# Talk:Combinatorial principles

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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":

1. CH, GCH, SCH (predates the name "combinatorial principle"?);
2. V=L, V=L[x] (not "combinatorial" enough?)
3. 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:

1. What is the definition of "combinatorial principle"? (This definitely needs a reference, but I do not know one)
2. Decide for each concrete statement whether it is or is not a "combinatorial principle". (Easy to find references callind diamond and square "combinatorial principles")