someday, if I wanted to write a general article about large cardinal properties in the abstract). --Trovatore 03:36, 15 July 2005 (UTC)
- The general convention is to separate out lists of links, when they become bulky, making two articles. Charles Matthews 15:18, 15 July 2005 (UTC)
Currently dabbling with this page
I think this page needs some reworking. For one thing, it should really be at large cardinal property ("largeness" is not a property of cardinals; various large cardinal properties are). Then large cardinal axiom should also be defined in boldface. Then we need a discussion of the various "intervals" of large cardinal properties: the "small" ones consistent with V=L, the larger ones that correspond to determinacy of pointclasses, still larger ones for which corresponding determinacy results are not yet known. A more precise description of consistency strength wouldn't hurt either. Woodin's abstract definition of large cardinal property could be mentioned, together with Steel's objections to it (unfortunately I don't think the latter have been published anywhere, so it might be tough to source). In the end I think the list should go to list of large cardinal properties; on length alone it's not unmanageable here, but it's kind of a different subject from the general discussion. --Trovatore 16:14, 5 November 2005 (UTC)
Thanks for refactoring this page; I"m struggling to understand Large Cardinals and the simple list of types that was at 'Large_Cardinal(s)' was singularly (heh) unhelpful. --hmackiernan
Yes, ditto - I found this impenetrable at quick read, even with moderately connected background. -- RJA
Possibility of Inconsistency
This artical should at least mention the possibility that some large cardinal axioms are inconsistent. —The preceding unsigned comment was added by 18.104.22.168 (talk • contribs) 00:27, 11 February 2006 .
- Perhaps, but it's tricky to find NPOV language. In my view it's possible, in a certain sense, that even Peano arithmetic is inconsistent (that is, we don't know apodeictically that PA is consistent). We also don't know apodeictically that the existence of rank-into-rank cardinals is consistent. I don't see a difference in kind between the two cases; it's a difference of degree.
- What meaning do you want to convey by the statement that it's "possible" that some LCAs are inconsistent? Are you claiming that there are possible worlds in which they really are inconsistent? Probably not, but then just what is the distinction you're making with the status of weaker theories? --Trovatore 04:21, 11 February 2006 (UTC)
The first sentence seems to be missing an "if". I don't know what exactly it should say, so I can't fix it. 22.214.171.124 15:14, 29 January 2007 (UTC)
- It already has an "if" in it. See: "In the mathematical field of set theory, a large cardinal property is a property of cardinal numbers, such that the existence of such a cardinal is not known to be inconsistent with ZFC and it has been proven that if ZFC is consistent, then ZFC is consistent with the nonexistence of such a cardinal.". JRSpriggs 06:14, 30 January 2007 (UTC)
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This page seems to be rather difficult to understand; articles should be written in language more useful for lay readers (for example, average adults with a high-school education or "some" college). 126.96.36.199 04:19, 7 April 2007 (UTC)
- To be honest, I don't think that audience has a chance of understanding the subject matter. Just the same, the opening paragraph does get into technicalities a little too fast; I'll think about whether I can improve it. But I think the best reasonable goal is that mathematicians in general who aren't set theorists, or maybe undergraduates with a strong introduction to set theory, can follow it in outline form. --Trovatore 08:12, 7 April 2007 (UTC)
- I agree with Trovatore. A reasonable goal is for it to be comprehensible to undergraduates with a strong introduction to set theory. Bayle Shanks (talk) 05:19, 19 October 2013 (UTC)
"More" or "fewer" sets?
Could this article have a short discussion on wether a large cardinal axiom generally implies "more" or "fewer" sets? I understand that the question appears as very unclear, but I can see no way to formulate it in technical terms. What I mean is that on the linguistic level "there exists a super-super cardinal" seems to imply that there are many MANY more sets than ZFC predicts. On the other hand, an added axiom is added rule that an object must obey in order for it to be a set. Not all ZFC axioms provide for the existence for additional sets, but rather prohibits concievable sets. Is this not true for large cardinal axioms as well?
Can one say that "the more cardinal axioms that are true, the taller but thinner the universe is" and vice versa "the more cardinal axioms whose negations are true, the shorter but fatter the universe is"? I hope I'm not making a complete fool of myself by asking this. I'm certainly no set theorist, just a guy becoming very curious about set theory at a mature age. YohanN7 (talk) 08:57, 20 August 2009 (UTC)
- OK, first of all, an axiom is not "an added rule that an object must obey" to be considered a set. It's an added rule that a structure must obey, to be considered a model of set theory. I don't think you can make sense of the idea that there are these candidate sets floating around, and some of them are excluded when you add more axioms. Any candidate subset of a set, is an actual subset of that set. This fact however cannot be captured by first-order logic.
- Large cardinal axioms certainly don't make the universe "taller but thinner". For example, Goedel's constructible universe, L, is as "tall" as it's possible to be (it has all the ordinals), but the interesting large cardinal axioms are false in L. The canonical example is that L thinks there are no measurable cardinals — not because measurable cardinals, as ordinals, are not elements of L. They are elements of L. But L lacks the measures or ultrafilters that witness their measurability. So here the large-cardinal axiom is (loosely speaking) working in the opposite direction from what you suggest.
- But it's in any case misleading to speak of axioms making the universe larger or smaller — after all, every consistent first-order theory (in a countable language) has a countable model. We know the universe is not countable, but we cannot capture that knowledge with a first-order theory. See Skolem's paradox. --Trovatore (talk) 10:03, 20 August 2009 (UTC)
- First of all, thanks Trovatore. It's not the first time you have answered my naive questions in a very serious manner and getting my questions quite right as I meant them. I will certainly follow your pointers by reading and trying to grasp as much as possible here in Wiki + "free" links for a start. It's going to take some time, thats for sure. I do have a couple of minor follow-up questions or two, but they will have to wait for a while. 188.8.131.52 (talk) 16:54, 20 August 2009 (UTC)
- To Trovatore: Might there be a true theory which has no countable standard model? Perhaps ZFC+I0, the rank-into-rank axiom?
- I know that the downward Löwenheim–Skolem theorem implies that if there is a set which is a standard model of ZFC+I0, then there is a countable set which is a standard model of it. But perhaps the universe satisfies ZFC+I0 but contains no transitive set which does so. JRSpriggs (talk) 09:23, 21 August 2009 (UTC)
submodel of L
- pointing out that (for example) there can be a transitive submodel of L that believes there exists a measurable cardinal, even though L itself does not satisfy that proposition.
- It's the smallest one that contains all the ordinals (and therefore the smallest transitive proper class model). But L contains transitive set models (of countable height) that satisfy "there exists a measurable". That follows by Shoenfield absoluteness, because the assertion "there exists a real coding a countable model that is wellfounded and satisfies φ" is , so L satisfies it if V does. Then you take the Mostowski collapse.
- Maybe we should emphasize that it's a set model? --Trovatore (talk) 19:58, 1 December 2010 (UTC)
added by Jaykov Foukzon, 30 August 2013:
- Earlier, Foukzon made a similar edit to List of large cardinal properties. I reverted it with the edit summary "revert edits by Foukzon -- COI, OR, result is trivial". The last sentence of his abstract says "Main result is: let k be an inaccessible cardinal and H[k] is a set of all sets having hereditary size less then k, then ~Con(ZFC+(V=H[k])).". Also see hereditarily countable set.
- If κ is a strong inaccessible, then the axiom of choice implies that Hκ=Vκ. This can be proved by a simple induction on the rank of the elements. Since it is obvious that V≠Vκ, then ZFC proves V≠Hκ and consequently ZFC+V=Hκ is inconsistent. So his main result is trivial and does not require Löb's theorem or any reflection principle. JRSpriggs (talk) 01:11, 31 August 2013 (UTC)
JRSpriggs01:11, 31 August 2013 (UTC)wrote:
"Since it is obvious that V≠Vκ, then ZFC proves V≠Hκ"" This JRSpriggs statement is obviously false and abnormal — Preceding unsigned comment added by 184.108.40.206 (talk) 16:43, 21 November 2013 (UTC)
Abbreviation V=Hκ is standard and means that Hκ is a model of ZFC. Such model obvious exist iff Con(ZFC+large cardinal properties). So main result obvious is: ~Con(ZFC+large cardinal properties).
- Spring Western Sectional Meeting
University of Colorado Boulder, Boulder,CO April 13-14, 2013 (Saturday-Sunday) Meeting #1089
- Fall Southeastern Sectional Meeting University of Louisville, Louisville, KY October 5-6, 2013 (Saturday - Sunday) Meeting #1092
- Since all those abstracts contain obvious errors of spelling and grammar, it is clear that they have not been subject to any editorial supervision and thus cannot be considered reliable. JRSpriggs (talk) 03:58, 18 November 2013 (UTC)
The article currently says, "A necessary condition for a property of cardinal numbers to be a large cardinal property is that the existence of such a cardinal is not known to be inconsistent with ZFC and it has been proven that if ZFC is consistent, then ZFC + "no such cardinal exists" is consistent."
Consider a property of cardinal numbers p, and the value of the predicate "large cardinal property", or LCP(p). According to the above quote from the article, LCP is not objective; the value of LCP(p) is not static, but changes over time, e.g. the instant that the aforementioned consistency proof is discovered, the value of LCP(p) changes from False to True.
Is this really what is meant by "large cardinal property"? I have no expertise but I suspect the term is actually meant to be objective, and that what the article meant to say was something like:
- A necessary condition for a property P of cardinal numbers to be a large cardinal property is that both the existence or the non-existence of cardinals with the property P would be consistent with ZFC. In practice, the phrase 'large cardinal property' tends to be used for properties P for which (a) it has been proven that if ZFC is consistent, then ZFC + "no cardinal exists for which P holds" is consistent, and also for which (b) the existence of a cardinal for which P holds is not known to be inconsistent with ZFC.
- There is no precise, general, abstract definition of "large cardinal property", that is generally agreed on by workers in the field. I wouldn't call it "subjective" exactly, or at least I wouldn't use that word. It's more that no one has really succeeded in discussing "large cardinals" in general, in a mathematically precise way.
- Woodin has an attempt at an abstract definition in his 2001 "Notices of the AMS" paper (see the references in the continuum hypothesis article for a link), but it was not generally accepted as the final word on the matter (and I don't know that he intended it to be — he just needed a nonce definition for the work he was doing there). Also, his project from that paper sort of fell apart, if I understand correctly.
- By the way, I don't necessarily agree with the "necessary condition" claim; for example, some workers consider the axiom of replacement to be a sort of large-cardinal axiom. --Trovatore (talk) 06:20, 19 October 2013 (UTC)
- The partial definition is an attempt to describe some properties which have been shared by all those things which have been widely called "large cardinals". Since this list of things changes over time as our knowledge changes, it is (as you say) subjective and empirical. As the lead says, "There is no generally agreed precise definition of what a large cardinal property is, ...". Expecting objectivity is perhaps unreasonable since this deals not with objects in one model, but with a host of models which which may or may not subsist in some indefinable sense. People differ on which models they are willing to consider seriously and what they may have proved about objects within them at any given time. JRSpriggs (talk) 18:52, 20 October 2013 (UTC)
"Generally very large"
Recalling a discussion, Talk:Continuum_hypothesis#Large_cardinal.3F, it might be appropriate to qualify the statement
- Cardinals with such properties are, as the name suggests, generally very "large" (for example, bigger than ℵ0, bigger than the cardinality of the continuum, etc.).
- Qualify it in what way? JRSpriggs (talk) 08:08, 12 August 2014 (UTC)