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Just a list?
Is this article intended to be just a list — If so should it be moved to "List of Combinatorial principles"? Or is this just an outline for a article on these principles? Paul August ☎ 17:25, May 31, 2005 (UTC)
Infinitary "combinatorial principles"
There is a set-theoretical notion of "combinatorial principle", which should probably get its own article combinatorial principle (set theory) or infinitary combinatorial principle (with a link from this article). I am not sure what the proper definition of these "set-theoretical" or "infinitary" combinatorial principles should be, and if this is defined in any textbooks. The archetypal combinatorial principle is probably Diamondsuit. Other combinatorial principles (such as club or square; there is also spade, stick, and many others) share the following properties:
- They postulate the existence of a certain object C...
- often C is a sequence of uncountable length, sometimes class length
- ... which is immensely helpful in analyzing a structure S
- (e.g. S is the set of subsets, or subsets of a certain kind, of a given set)
- "analyzing" often refers to constructions by transfinite induction.
I am not sure if all the following (which have the properties I mentioned) are called "combinatorial principles":
- CH, GCH, SCH (predates the name "combinatorial principle"?);
- V=L, V=L[x] (not "combinatorial" enough?)
- existence of 0# (not uncountable?)
--Aleph4 21:46, 16 December 2006 (UTC)
- It seems to me that CH, GCH, and SCH probably count as being in the same ballpark. V=L implies all the things we usually want to call infinitary combinatorial principles, but it seems a little on the abstract side to say that it is one itself. The existence of 0# strikes me as going in the opposite direction: It should be true, whereas diamond and square and so on should be false. The sequences posited by diamond and square are dei ex machina, things that just happen to guess right without any reason that they ought to be able to. 0# on the other hand is a "concrete" sort of thing that ought to exist if the universe is big enough and rich enough. --Trovatore 23:57, 16 December 2006 (UTC)
I should clarify that I have really two questions:
- What is the definition of "combinatorial principle"? (This definitely needs a reference, but I do not know one)
- Decide for each concrete statement whether it is or is not a "combinatorial principle". (Easy to find references callind diamond and square "combinatorial principles")
(An answer to "1" should also answer "2", but perhaps enough answers to "2" will help us find an answer to "1".)
I do not think that "truth" is relevant in answering "1" (besides the fact that "truth" in set theory is POV). Both ZFC and NF are "set-theoretical axiom systems", even though you and I "know" ZFC is true and NF is not. But you are right that the uniqueness of 0# puts it in a different category than diamond and square. --Aleph4 17:40, 17 December 2006 (UTC)
- As far as I know, which isn't very far, there is no generally accepted abstract definition of "infinitary combinatorial principle". I wasn't suggesting that truth per se should be a defining aspect of it (certainly there could be true combinatorial principles and also false ones) but trying, perhaps not very successfully, to explain my sense that the existence of 0# was not merely not an example of things like diamond and square, but an example of something opposite to them. And that sense could, of course, itself be wrong. --Trovatore 17:54, 17 December 2006 (UTC)
In the seventies when we learned to use diamond, we called it, and its variants "combinatorial principles" with no exact definition for the phrase (BTW it was never called diamondsuit, just diamond). I guess, Aleph4's description covers what we felt under this expression, so for example, Shelah's various black box and guessing club principles ARE combinatorial principles, I would also include MA, too, and PFA, MM are certainly not. Kope 14:46, 8 January 2007 (UTC)