Talk:Large cardinal

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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.

Kunen might disagree.--174.110.141.178 (talk) 01:42, 3 December 2011 (UTC)[reply]

Move?

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)[reply]

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)[reply]

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)[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 by 140.247.29.136 (talk • contribs) 00:27, 11 February 2006 .

added by Jaykov Foukzon, 30 August 2013‎ *[1] "Generalized Lob's Theorem.Strong Reflection Principles and Large Cardinal ‎Axioms"

[2]
[3]

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)[reply]

Opening sentence

The first sentence seems to be missing an "if". I don't know what exactly it should say, so I can't fix it. 153.42.34.134 15:14, 29 January 2007 (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.". JRSpriggs 06:14, 30 January 2007 (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). 69.140.164.142 04:19, 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. --Trovatore 08:12, 7 April 2007 (UTC)[reply]

"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)[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.

How's that possible? Isn't L the smallest transitive model of ZFC?—Emil J. 19:18, 1 December 2010 (UTC)[reply]

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

\Sigma _{2}^{1}

, 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)[reply]

Ah, stupid me, I didn't think of set models. Yes, I think it might be helpful to state it more explicitly.—Emil J. 12:21, 2 December 2010 (UTC)[reply]