Stationary sets
tc...@lsa.umich.edu
always failed. The definition is not hard to parse, but parsing is a far
cry from understanding. What's the motivation for the definition and why
are stationary sets so important? What kinds of theorems can be proved
with them that might be interesting to people who don't know or care about
stationary sets for their own sake?
--
Tim Chow tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences
John Jones
JJ
<tc...@lsa.umich.edu> wrote in message
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tc...@lsa.umich.edu
John Jones <scoob...@btopenworld.com> wrote:
>Please give a definition, as some people can give an intuitive answer.
See for example http://planetmath.org/encyclopedia/StationarySet.html
Daryl McCullough
tc...@lsa.umich.edu says...
>
>Occasionally I have tried to understand what stationary sets are but have
>always failed. The definition is not hard to parse, but parsing is a far
>cry from understanding. What's the motivation for the definition and why
>are stationary sets so important? What kinds of theorems can be proved
>with them that might be interesting to people who don't know or care about
>stationary sets for their own sake?
There is a discussion of stationary sets by Jech available
online:
http://www.math.psu.edu/jech/preprints/stat.ps
--
Daryl McCullough
Ithaca, NY
John Jones
opportunity already, or bygumminy otherwise I think you might be an ignorant
plonker.
JJ
<tc...@lsa.umich.edu> wrote in message
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John Jones
Daryl McCullough <da...@atc-nycorp.com> wrote in message
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Right then. I'll go and have a look you twat.
Roger Bishop Jones
> Occasionally I have tried to understand what stationary sets are but have
> always failed. The definition is not hard to parse, but parsing is a far
> cry from understanding. What's the motivation for the definition and why
> are stationary sets so important? What kinds of theorems can be proved
> with them that might be interesting to people who don't know or care about
> stationary sets for their own sake?
I doubt that I understand this any better than you Tim, but
I have recently been looking at a paper by Patrick Dehornoy
on "recent progress on the continuum hypothesis", and he
remarks that:
"A prominent technical role is played by the stationary subsets of aleph-1"
The concept appears in the definition of certain variants of Martin's
Axiom which figure in important results and conjectures suggesting
that CH is false.
--
Roger Jones
rbj at rbjones dot com
David C. Ullrich
<scoob...@btopenworld.com> wrote:
>If you don't tell me what you understand of it right now, and you've had the
>opportunity already, or bygumminy otherwise I think you might be an ignorant
>plonker.
>JJ
You're unable to click on a link and read a definition? Ok, I'll tell
you what "stationary" means: A subset S of a cardinal k is
stationary if S intersects every closed and unbounded subset of k.
There, that's the definition. Now you go ahead and give us your
"intuitive" explanation of what the motivation for the definition is
and why stationary sets are important.
Make sure to do that right now, btw.
><tc...@lsa.umich.edu> wrote in message
>news:3f18530a$0$3928$b45e...@senator-bedfellow.mit.edu...
>> In article <bf9jp4$hpa$1...@hercules.btinternet.com>,
>> John Jones <scoob...@btopenworld.com> wrote:
>> >Please give a definition, as some people can give an intuitive answer.
>>
>> See for example http://planetmath.org/encyclopedia/StationarySet.html
>> --
>> Tim Chow tchow-at-alum-dot-mit-dot-edu
>> The range of our projectiles---even ... the artillery---however great,
>will
>> never exceed four of those miles of which as many thousand separate us
>from
>> the center of the earth. ---Galileo, Dialogues Concerning Two New
>Sciences
>
************************
David C. Ullrich
Alan Stern
> Occasionally I have tried to understand what stationary sets are but have
> always failed. The definition is not hard to parse, but parsing is a far
> cry from understanding. What's the motivation for the definition and why
> are stationary sets so important? What kinds of theorems can be proved
> with them that might be interesting to people who don't know or care about
> stationary sets for their own sake?
One of the first and most important results concerning stationary sets
is Fodor's theorem. For those who aren't familiar with it, one of its
consequences is that whenever K is an uncountable regular cardinal and
f:K -> K is a regresssive (or "press-down") function, there is a
stationary set S contained in K such that f is constant on S. ("f is
regressive" means that for a > 0, f(a) < a.) This is such a general
result, it's importance ought to be apparent.
Another way to think about stationary sets is in terms of the filter
generated by the closed unbounded sets, an important object in itself.
To say that S is stationary is the same as saying that S is not in
the dual ideal, i.e., that S is not contained in the complement of a
club set. Hence S isn't "small" in the sense of this filter.
Alan Stern
George Cox
>
> Occasionally I have tried to understand what stationary sets are but have
> always failed. The definition is not hard to parse, but parsing is a far
> cry from understanding. What's the motivation for the definition and why
> are stationary sets so important? What kinds of theorems can be proved
> with them that might be interesting to people who don't know or care about
> stationary sets for their own sake?
Very loosely speaking: being stationary is like having non zero measure.
GC
John Jones
Look. There are no subsets. That kills it stone dead. The rest was fluff on
account of that.
JJ
David C. Ullrich <ull...@math.okstate.edu> wrote in message
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John Jones
JJ
Roger Bishop Jones <s...@signa.ture> wrote in message
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David C. Ullrich
<scoob...@btopenworld.com> wrote:
>'A subset S'?
>Look. There are no subsets.
Whee!
You know, you really have no idea whatever what you're
talking about. Thanks for the "inutitive explanation" of the
definition of "stationary"...
************************
David C. Ullrich
John Jones
collection then a set of cows would be a herd. So the sets of cows is not a
subset of herds. There are no relationships between sets. We confirm a
relationship, we don't confirm a relationship between sets. But set theory
claims relationship. Sets are the names of the sets. We can't even say that
there are sets, as sets are merely names-, they point, and have no
properties. Set theory is dishonest.
JJ
David C. Ullrich <ull...@math.okstate.edu> wrote in message
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fishfry
"John Jones" <scoob...@btopenworld.com> wrote:
> There are no subsets because a set is not a collection. If a set was a
> collection then a set of cows would be a herd.
Suppose I drive by a field full of cattle. Are the cows seen but not
herd?
David C. Ullrich
<scoob...@btopenworld.com> wrote:
>There are no subsets because a set is not a collection.
I see<giggle>. Reminds me of a recent post to sci.astronomy
that explained why there are no planets.
************************
David C. Ullrich