Talk:Large cardinal

message: 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)
thread_title: Untitled

message: 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)
replies: text: 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)
thread_title: Move?

message: 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)
thread_title: Currently dabbling with this page

message: 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 .
: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)
thread_title: Possibility of Inconsistency

message: 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)
replies: text: 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)

text: 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)

text: 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)
thread_title:  "Partial Definition" 

message: added by Jaykov Foukzon, 30 August 2013‎:
* "Generalized Lob's Theorem.Strong Reflection Principles and Large Cardinal ‎Axioms"
*
*
-- 
: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)
replies: text: 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)
thread_title:  Foukzon's result 

message: 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.).
occurring in the lead. YohanN7 (talk) 01:16, 12 August 2014 (UTC)
replies: text: Qualify it in what way? JRSpriggs (talk) 08:08, 12 August 2014 (UTC)
replies: text: The former version suggested that every large cardinal is larger than . This need not be the case, even if it is probably true for most large cardinals, since  cannot be strongly inaccessible. I don't know how to formulate these things concisely. YohanN7 (talk) 12:15, 12 August 2014 (UTC)
thread_title:  "Generally very large" 

message: 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)
replies: text: 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)
thread_title:  Add a "History" section