V = L
William Elliot
It is constructive.
It proves AxC and GCH.
It vanquishes the intangible inaccessibles or does it?
If it doesn't then does consistency of set theory implies
consistency of set theory + V = L + inaccessibles?
In view of the following results, I'd say no.
Consistency of set theory implies
consistency of set theory + no inaccessibles.
It cannot be proved
consistency of set theory implies
consistency of set theory + inaccessibles.
--
No matter how big a number, by Loewenheim-Skolem's paradox
it's but a facade for a countable number.
----
Dave L. Renfro
> How wonderfully simple is V = L.
> It is constructive.
> It proves AxC and GCH.
> It vanquishes the intangible inaccessibles or does it?
I don't know much about this stuff, but I have a nice
book by Keith J. Devlin (see below) that you might find
very interesting and useful. It's written in a folksy
expository manner that should make it accessible to most
anyone with a good undergraduate math background. Devlin
explains the axiom of constructibility (essentially
due to Godel), why it's natural (to him, at least), and
shows how it resolves all the (in 1977, at least) major
ZFC-independent results of interest to mathematicians.
Since it's in the Springer-Verlag Lecture Notes in
Mathematics series, it should be in just about every
research university's library. Or just order it by
interlibrary loan. I'm not sure if public libraries
can get ahold of a copy by interlibrary loan, but I
suppose you can try if you don't have access to a
university library.
Keith J. Devlin, "The Axiom of Constructibility: A Guide
for the Mathematician", Lecture Notes in Mathematics" #617,
Springer-Verlag, 1977, viii + 95 pages.
[MR 58 #5207; Zbl 369.02043]
http://www.emis.de/cgi-bin/MATH-item?0369.02043
http://www.amazon.com/gp/product/0387085203
Dave L. Renfro
Gene Ward Smith
Dave L. Renfro wrote:
> William Elliot wrote (in part):
>
> > How wonderfully simple is V = L.
> Devlin explains the axiom of constructibility (essentially
> due to Godel), why it's natural (to him, at least), and
> shows how it resolves all the (in 1977, at least) major
> ZFC-independent results of interest to mathematicians.
The only problem with V = L is believing it is true. Possibly that
could be replaced with believing only constructable sets are truly
meaningful or something. If Devlin gives arguments why we should think
V really is L, that might be interesting.
> I'm not sure if public libraries
> can get ahold of a copy by interlibrary loan, but I
> suppose you can try if you don't have access to a
> university library.
Linking public and university libraries is a growing trend. Here in San
Jose the SJSU and main city libraries have physically merged. It's also
a part of the Link+ interlibrary loan system, which allows people in
many public libraries to obtain (for free) books from university
library systems. So I'd by no means rule the possibility out.
Aatu Koskensilta
> How wonderfully simple is V = L.
> It is constructive.
It isn't any more constructive than the rest of ZFC axioms.
Constructively the constructible universe isn't particularly well
behaved or interesting.
> It proves AxC and GCH.
Yeah, but for wrong reasons.
> It vanquishes the intangible inaccessibles or does it?
No. V=L is consistent with any number of small large cardinal axioms.
--
Aatu Koskensilta (aatu.kos...@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Gerry Myerson
William Elliot <ma...@hevanet.remove.com> wrote:
> How wonderfully simple is V = L.
> It is constructive.
> It proves AxC and GCH.
> It vanquishes the intangible inaccessibles or does it?
But if you subtract V from both sides, you get 0 = XXXXV,
which is a bit of a problem.
--
Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email)
Jonathan Hoyle
> How wonderfully simple is V = L.
> It is constructive.
Yes, although awfully limited.
> It proves AxC and GCH.
True.
> It vanquishes the intangible inaccessibles or does it?
Not true. Inaccessible cardinals are consistent with V=L, although
much larger cardinals (such as those larger than 0#) are not.
> If it doesn't then does consistency of set theory implies
> consistency of set theory + V = L + inaccessibles?
Yes, or better stated: if "ZFC + there exists inaccessible cardinals"
is consistent, then so is "ZFC + V=L + there exists inaccessible
cardinals".
> In view of the following results, I'd say no.
>
> Consistency of set theory implies
> consistency of set theory + no inaccessibles.
That's true as well.
> It cannot be proved
> consistency of set theory implies
> consistency of set theory + inaccessibles.
That is false. Con(ZFC+inaccessibles) = Con(ZFC+no inaccessibles)
> No matter how big a number, by Loewenheim-Skolem's paradox
> it's but a facade for a countable number.
I don't understand this sentence. Please rephrase.
martin cohen
> In article <Pine.BSI.4.58.06...@vista.hevanet.com>,
> William Elliot <ma...@hevanet.remove.com> wrote:
>
>
>>How wonderfully simple is V = L.
>> It is constructive.
>> It proves AxC and GCH.
>> It vanquishes the intangible inaccessibles or does it?
>
>
> But if you subtract V from both sides, you get 0 = XXXXV,
> which is a bit of a problem.
>
But can you use this to prove P = NP???
Rupert
William Elliot wrote:
> How wonderfully simple is V = L.
> It is constructive.
> It proves AxC and GCH.
> It vanquishes the intangible inaccessibles or does it?
>
No, it doesn't.
> If it doesn't then does consistency of set theory implies
> consistency of set theory + V = L + inaccessibles?
No, but the consistency of an inaccessible implies the consistency of
V=L plus an inaccessible.
Rupert
Jonathan Hoyle wrote:
> William Elliot wrote:
> > How wonderfully simple is V = L.
> > It is constructive.
>
> Yes, although awfully limited.
>
> > It proves AxC and GCH.
>
> True.
>
> > It vanquishes the intangible inaccessibles or does it?
>
> Not true. Inaccessible cardinals are consistent with V=L, although
> much larger cardinals (such as those larger than 0#) are not.
>
> > If it doesn't then does consistency of set theory implies
> > consistency of set theory + V = L + inaccessibles?
>
> Yes, or better stated: if "ZFC + there exists inaccessible cardinals"
> is consistent, then so is "ZFC + V=L + there exists inaccessible
> cardinals".
>
> > In view of the following results, I'd say no.
> >
> > Consistency of set theory implies
> > consistency of set theory + no inaccessibles.
>
> That's true as well.
>
> > It cannot be proved
> > consistency of set theory implies
> > consistency of set theory + inaccessibles.
>
> That is false. Con(ZFC+inaccessibles) = Con(ZFC+no inaccessibles)
>
No, it's true. If ZFC+Con(ZFC) is consistent, then in ZFC we cannot
prove Con(ZFC)->Con(ZFC+IC). ZFC+IC has stronger consistency strength
than ZFC.
William Elliot
Newsgroups: sci.math
Subject: Re: V = L
> Dave L. Renfro wrote:
> > William Elliot wrote (in part):
>
> > > How wonderfully simple is V = L.
> > Devlin explains the axiom of constructibility (essentially
> > due to Godel), why it's natural (to him, at least), and
> > shows how it resolves all the (in 1977, at least) major
> > ZFC-independent results of interest to mathematicians.
> The only problem with V = L is believing it is true. Possibly that
> could be replaced with believing only constructable sets are truly
> meaningful or something. If Devlin gives arguments why we should
> think V really is L, that might be interesting.
Like the axiom of foundation, only grounded sets are truly meaningful.
----
William Elliot
> William Elliot wrote:
> > How wonderfully simple is V = L.
> > It is constructive.
>
> It isn't any more constructive than the rest of ZFC axioms.
> Constructively the constructible universe isn't particularly well
> behaved or interesting.
>
so lets remove axiom of foundations.
> > It proves AxC and GCH.
>
> Yeah, but for wrong reasons.
>
What's wrong about them?
> > It vanquishes the intangible inaccessibles or does it?
>
> No. V=L is consistent with any number of small large cardinal axioms.
>
Does no inaccessibles imply V = L?
Isn't it the case that the two propositions are independent?
William Elliot
> William Elliot <ma...@hevanet.remove.com> wrote:
>
> > How wonderfully simple is V = L.
> > It is constructive.
> > It proves AxC and GCH.
> > It vanquishes the intangible inaccessibles or does it?
>
> But if you subtract V from both sides, you get 0 = XXXXV,
> which is a bit of a problem.
The bit of a problem is there's no Roman numerals for zero.
So you get
= XLV
Aatu Koskensilta
> On Mon, 24 Jul 2006, Aatu Koskensilta wrote:
>> William Elliot wrote:
>> Yeah, but for wrong reasons.
> What's wrong about them?
V=L implies AC because if V=L there is a definable well-ordering of the
universe (of a particularly simple sort). That's perverse and wreaks
havoc in descriptive set theory.
>>>It vanquishes the intangible inaccessibles or does it?
>> No. V=L is consistent with any number of small large cardinal axioms.
> Does no inaccessibles imply V = L?
No.
> Isn't it the case that the two propositions are independent?
Yes.
Aatu Koskensilta
> How wonderfully simple is V = L.
The V=L hypothesis is not only vimple, but totavvy sivvy.
Jonathan Hoyle
> > > consistency of set theory implies
> > > consistency of set theory + inaccessibles.
> >
> > That is false. Con(ZFC+inaccessibles) = Con(ZFC+no inaccessibles)
> >
>
> No, it's true. If ZFC+Con(ZFC) is consistent, then in ZFC we cannot
> prove Con(ZFC)->Con(ZFC+IC). ZFC+IC has stronger consistency strength
> than ZFC.
I was under the impression that IC was undecidable with respect to ZFC,
and thus neither ZFC+IC nor ZFC+~IC could generate a contradiction
unless one already existed in ZFC. (If such a contradiction could be
generated by either ZFC+IC or ZFC+~IC, then IC cannot be said to be
undecidable wrt ZFC.)
However, I certainly could be in error on this matter. Could you
clarify this more, Rupert? Thanks in advance.
Regards,
Jonathan Hoyle
Jonathan Hoyle
> book by Keith J. Devlin (see below) that you might find
> very interesting and useful. It's written in a folksy
> expository manner that should make it accessible to most
> anyone with a good undergraduate math background. Devlin
> explains the axiom of constructibility (essentially
> due to Godel), why it's natural (to him, at least), and
> shows how it resolves all the (in 1977, at least) major
> ZFC-independent results of interest to mathematicians.
I have another of Devlin's books "The Joy of Sets" which gives a good
overview of Set Theory and even continues on to some theories which do
not contain the Axiom of Foundation.
William Elliot
> William Elliot wrote:
> > On Mon, 24 Jul 2006, Aatu Koskensilta wrote:
> >> William Elliot wrote:
> >>>[V=L] proves AxC and GCH.
> >> Yeah, but for wrong reasons.
> > What's wrong about them?
>
> V=L implies AC because if V=L there is a definable well-ordering of the
> universe (of a particularly simple sort). That's perverse and wreaks
> havoc in descriptive set theory.
>
Perverse? Is descriptive set theory as perverse as descriptive geometry?
It was so perverse that when I looked into it, I remembered nothing about
it, not even a basic notion or two. What care be there that it bring
havoc to descriptive set theory it it's of so little notice I and a
multitude of others, who have even heard about constructionism, have never
heard of descriptive set theory?
Anyway, be it as you wish, you've set the stage to make an introduction of
descriptive set theory to us hoards of ignorant mathematicians.
> >>>It vanquishes the intangible inaccessibles or does it?
> >> No. V=L is consistent with any number of small large cardinal axioms.
> > Does no inaccessibles imply V = L?
>
> No.
>
Sigh, nothing is easy in this world of infinitely abundant
hyper-complexities. What be there amiss to wish to simply live in a
well ordered universe? I see nothing awry about a well ordered universe.
Why do you?
> > Isn't it the case that the two propositions are independent?
>
> Yes.
I'd be surprised as a polynomial relation between pi and e,
if it was else.
--
Tho any pervert can mess up, only Bush can reek world wide havoc.
David C. Ullrich
<ma...@hevanet.remove.com> wrote:
>On Tue, 25 Jul 2006, Aatu Koskensilta wrote:
>
>> William Elliot wrote:
>> > On Mon, 24 Jul 2006, Aatu Koskensilta wrote:
>> >> William Elliot wrote:
>
>> >>>[V=L] proves AxC and GCH.
>> >> Yeah, but for wrong reasons.
>> > What's wrong about them?
>>
>> V=L implies AC because if V=L there is a definable well-ordering of the
>> universe (of a particularly simple sort). That's perverse and wreaks
>> havoc in descriptive set theory.
>>
>Perverse? Is descriptive set theory as perverse as descriptive geometry?
>It was so perverse that when I looked into it, I remembered nothing about
>it, not even a basic notion or two. What care be there that it bring
>havoc to descriptive set theory it it's of so little notice I and a
>multitude of others, who have even heard about constructionism, have never
>heard of descriptive set theory?
Good point. If _you_ have never heard of it it can't be of any
importance.
>Anyway, be it as you wish, you've set the stage to make an introduction of
>descriptive set theory to us hoards of ignorant mathematicians.
I like that, "hoards of ignorant mathematicians". What's the word
for a sentence that demonstrates its own truth? (Unless you were
referring to piles of mathematicians that someone's stored for
a rainy day somewhere you meant "hordes".)
>
>> >>>It vanquishes the intangible inaccessibles or does it?
>> >> No. V=L is consistent with any number of small large cardinal axioms.
>> > Does no inaccessibles imply V = L?
>>
>> No.
>>
>Sigh, nothing is easy in this world of infinitely abundant
>hyper-complexities. What be there amiss to wish to simply live in a
>well ordered universe? I see nothing awry about a well ordered universe.
>Why do you?
>
>> > Isn't it the case that the two propositions are independent?
>>
>> Yes.
>
>I'd be surprised as a polynomial relation between pi and e,
>if it was else.
>--
>Tho any pervert can mess up, only Bush can reek world wide havoc.
"wreak". (Yes, Bush also reeks.)
************************
David C. Ullrich
William Elliot
> <ma...@hevanet.remove.com> wrote:
> >On Tue, 25 Jul 2006, Aatu Koskensilta wrote:
> >> William Elliot wrote:
> >> > On Mon, 24 Jul 2006, Aatu Koskensilta wrote:
> >> >> William Elliot wrote:
> >
> >> >>>[V=L] proves AxC and GCH.
> >> >> Yeah, but for wrong reasons.
> >> > What's wrong about them?
> >>
> >> V=L implies AC because if V=L there is a definable well-ordering of the
> >> universe (of a particularly simple sort). That's perverse and wreaks
> >> havoc in descriptive set theory.
> >>
> >Perverse? Is descriptive set theory as perverse as descriptive geometry?
> >It was so perverse that when I looked into it, I remembered nothing about
> >it, not even a basic notion or two. What care be there that it bring
> >havoc to descriptive set theory it it's of so little notice I and a
> >multitude of others, who have even heard about constructionism, have never
> >heard of descriptive set theory?
>
> Good point. If _you_ have never heard of it it can't be of any
> importance.
>
> >Anyway, be it as you wish, you've set the stage to make an introduction of
> >descriptive set theory to us hoards of ignorant mathematicians.
>
> I like that, "hoards of ignorant mathematicians". What's the word
> for a sentence that demonstrates its own truth? (Unless you were
> referring to piles of mathematicians that someone's stored for
> a rainy day somewhere you meant "hordes".)
>
Alright, lest you join the hoard, what in a nutshell is descriptive set
theory?
G. A. Edgar
Descriptive set theory:
open sets, closed sets, Borel sets, G_delta sets, analytic sets,
projective sets, etc.
Perhaps you have heard of some of these, even without the name
"descriptive set theory".
> and if you have, is it of any importance to you?
Yes, each of these (except projective sets) have appeared in one or
more of my papers.
--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/
Dave L. Renfro
> Yes, each of these (except projective sets) have
> appeared in one or more of my papers.
I think it's a good bet that projective sets also
appeared in the same papers. ;)
William ... Given your interest in topology and
foundational issues, I'm rather surprised that you
know very little about descriptive set theory, since
this is one of the main areas that topology and set
theory owes its development to.
The following are probably the three most relevant texts
on the subject. (Before reading your comments elsewhere
in this thread, I would have guessed that you'd have
at least one of them on your bookshelves.) The last
third of Jech's "Set Theory" -- his Academic Press
graduate level text -- is also a good reference.
Alexander S. Kechris, "Classical Descriptive Set Theory",
Graduate Texts in Math #156, Springer-Verlag, 1995.
Yiannis N. Moschovakis, "Descriptive Set Theory",
North-Holland, 1980.
S. M. Srivastava, "A Course on Borel Sets", Graduate
Texts in Math #180, Springer-Verlag, 1998.
I wrote a historical survey of certain aspects of
descriptive set theory in the following post.
"How bad can 2^(Aleph_0) be?" history question (June 5, 2001)
http://groups.google.com/group/sci.math/msg/82a5ec796aeb1073
Dave L. Renfro
William Elliot
Newsgroups: sci.math
Subject: Re: V = L
> Descriptive set theory:
> open sets, closed sets, Borel sets, G_delta sets, analytic sets,
> projective sets, etc.
> Perhaps you have heard of some of these, even without
> the name "descriptive set theory".
All except what's an analytic set? Same as measurable set? But aren't
they Borel sets?
Also what's a projective set?
How come filters, also collectively being a family of subsets of P(Space),
isn't descriptive?
> > and if you have, is it of any importance to you?
> Yes, each of these (except projective sets) have appeared
> in one or more of my papers.
It's quit hard to avoid open, closed, Borel sets or even G_delta.
But then again, how could a well ordered universe wreak havoc with
this form of set theory, which as best as I can glean from Wikipedia,
is mostly for subsets of reals used in analysis.
----
William Elliot
> > Yes, each of these (except projective sets) have
> > appeared in one or more of my papers.
> I think it's a good bet that projective sets also
> appeared in the same papers. ;)
> William ... Given your interest in topology and
> foundational issues, I'm rather surprised that you
> know very little about descriptive set theory, since
> this is one of the main areas that topology and set
> theory owes its development to.
Never mentioned in all of my studies.
> The following are probably the three most relevant texts
> on the subject. (Before reading your comments elsewhere
> in this thread, I would have guessed that you'd have
> at least one of them on your bookshelves.) The last
> third of Jech's "Set Theory" -- his Academic Press
> graduate level text -- is also a good reference.
No I don't. I've an assortment of papers and two topology books,
Steen's "Counter Examples..." and Kelley's classic on topology.
> Alexander S. Kechris, "Classical Descriptive Set Theory",
> Graduate Texts in Math #156, Springer-Verlag, 1995.
Why should I be interested in this? I've looked into interior algebras
which are mildly interesting and vaguely related to descriptive set
theory.
The question that started this is why does V = L, ie a well order
universe, wreak havoc with descriptive set theory. From the brief synopsis
given by Wikipedia, that is far from apparent.
> Yiannis N. Moschovakis, "Descriptive Set Theory",
> North-Holland, 1980.
> S. M. Srivastava, "A Course on Borel Sets", Graduate
> Texts in Math #180, Springer-Verlag, 1998.
> I wrote a historical survey of certain aspects of
> descriptive set theory in the following post.
> "How bad can 2^(Aleph_0) be?" history question (June 5, 2001)
> http://groups.google.com/group/sci.math/msg/82a5ec796aeb1073
Whew. What those guys went thru trying to show CH.
Nice historical review which is never mentioned in texts.
This was a runner up to showing the independence of CH?
Am I to suppose it was a great relief to them to know CH
had joined the independent party?
Herein however is a puzzle that's been floating about in this thread
and the other "How many ordinals":
If, for example CH, is shown to be independent of ZFC, then is
not ZFC + CH and ZFC + ~CH both immediately as consistent as ZFC?
Does this ruse work? If ZFC + CH is inconsistent
then ZFC + CH |- contradiction
consequently ZFC |- ~CH
contrary to independence.
----
Denis Feldmann
If it is *shown* (i.e has a formal proof in ZFC), yes (this is the case
for CH, since Cohen's work). But A can be independent from ZFC and still
ZFC+A not necessarily equiconsistent with ZFC: existence of an
inaccessible cardinal is clearly not provable in ZFC (if it can be
proved, it follows that ZFC has an inner model, so is inconsistent from
Godel) ; the negation is probably not provable in ZFC, but precisely
because ZFC + "exists an inaccessible" is strictly stronger than ZFC,
the possibility will always exist (until a proof is found, or ZFC is
proved unconsistent :-))
G. A. Edgar
> > Descriptive set theory:
> > open sets, closed sets, Borel sets, G_delta sets, analytic sets,
> > projective sets, etc.
>
> > Perhaps you have heard of some of these, even without
> > the name "descriptive set theory".
>
> All except what's an analytic set? Same as measurable set? But aren't
> they Borel sets?
No.
Lebesgue has a paper with a famous mistake, when he claimed if you
project a Borel set in the plane onto the x-axis, the result is a
Borel set again. It may not be. But it is an analytic set
(that could be the definition of analytic set in the line);
and it is Lebesgue measurable, even if it is not Borel.
>
> Also what's a projective set?
keep repeating this operation (in Euclidean spaces of any dimension:
projection, complementation). These results are called projective sets.
There are projective sets that are not analytic. Whether all projective
sets are Lebesgue measurable is independent of ZFC.
>
> How come filters, also collectively being a family of subsets of P(Space),
> isn't descriptive?
>
> > > and if you have, is it of any importance to you?
>
> > Yes, each of these (except projective sets) have appeared
> > in one or more of my papers.
>
> It's quit hard to avoid open, closed, Borel sets or even G_delta.
> But then again, how could a well ordered universe wreak havoc with
> this form of set theory, which as best as I can glean from Wikipedia,
> is mostly for subsets of reals used in analysis.
>
That "wreak havoc" comment was one person's strange opinion.
Don't worry about it.
Descriptive set theory is fine, either with or without V=L.
William Elliot
Newsgroups: sci.math
Subject: Re: V = L
William Elliot a crit :
> > If, for example CH, is shown to be independent of ZFC, then is
> > not ZFC + CH and ZFC + ~CH both immediately as consistent as ZFC?
> If it is *shown* (i.e has a formal proof in ZFC), yes (this is the
> case for CH, since Cohen's work).
What a hair splitting head ache that was. Ok, the in the following:
> > Does this ruse work? If ZFC + CH is inconsistent
> > then ZFC + CH |- contradiction
> > consequently ZFC |- ~CH
> > contrary to independence.
it was slipped in that independence was proved in ZFC.
> But A can be independent from ZFC
> and still ZFC+A not necessarily equiconsistent with ZFC: existence of
> an inaccessible cardinal is clearly not provable in ZFC (if it can
> be proved, it follows that ZFC has an inner model, so is inconsistent
> from Godel) ; the negation is probably not provable in ZFC, but
> precisely because ZFC + "exists an inaccessible" is strictly stronger
> than ZFC, the possibility will always exist (until a proof is found,
> or ZFC is proved unconsistent :-))
Whew, what language, meta-language hair spliting. Ok, I think I got it.
Thanks. I may remain content with
Consistency of set theory implies
consistency of set theory + no inaccessibles.
It cannot be proved
consistency of set theory implies
consistency of set theory + inaccessibles.
In other words, it's sure ground to deny their existence (hurray says
Occam) while to assert their existence is unwarrenteed, uninsured
speculation.
Most humorous is it's metaphsical implications
god(s) not exist, is logically consistent
god(s) exist, is without logical base.
But then again, that's what they have always proclaimed:
faith beyond reason.
----
William Elliot
Newsgroups: sci.math
Subject: Re: V = L
> > > Descriptive set theory:
> > > open sets, closed sets, Borel sets, G_delta sets, analytic sets,
> > > projective sets, etc.
>
> > > Perhaps you have heard of some of these, even without
> > > the name "descriptive set theory".
>
> > All except what's an analytic set? Same as measurable set? But
> > aren't they Borel sets?
> No.
> Lebesgue has a paper with a famous mistake, when he claimed if you
> project a Borel set in the plane onto the x-axis, the result is a
> Borel set again. It may not be. But it is an analytic set
> (that could be the definition of analytic set in the line);
> and it is Lebesgue measurable, even if it is not Borel.
> > Also what's a projective set?
> keep repeating this operation (in Euclidean spaces of any dimension:
> projection, complementation). These results are called projective
> sets. There are projective sets that are not analytic. Whether all
> projective sets are Lebesgue measurable is independent of ZFC.
It's more than ample to convince me I'm just a student and not a
mathematican.
> > It's quit hard to avoid open, closed, Borel sets or even G_delta.
> > But then again, how could a well ordered universe wreak havoc with
> > this form of set theory, which as best as I can glean from
> > Wikipedia, is mostly for subsets of reals used in analysis.
> That "wreak havoc" comment was one person's strange opinion.
> Don't worry about it.
Whew.
Now if only we lived in a well order universe...
> Descriptive set theory is fine, either with or without V=L.
So is descriptive geometry. Wikipedia synopises clarified what it is.
Seems most applicable to architectural design.
-- Totalitarianism
Society is partially ordered. Can society be well ordered?
By use of the axiom of choice, a partial order can be extended
to a total order. Therefore society can be totally ordered.
However do recall, total order may not be well ordered.
----
Ulysse from CH
<ed...@math.ohio-state.edu.invalid> wrote:
>> (...) what's an analytic set? Same as measurable set? But aren't
>> they Borel sets?
>
>No.
>Lebesgue has a paper with a famous mistake, when he claimed if you
>project a Borel set in the plane onto the x-axis, the result is a
>Borel set again. It may not be. But it is an analytic set
>(that could be the definition of analytic set in the line);
>and it is Lebesgue measurable, even if it is not Borel.
>
Bourbaki has an interesting theory of analytic sets (1) - called
Souslin sets. In fact there is a topological characterization of such
sets: metrizable spaces which are continuous images of polish
spaces (2) are called Souslin spaces (note that this def. does not
use the concept of a Borel set - Wikipedia's def. of an analytic set
does) and in any top. space Souslin sets are just subsets becoming
a Souslin space under the induced top. (It is proven that continuous
images of Borel subsets of a Souslin space in a metrizable one are
Souslin, showing equivalence of def's.) There is a similar concept
of a Lusin space with following properties: their def. is a bit more
complicated, but in fact it turns out that Lusin spaces are just
the metrizable images of polish spaces by continuous *bijections*.
And in a Lusin space (like |R^n) the Borel sets are exactly the
subsets having a Lusin indiced top. (long proof). Proof that Souslin
sets are measurable is given: to be more precise, there are defined
concepts of "capacity" and of "capacitable set"; well known
results in measure theory show that for Radon measures (measures
defined on Borel sets, then completed) the outer measure is a capacity
and capacitable <=> measurable; it is just proven that Souslin sets
are capacitable - if they are relatively compact; this restriction may
probably be removed in a locally compact metrizable space that
is "countable at infinity" (one-point compactification 1st countable)
(1) in chapt. 9 of "General Topology" (Utilisation des nombres réels
en topologie générale", §9 (Souslin spaces, Borel sets)
(2) a polish space is by def. a 2nd countable topol. space for which
there exists a metric defining the top. and making the space a
*complete* metric space (example: |R \ |Q is polish !)
>
>> Also what's a projective set?
>keep repeating this operation (in Euclidean spaces of any dimension:
>projection, complementation). These results are called projective sets.
>There are projective sets that are not analytic. Whether all projective
>sets are Lebesgue measurable is independent of ZFC.
>
And what about the statement "there are Lebesgue measurable sets
which are not projective" ?
>
>(...)
G. A. Edgar
> And what about the statement "there are Lebesgue measurable sets
> which are not projective" ?
easily seen to be true by cardinality
Ulysse from CH
<uly...@usager-anonyme.ch> wrote:
>On Thu, 27 Jul 2006 08:19:22 -0400, "G. A. Edgar"
><ed...@math.ohio-state.edu.invalid> wrote:
>> (...)
>>Lebesgue has a paper with a famous mistake, when he claimed if you
>>project a Borel set in the plane onto the x-axis, the result is a
>>Borel set again. It may not be. But it is an analytic set
>>(that could be the definition of analytic set in the line);
>>and it is Lebesgue measurable, even if it is not Borel.
>Bourbaki has an interesting theory of analytic sets (1) - called
>Souslin sets. In fact there is a topological characterization of such
>sets: metrizable spaces which are continuous images of polish
>spaces (2) are called Souslin spaces (...)
>and in any top. space Souslin sets are just subsets becoming
>a Souslin space under the induced top. (...)
>(1) in chapt. 9 of "General Topology" (Utilisation des nombres réels
>en topologie générale", §9 (Souslin spaces, Borel sets)
>(2) a polish space is by def. a 2nd countable topol. space for which
>there exists a metric defining the top. and making the space a
>*complete* metric space (example: |R \ |Q is polish !)
>(...)
>
After having written this, I realized that I do not know an example of
a Souslin (= analytic) set which isn't Borel ! Can someone give me
one - that is an example showing that Lebesgue was indeed in error ?
Wikipedia apparently does not give one, and if Bourbaki gave one
in the mentioned text (except may-be in the exercises) I probably
would have copied it into my notes.
I want to add an interesting fact (proven in the Bourbaki text).
If in a metrizable space X, a subset S and X\S are both Souslin,
then S is Borel. From this, it follows that if S is Souslin and >not<
Borel, then X\S is Lebesgue measurable and not Souslin (when
X is |R or |R^n). That there are non-Borel measurable sets in |R
seems to be provable by a simple cardinality argument: the Cantor
set Z has measure 0, so all its subsets are measurable (of measure 0),
and card(Z) = c (power of the continuum), but the set of all Borel
sets has (?) only card. c (< 2^c) - IIRC this has been said recently
in sci.math. What about the card. of the set of all Souslin sets
in |R ? and that of the set of projective sets ?
Dave L. Renfro
> the Cantor set Z has measure 0, so all its subsets are
> measurable (of measure 0), and card(Z) = c (power of
> the continuum), but the set of all Borel sets has (?)
> only card. c (< 2^c) - IIRC this has been said recently
> in sci.math. What about the card. of the set of all
> Souslin sets in |R ? and that of the set of projective sets ?
The same argument for the Borel sets that uses transfinite
induction on the omega_1 length Borel hierarchy works for
the projective sets, since they also form a hierarchy of
length omega_1. You can find proofs for the Borel hierarchy
result in Devlin's undergraduate set theory text and in the
excellent text "Sets: Naive, Axiomatic and Applied" by
D. Van Dalen, H. C. Doets, and H. De Swart.
Dave L. Renfro
G. A. Edgar
> a Souslin (= analytic) set which isn't Borel ! Can someone give me
> one - that is an example showing that Lebesgue was indeed in error ?
http://groups.google.com/group/sci.math/browse_frm/thread/a048781739a61b
14/766042686db74622
Dave L. Renfro
> The same argument for the Borel sets that uses transfinite
> induction on the omega_1 length Borel hierarchy works for
> the projective sets, since they also form a hierarchy of
> length omega_1. You can find proofs for the Borel hierarchy
> result in Devlin's undergraduate set theory text and in the
> excellent text "Sets: Naive, Axiomatic and Applied" by
> D. Van Dalen, H. C. Doets, and H. De Swart.
To follow-up on my earlier comments, here are two
places where the proof that there are c Borel sets
is carried out at a fairly elementary level:
"Basic Set Theory" by M. A. Lifshits and Nikolai
Konstantinovich Vereshchagin
Use "Search in this book" = "Borel", choose pp. 101-103
http://books.google.com/books?vid=ISBN0821827316
"The Joy of Sets: Fundamentals of Contemporary
Set Theory" by Keith J Devlin
Use "Search in this book" = "Borel", choose pp. 101 & 102
http://books.google.com/books?vid=ISBN0387940944
If anyone is interested in some on-line notes for
descriptive set theory, the following (105 pages)
appears to be very nice:
David Marker, Course Notes for Math 512: Descriptive
Set Theory, Fall 2002, University of Illinois at Chicago
http://www.math.uic.edu/~marker/math512/dst.pdf
Dave L. Renfro
Jonathan Hoyle
> by a simple cardinality argument: the Cantor set Z has measure 0,
> so all its subsets are measurable (of measure 0),
I didn't follow this part. Why should all of the subsets of Z be
measurable? It seems to me that there are non-measurable subsets.
Dave L. Renfro
>> That there are non-Borel measurable sets in |R
>> seems to be provable by a simple cardinality argument:
>> the Cantor set Z has measure 0, so all its subsets are
>> measurable (of measure 0),
Jonathan Hoyle wrote:
> I didn't follow this part. Why should all of the subsets
> of Z be measurable? It seems to me that there are
> non-measurable subsets.
Every subset of a (Lebesgue) measure zero set is
measurable, with a measure equal to zero.
There are several equivalent definitions of Lebesgue
measurability, but I think this result is immediate
for all of them.
Dave L. Renfro
Jonathan Hoyle
> > of Z be measurable? It seems to me that there are
> > non-measurable subsets.
>
> Every subset of a (Lebesgue) measure zero set is
> measurable, with a measure equal to zero.
>
> There are several equivalent definitions of Lebesgue
> measurability, but I think this result is immediate
> for all of them.
Ah yes, a brain cramp on my part. I overlooked the "measure zero"
part.
Yes, of course, you are correct. (Sometimes Fridays are not any better
than Mondays for "thinking.) :-/
William Elliot
> ><ed...@math.ohio-state.edu.invalid> wrote:
> >>Lebesgue has a paper with a famous mistake, when he claimed if you
> >>project a Borel set in the plane onto the x-axis, the result is a
> >>Borel set again. It may not be. But it is an analytic set
> >>(that could be the definition of analytic set in the line);
> >>and it is Lebesgue measurable, even if it is not Borel.
> >Bourbaki has an interesting theory of analytic sets (1) - called
> >Souslin sets. In fact there is a topological characterization of such
> >sets: metrizable spaces which are continuous images of polish
> >spaces (2) are called Souslin spaces (...)
Steen's "Counter examples in Topology" p173.
A Souslin space is a counterexample (if one exists) to Souslin's
conjecture:
a linear ordered space must be separable whenever
it satisfies the countable chain condition
(every disjoint collection of open sets is countable).
Sousline's conjecture is consistence with and independent of the axioms of
set theory. (Tho not clarified, I would assume AxC is included as
topologists tend to be prochoice.)
G. A. Edgar
William Elliot <ma...@hevanet.remove.com> wrote:
> On Fri, 28 Jul 2006, Ulysse from CH wrote:
> > ><ed...@math.ohio-state.edu.invalid> wrote:
>
> > >>Lebesgue has a paper with a famous mistake, when he claimed if you
> > >>project a Borel set in the plane onto the x-axis, the result is a
> > >>Borel set again. It may not be. But it is an analytic set
> > >>(that could be the definition of analytic set in the line);
> > >>and it is Lebesgue measurable, even if it is not Borel.
> > >Bourbaki has an interesting theory of analytic sets (1) - called
> > >Souslin sets. In fact there is a topological characterization of such
> > >sets: metrizable spaces which are continuous images of polish
> > >spaces (2) are called Souslin spaces (...)
>
> Steen's "Counter examples in Topology" p173.
> A Souslin space is a counterexample (if one exists) to Souslin's
> conjecture:
> a linear ordered space must be separable whenever
> it satisfies the countable chain condition
> (every disjoint collection of open sets is countable).
This is a different "Souslin space" than the ones related to
descriptive set theory. There were only two publications from Suslin
during his short life: one (with Luzin) on Borel sets and such, one a
problem on characterizing the line.
William Elliot
> William Elliot <ma...@hevanet.remove.com> wrote:
>
> > Steen's "Counter examples in Topology" p173.
> > A Souslin space is a counterexample (if one exists) to Souslin's
> > conjecture:
> > a linear ordered space must be separable whenever
> > it satisfies the countable chain condition
> > (every disjoint collection of open sets is countable).
>
> This is a different "Souslin space" than the ones related to
> descriptive set theory.
>
> There were only two publications from Suslin during his short life: one
> (with Luzin) on Borel sets and such, one a problem on characterizing the
> line.
>
M. Souslin is from Moscow university. That's all I could find out as
links to his biographies were faulty and he's not in the two math
biography sites I know. Another Abel lost to the world?
G. A. Edgar
> M. Souslin is from Moscow university. That's all I could find out as
> links to his biographies were faulty and he's not in the two math
> biography sites I know. Another Abel lost to the world?
Suslin died in 1917 in the typhoid epidemic following the Russian
Revolution. I believe he was 21. [Perhaps you mean to compare to
Galois, not Abel.]
Dave L. Renfro
>> M. Souslin is from Moscow university. That's all I could find
>> out as links to his biographies were faulty and he's not in the
>> two math biography sites I know. Another Abel lost to the world?
G. A. Edgar wrote:
> Suslin died in 1917 in the typhoid epidemic following the Russian
> Revolution. I believe he was 21. [Perhaps you mean to compare
> to Galois, not Abel.]
Actually, Suslin died in 1919. While I suspect William Elliot
did mean to compare Suslin to Galois, I came across the following
passage while looking up some things for the post I just made
in the sci.math thread "Mikhail Y. Suslin and Lebesgue's error":
"The whole episode [regarding Suslin] recalls a pivotal
mistake by Augustin-Louis Cauchy and the clarification due
to the young Niels Abel that led to the concept of uniform
convergence, even to an unjustifiable (for all)(there exists)
to (there exists)(for all) switch and Abel's untimely death."
This passage appears just before Theorem 12.2 (p. 148) in
Akihiro Kanamori's "The Higher Infinite", Springer-Verlag, 1994.
Dave L. Renfro
Rupert
Jonathan Hoyle wrote:
> > > > It cannot be proved
> > > > consistency of set theory implies
> > > > consistency of set theory + inaccessibles.
> > >
> > >
> >
> > No, it's true. If ZFC+Con(ZFC) is consistent, then in ZFC we cannot
> > prove Con(ZFC)->Con(ZFC+IC). ZFC+IC has stronger consistency strength
> > than ZFC.
>
> I was under the impression that IC was undecidable with respect to ZFC,
> and thus neither ZFC+IC nor ZFC+~IC could generate a contradiction
> unless one already existed in ZFC.
This is true provided ZFC+IC is consistent, which is conjectured to be
the case. However, it cannot be proved to be true in ZFC+Con(ZFC),
assuming that ZFC+Con(ZFC) is consistent.
> (If such a contradiction could be
> generated by either ZFC+IC or ZFC+~IC, then IC cannot be said to be
> undecidable wrt ZFC.)
>
> However, I certainly could be in error on this matter. Could you
> clarify this more, Rupert? Thanks in advance.
>
Con(ZFC+IC) might well be true, and most set theorists believe it to be
true, but it cannot be proved in ZFC+Con(ZFC), assuming that
ZFC+Con(ZFC) is consistent. This can be seen by using Goedel's second
incompleteness theorem. You see, Con(ZFC+IC) implies the consistency of
ZFC+Con(ZFC). So, if ZFC+Con(ZFC) could prove Con(ZFC+IC), it could
prove its own consistency. Then, by Goedel's second incompleteness
theorem, it would be inconsistent.
So the situation is: if ZFC is consistent, then it certainly cannot
prove IC. Can it refute IC? Some people think there is some possibility
that it can, but most people believe it can't. However, this is
something you have to "take on faith". You can't prove it in
ZFC+Con(ZFC), or ZFC+Con(ZFC)+Con(ZFC+Con(ZFC)), etc., assuming these
theories are consistent. You have a similar phenomenon with each new
large cardinal axiom - its consistency has to be either taken on faith
or remain conjectural.
> Regards,
>
> Jonathan Hoyle
William Elliot
> In article <Pine.BSI.4.58.06...@vista.hevanet.com>,
> William Elliot <ma...@hevanet.remove.com> wrote:
>
> > M. Souslin is from Moscow university. That's all I could find out as
> > links to his biographies were faulty and he's not in the two math
> > biography sites I know. Another Abel lost to the world?
>
> Suslin died in 1917 in the typhoid epidemic following the Russian
> Revolution. I believe he was 21. [Perhaps you mean to compare to
> Galois, not Abel.]
>
Thanks G.A., that was what I was wanting to know.