Talk:Combinatorial principles

message: 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)
replies: text: 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)

text: 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)
thread_title:  Infinitary "combinatorial principles" 

message: What is the status of these principles from an axiomatic point of view? (I think this would be a useful specification to include in the article.)--5.2.200.163 (talk) 14:45, 7 September 2017 (UTC)
thread_title:  Principles or axioms? 

message: Why call it the sum of infinite polynomial when polynomials are defined as finite and what it's describing already has a name, series? It's called series everywhere else from what I can tell. Example: http://mathworld.wolfram.com/GeneratingFunction.html  — Preceding unsigned comment added by Itchyjunk (talk • contribs) 14:40, 17 November 2019 (UTC)
thread_title:  Generating function