New location
May 24, 2013 at 13:53 | Posted in Uncategorized | 1 CommentFor those who may be interested, I’m migrating this blog to
http://eteisworth.blogspot.com/
because I can use mathjax more easily over there.
The Story of Generators – 3 (Existence)
October 2, 2012 at 15:32 | Posted in Uncategorized | Leave a commentThe Exact Upper Bound Argument
Let us assume that is as usual, with
. If we want to produce a generator for
, we saw last time that what we need is a universal sequence for
that in addition possesses an exact upper bound modulo
.
I don’t want to make an excursion into the theory of exact upper bounds, as there have been many high quality write-ups of this material already: in addition to Section 2.1 of the Abraham/Magidor article, there are also some materials due to Kojman that do an excellent job of exposing this material. The point is that one can use results of Shelah to modify the universal sequence so that it ends up with an exact upper bound, and then we get generators by the previous post.
Instead of re-hashing such arguments, I offer the following direct route from universal sequences to generators. The proof takes advantage of the ideal , and we will see similar arguments shortly when we talk about transitivity of generators. The argument is based on the exposition of generators given in the Burke-Magidor pcf paper. It turns out that framing things in terms of
simplifies things considerably.
From universal sequences to generators
Theorem 1 Suppose
is a progressive set of regular cardinals and
. Then there is a generator for
.
Proof: We can assume that . (If either
or
are in
, then they have generators consisting of singletons.)
Let be a universal sequence for
, that is, a sequence such that
-
, and
-
is cofinal in
whenever
is an ultrafilter on
satisfying
.
Such sequences exist by Theorem 4.2 of the Abraham/Magidor article, or see Lemma 2.1 on page 327 of Cardinal Arithmetic.
Let . Since we have assumed
, we know there is a stationary set
lying in the ideal
. (See the first section of [Sh:420], or Theorem 3.18 in my own Handbook article.)
This means that there is a family and a club
such that
-
is a closed (possibly bounded) subset of
- if
then
, and
- if
then
is club in
of order-type
.
For each with
, we define
If , we set
.
Since is
-directed, we can find a single function
bounding the collection
modulo
.
Now we let be an elementary submodel of
for some sufficiently large regular cardinal
such that
-
,
,
,
,
,
, and
are all in
-
-
is an ordinal
This is possible because is stationary in
.
We define
We prove that is a generator for
. Part of this is simple, as by choice of
we know
. To finish, we must establish the following:
Proposition 2 If
is an ultrafilter on
with
, then
.
The proof is not difficult, but the following lemma is critical.
Lemma 3
.
Proof:
This is where the assumption pays dividends. Note that
as
, and therefore we know that
is club in
with order-type
.
Given in
, we know that
is an initial segment of
, and both are initial segments of
. Thus
Given and
, we know there is a
with
. Since
is cofinal in
, we know
is an ordinal
in
and
. Thus
, and we see
In particular, if , then there is an
such that
Since is club in
with order-type
, we can find a single
such that for all
,
In particular,
The set on the right is definable from parameters available in (both
and
are there), and therefore
.
Let us return now to the proof of Proposition 2.
Proof: By way of contradiction, suppose is an ultrafilter on
forming a counterexample. Since
is in
, we can assume that
is in
as well (this is the key point).
This sequence is universal for
and therefore the collection
is cofinal in
. Since
, we know furthermore that the sequence
is
increasing.
Thus, there is an such that
whenever
. Again, we may assume that
is in
(and hence less than
).
Choose greater than
and let
. Then
and
. In particular,
and
So we have
One the other hand, since we know
But now we have obtained a contradiction.
Proposition 2 taken together with the fact that easily establishes that
is a generator for
, so we are done.
The Story of Generators – 2
October 1, 2012 at 14:22 | Posted in Uncategorized | Leave a commentFinding Generators
Theorem 1 Suppose
is a progressive sequence of regular cardinals
![]()
is a universal sequence for
![]()
is an exact upper bound for
modulo
![]()
![]()
Then for any ultrafilter
on
,
What does this have to do with generators? Just note the following:
Corollary 2 Under the assumptions of the theorem, we have
, hence
is a generator for
.
Proof: If is an ultrafilter on
containing
, then either
meets
or it does not. In the first case, the cofinality of
is less than
by definition of
, and in the second case the cofinality is exactly
by the conclusion of the theorem. In either case, the cofinality is at most
and so
is in
and we have
For the other inclusion, suppose is in
but not in
. Let
be an ultrafilter on
containing
but disjoint to
. Since the cofinality of
is
, we have contradicted the conclusion of the theorem.
Digression on exact upper bounds
Before proving Theorem~1, we need to say a few words about exact upper bounds because different authors treat them in slightly different ways. Let us assume that is an exact upper bound for
mod
just as in the statement of the theorem. It is easy to see
and so our is equal mod
to a function
satisfying
If we define , then
and
are equal modulo the ideal
, and for any ultrafilter
on
disjoint to
, we have
if and only if
.
The point of the above is that we can replace by
and not change anything, so we may as well assume that our function
satisfies
Proof of Theorem
Proof: Suppose first that is an ultrafilter on
containing
but disjoint to
. The sequence
is
-increasing, so if we can show it is cofinal in
we will know that the cofinality of
is exactly
.
Suppose . Then by setting
equal to zero outside of
we produce a function
that is equal to
mod
and less than
everywhere. Our assumptions on
then give us an
such
and since , we achieve
as required.
For the other direction, suppose by way of contradiction that is an ultrafilter on
satisfying
with
.
Outside of the set , we have
and so
is
-equivalent to a function
. Since
is universal for
, there is an
such that
and hence
as well.
But is an exact upper bound for
mod
, and so
Since the cofinality of is
we know
, and therefore
Putting all of this together yields
and this is a contradiction.
The Story of Generators 1
October 1, 2012 at 12:10 | Posted in Uncategorized | Leave a commentIntroduction
I’m not sure exactly where things left off, so I’ll just begin with a series of posts on generators for pcf. I will also try to keep the posts short so that I can ease back into the routine of writing them.
Let us assume that is a progressive set of regular cardinals, and
. I will assume we know already some of the basics about the ideal
and I will use these facts without much comment.
What we want to look at is the existence of a generator for : we will sketch the proof that the ideal
is generated over
by a single set
. And I do mean “sketch”, as the details are worked out nicely in Section 4 of the Abraham/Magidor article in the Handbook of Set Theory.
Universal Sequences
We’ll start with the following definition:
Definition 1 Suppose
. A sequence
of functions in
is a universal sequence for
if
is
-increasing, and
is cofinal in
whenever
is an ultrafilter on
with
.
Universal sequences are tightly related to the existence of generators for pcf, as we shall see. I want to point out the following result:
Theorem 2 Suppose
is a progressive set of regular cardinals and
Then the following statements are equivalent:
- There is a universal sequence for
.
- There is a family
such that for any ultrafilter
, if
has cofinality
then
remains unbounded in
.
- There is a family
of subsets of
such that
modulo
, and
is the ideal generated by
together with the sets
.
The above is basically Fact 2.2 on page 13 of Cardinal Arithmetic, and it follows quite easily from the work done in the first section of the book. (The 3rd statement says, in the notation of the book, that is semi-normal.) Abraham and Magidor develop basic pcf theory in a slightly different order, and deriving the above result from the material they present prior to defining universal sequences is a bit more difficult.
Of course, the main point is the following result:
Theorem 3 If
is a progressive set of regular cardinals, then every
has a universal sequence.
Again, we will not prove the above as this is Theorem 4.2 of the Abraham-Magidor article, and their proof is quite clear.
What have we learned from the above? We have outlined the first steps towards proving that generators exist. Putting the two theorems presented here together, we see that if , then
is pretty simply generated over
.
In our next post, we’ll see how “tuning up” a universal sequence leads to the existence of generators.
Miss me?
May 21, 2012 at 11:05 | Posted in Uncategorized | 1 CommentWill be back posting this week. The long break was due to being busy with other aspects of my career, but things have settled back down and my sabbatical starts in three weeks.
Dictionary
March 30, 2012 at 14:40 | Posted in Uncategorized | Leave a comment
I took a look at the Abraham/Magidor article in the Handbook of Set Theory, and what they discuss in Section 5 of their article is within epsilon of what Shelah uses in Section 1 of Chapter VIII of The Book. There are minor technical differences,but their writing is so much clearer than Shelah’s that I’m tempted to prove the main results of Chapter VIII section 1 using their version of things (minimally obedient universal sequences and -presentable models) instead of Shelah’s (“suppose (a)-(e) of 1.2 hold”).
What this means in practical terms is that I’m going to be doing some translation of Section 1 of Chapter VIII into the language of the Handbook and see how well the proofs go through.
Tedious, but probably worthwhile in the interest of making the material accessible!
Some Assumptions
March 28, 2012 at 13:33 | Posted in Uncategorized | Leave a commentI have been contemplating exactly where to begin the discussion of these matters, and I think that I want to go back to Chapter VIII of The Book, and look at the first section of it because the arguments there keep appearing in later works.
In this post, I’m going to write down some assumptions [(a)-(e) of Claim 1.2] that will be used in the next few posts.
The Assumptions
We assume the following
-
is a set of regular cardinals satisfying
- For every
, we let
satisfy
-
,
-
is strictly increasing modulo
,
- if
and
, then for each
we have
- for every
and
, there is a
such that
(everwhere)
-
-
is a sufficiently large regular cardinal,
is a well-ordering of
-
is an increasing continuous sequence of elementary submodels of
such that
-
-
and
.
I’m going to do some “dictionary work” to get the official names for such objects. I just want to check which terms have become standard, and I’ll use the Abraham/Magidor Handbook article as the final word.
Quick update
March 21, 2012 at 21:06 | Posted in Uncategorized | Leave a commentWell, I think hitting the medium-term goals is going to be much harder more interesting than I thought! I’ll start posting pieces of what I know starting next week after our Spring Break is over…
Medium Term Goals
March 6, 2012 at 11:35 | Posted in Uncategorized | Leave a commentI feel the urge to return to the main work of this blog, namely working through the details of the more exotic portions of “The Book” and its continuations. I also happen to have an ideal body of results to attack:
As noted earlier, Shelah’s second proof of the “cov vs. pp Theorem” ([Sh:400] 3.5) contains an error (acknowledged in [Sh:513]), and although it is possible to effect a repair of most of it, the full version is still unproven. Now if we look ahead to [Sh:410] (one of the first papers continuing The Book) we find that a great many proofs of theorems end with a sentence saying roughly “now repeat the proof of [Sh:400, 3.5]” (I count at least four instances of this in Section 2 of the paper).
So on the face of it, it’s not clear how many of these results are actually valid. It may be that the weaker version of [Sh:400, 3.5] is strong enough to push through the arguments, but since there are very few details it’s hard to tell.
This looks like a nice task for me to tackle in the blog, so that’s probably what I will do.
On ADS (Part 4)
February 10, 2012 at 14:40 | Posted in Uncategorized | Leave a comment
So what does it mean when we say “almost works”? We’ll start with a lemma:
Lemma 1 For a fixed
, the set
contains at most one element from each set
, and hence is at most countable.
Proof: Suppose by way of contradiction what and
are distinct members of
for which
Then
and this contradicts the fact that is a transversal for
.
Now given , we are going to define
to be those
for which
has failed to work, that is,
Lemma 2 The set
is at most countable.
Proof: If not, find for which the set
is uncountable, and set
Then for each , we have
which contradicts the preceding lemma.
So for a given , the function
will “disjointify”
from all but countably many
with
: the function
“almost works”.
Notice that if and only if
, so that we can define a graph
on
by connecting
and
if and only if
. By the preceding lemma, every vertex of the graph has at most countably many edges coming out of it. An easy argument tells us that the connected components of our graph
are at most countable as well.
Now if and
are members of different connected components of
then
will disjointify
and
. But each connected component of
can be disjointified easily (by induction) as it is at most countable.
Thus, it is straightforward now to “correct” to a function
which will work everywhere.
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