Talk:Large cardinal property

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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.

Contents 1 Move? 2 Currently dabbling with this page 3 Thank you 4 Possibility of Inconsistency 5 Opening sentence

[edit] 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)

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)

[edit] 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)

[edit] 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

[edit] 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 .

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)

[edit] 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)

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). 69.140.164.142 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)