This talk page is a forum for discussing general issues regarding the development of Cantor's Attic. Please post comments below in each section. Sign your comment with four tilde symbols ~~~~ to leave your username and timestamp.
Upper, Middle and Lower Attics
Please discuss the division into upper, middle and lower attics. We could unify these into one presentation. Should we? The concepts do partition nicely into those categories, which is why I started this way. Perhaps we should find better names? JDH 18:25, 29 December 2011 (PST)
I changed "subattic" to "the parlour". Also, how about replacing "upper attic" with "the belfry"? JDH 19:30, 29 December 2011 (PST)
I've now added more rooms to our house: the library, for references, the playroom, for a fun account of the various paradoxes of infinity, and the cellar, where we might collect foundational or background information in support of the material on the main attic pages. JDH 19:25, 4 January 2012 (PST)
Pages on general set-theoretic Background?
Please discuss whether we should include extensive background supporting information on general set-theoretic topics, such as ultrafilters, measures, ultrapowers, extenders, forcing, and so on. JDH 18:25, 29 December 2011 (PST)
Open questions forum?
Please discuss whether Cantor's Attic should try to develop a forum for posting open questions related to large cardinals. We could easily create pages that compile this information, and we could then keep track of who asked which questions and when they were answered, providing links to papers and so on. Should we do this? JDH 18:27, 29 December 2011 (PST)
Adding non-large cardinal set-theoretic statements with LC strength?
What is the general opinion about whether non-LC assertions, but which have LC strength, such as PFA or MM or AD, belong here at Cantor's Attic? It would seem to be one of the major applications of the large cardinal hierarchy to study such assertions and to prove the equiconsistencies, so my tendency is to include them, even though they aren't explcitly large cardinal assertions. At the same time, I can see how their inclusion destroys the idea that what we are doing is classifying the notions of infinity. JDH 19:12, 29 December 2011 (PST)
How to handle ZF cardinals?
The cardinality concept in the ZF context as opposed to ZFC is rich and has very different features. We should definitely include pages with this information. What is the best way to organize it? We can of course have a summary page cardinal_general describing the main differences. But then the various large cardinal notions themselves admit non-AC forms. e.g. $\omega_1$ can be measurable, different characterizations of strongly compact are no longer equivalent without AC, etc. etc. Should this information be on the same page as the large cardinal itself? Or perhaps we should add a "back attic" containing pages with information about the non-AC context. JDH 08:37, 4 January 2012 (PST)
I believe that a general section about cardinalities and how they are defined without the axiom of choice. There should be a page listing the various "strange" cardinals in the lack of choice, D-finite, T-finite, Amorphous, etc. J. Truss wrote a paper classifying seven types of D-finite subsets, we can use those as guidelines. Another page is in order for what sort of things may fail without the axiom of choice (e.g. undefinable cardinality function); in the case of large cardinals I am not too sure, perhaps have a page for large cardinals without the axiom of choice with sections for each cardinal that we can say something about (e.g. measurable, and so on). If it gets too crowded we can start specific pages (e.g. inaccessible cardinals should have a separate page since we can say quite a lot on the various definitions of inaccessible in ZF). Asaf K 17:32, 4 January 2012 (UTC)
Joel, I think that cardinal general should be "merged" with cardinal into Cardinality. I have worked (partially) on a general definition, from which there should be a subsection (or an additional page) for well orderable cardinals; and a subsection (or another page) about non-well orderable cardinals. Only now after finishing somewhat of the introduction I see that such general introduction is given on cardinal, so I'm not too sure that having it twice is a good idea. I'll finish my edit and then we can decide whether or not merge the pages - and if so how to do it. --Asafk 11:05, 4 January 2012 (PST)
Let's keep cardinal for the ZFC account, and just link to the ZF cardinal page, whatever it is called. I think most set theorists understand the word cardinal to mean initial ordinal, unless it is explicitly stated that it is a non-AC context. So there is no need to force the general ZF definition into the ZFC context, where things are simpler. With the names, one should try to use the term that will be linked, which is why I prefer cardinal to Cardinality, since this is the word that will be typed in other pages and which can be linked easily. By this measure, cardinal general should be moved. How about: ZF cardinal? Another issue is that so far, the names are all a kind of cardinal, such as inaccessible, supercompact and so on, so that one can write e.g. uncountable regular cardinal, and cardinality doesn't quite fit with this. JDH 11:11, 4 January 2012 (PST)
I am fine with keeping it that way, I just think that Cardinality should be a page introducing the concept as a whole (perhaps we can put a refer to the cardinal page with "For the basic definition in ZFC ..." on the top). As for the non-well ordered cardinals, I suppose we can have a page perhaps called Choiceless Cardinals; In the meantime I will add pages for Dedekind-finite cardinals and Hartog number.--Asafk 11:30, 4 January 2012 (PST)