This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of
mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join
the discussion and see a list of open tasks.MathematicsWikipedia:WikiProject MathematicsTemplate:WikiProject Mathematicsmathematics articles
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.
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. --
Trovatore16:14, 5 November 2005 (UTC)reply
Thank you
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 by140.247.29.136 (
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? --
Trovatore04:21, 11 February 2006 (UTC)reply
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.".
JRSpriggs06:14, 30 January 2007 (UTC)reply
This is now called the Partial Definition. Currently it 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." --
SteveWitham (
talk)
20:03, 27 January 2018 (UTC)reply
The form of this sentence is, "A necessary condition for an A to be a B is that (C and it has been proven that (if D, then E))." I originally added this comment to show that it was ungrammatical; now I see that it's grammatical. --
SteveWitham (
talk)
20:03, 27 January 2018 (UTC)reply
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).
69.140.164.14204:19, 7 April 2007 (UTC)reply
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. --
Trovatore08:12, 7 April 2007 (UTC)reply
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)reply
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)reply
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.
213.67.197.241 (
talk)
16:54, 20 August 2009 (UTC)reply
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)reply
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.
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)reply
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).
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)reply
Subjective concept?
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.
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)reply
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.).
The former version suggested that every large cardinal is larger than 2ℵ0. This need not be the case, even if it is probably true for most large cardinals, since 2ℵ0 cannot be strongly inaccessible. I don't know how to formulate these things concisely.
YohanN7 (
talk)
12:15, 12 August 2014 (UTC)reply
Add a "History" section
One way to make the topic more accessible (heh) to not-already-experts is to have a History section. I am reading New Scientist: The Collection vol. 4 number 4, article "It doesn't add up.", box "A ladder of infinities." Which says that, "In 1908... Felix Hausdorff conceived the idea of 'large cardinals.'" Wikipedia's
Felix_Hausdorff article mentions
inaccessible_cardinals, and quotes him saying that if they exist, they must be of "exorbitant size". --
SteveWitham (
talk)
20:33, 27 January 2018 (UTC)reply
So, what are inaccessible cardinals and large cardinals? (Why do inaccessible cardinals appear in this article as well? Is it because inaccessible = large? If not, what is the relationship?) Did Hausdorff invent large cardinals? If so, why? If not, who did? What problem or edge was he exploring or what need was he trying to fill? 1908 was a long time ago. What has happened with (inaccessible and) large cardinals since? --
SteveWitham (
talk)
20:33, 27 January 2018 (UTC)reply
Spend some time looking at
List of large cardinal properties. Being inaccessible is a large cardinal property, almost the weakest and thus most inclusive such property. That is, any of the properties listed after "inaccessible cardinal" are such that a cardinal having that property will also be inaccessible. However, merely being a very large cardinal does not guarantee that the cardinal will have a large cardinal property. For example, if Γ is an
I0-cardinal, then its
successor cardinal Γ+ will not even be weakly inaccessible despite being enormously large.
JRSpriggs (
talk)
08:25, 28 January 2018 (UTC)reply