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| Daryl McCullough... |
 Posted: Wed Nov 04, 2009 7:27 am |
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Bill Taylor says...
[quote]
stevendaryl3... at (no spam) yahoo.com (Daryl McCullough) wrote:
That isn't true. Real numbers did not have names before Cantor.
Oh heavens, of course they did!
[/quote]
No, they did not. *Some* reals had names.
[quote]All the integers and ratiobnals had them, and also all the reals
we ever use in applications, like pi, e, sin^(-1)(2/7) and so on.
Points and lines of Euclidean geometry didn't have names.
Oh heavens, of course they did!
[/quote]
No, they did not! You are completely wrong on this.
[quote]I'll ignore the trivial fact that all Euclid's diagrams had letters
[/quote]
You don't understand the difference between a *variable* and a
unique name? When someone says "Let A be a point, and let R be
a line running through A", "A" and "R" are not intended to be
unique names. They are labels we are applying *locally*. Tomorrow,
I might call a *different* object "A".
[quote]No-one doubts he [Descartes] was doing Euclidean geometry, and
extending it, and all his points were real number pairs, see just
above.
[/quote]
Not every real has a name, then not every pair of reals has a name.
This is *provable*, Bill. It's a mathematical theorem. Yes, the
proof may have not existed before Cantor, but that doesn't mean
that it became true when Cantor proved it. It was true beforehand,
even if people didn't know it.
[quote]All the integers and ratiobnals had them, and also all the reals
we ever use in applications, like pi, e, sin^(-1)(2/7) and so on.
[/quote]
That is completely ridiculous. Of course, if a real comes up
in an application, then you *give* it a name. That's where "e"
came from. That's where "pi" came from.
Are you just claiming the tautology that if X is a real, then
I am free to name it "Fred", and from then on, I can define it
via "that unique real that I have named 'Fred'"? Sure, by that
notion, every real can have a name.
--
Daryl McCullough
Ithaca, NY |
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| Bill Taylor... |
| Posted: Wed Nov 04, 2009 6:34 pm |
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Well, I think we've got to the point where we're starting to repeat
ourselves, and not really listening to what the other has to say.
So maybe this will be brief...
stevendaryl3... at (no spam) yahoo.com (Daryl McCullough) wrote:
[quote]In the case of reals, presumably
if every real is definable at some level,
[/quote]
Yes.
[quote]then the collection of all such levels has a supremum, alpha.
[/quote]
No; as I said, w_1^CK doesn't really exist in the defineability sense.
[quote]The alternative, it seems to me, is to say that there is no *set*
of all reals
[/quote]
Correct; taking things strictly.
[quote](because every level produces new reals, and the
levels are never completed).
[/quote]
Ditto. As I said, it's powerset that is the real villain, but it can
be adopted for smoothness, provided one is subsequently careful,
meaning avoiding AC (and similar).
[quote]Then you *are* saying that the reals don't form a set.
Well, then AC is definitely not the problem, separation is.
[/quote]
Powerset.
[quote]we talk about the set of all reals r such that Phi(r), that's
an illegitimate operation if the reals don't form a set.
[/quote]
You're speaking of Unlimited Comprehension, not Separation.
[quote]You are saying that AC
is bad because it produces undefinable sets of reals. But it
doesn't *unless* there are already undefinable reals.
[/quote]
This is clearly wrong, and I suspect not what you meant to say.
It would be perfectly possible to work entirely with definable reals,
but (shun to) have undefineable collections of them.
Compare - all naturals are defineable, but (you said above)
there may be undefineable collections of them (i.e. reals).
[quote]AC would
not be the *source* of the undefinability. It's the innocent bystander.
[/quote]
I would rather put it, the executoror enforcer of an immoral law:
powerset.
-- Worried William |
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| Marshall... |
| Posted: Wed Nov 04, 2009 6:48 pm |
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On Nov 4, 8:34 pm, Bill Taylor <w.tay... at (no spam) math.canterbury.ac.nz> wrote:
[quote]
I would rather put it, the executoror enforcer of an immoral law:
powerset.
[/quote]
Pardon me; am I to understand that you are objecting to the
powerset axiom?
Marshall |
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| Peter Webb... |
| Posted: Thu Nov 05, 2009 6:04 am |
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"Marshall" <marshall.spight at (no spam) gmail.com> wrote in message
news:40b03443-d7d3-4ce2-9cdc-f8e2f5ba4309 at (no spam) u16g2000pru.googlegroups.com...
On Nov 4, 8:34 pm, Bill Taylor <w.tay... at (no spam) math.canterbury.ac.nz> wrote:
[quote]
I would rather put it, the executoror enforcer of an immoral law:
powerset.
[/quote]
Pardon me; am I to understand that you are objecting to the
powerset axiom?
Marshall
__________________________
I was there when the crime occurred, and it was clearly the Axiom of
Infinity who was the mastermind. Without him there, all the other axioms are
completely well behaved. Even the Axiom of Choice is always true, even if
you can't quite prove it ... I say get rid of the Axiom of Infinity, and our
problems with disruly theorems will disappear completely. |
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| Daryl McCullough... |
| Posted: Thu Nov 05, 2009 9:08 am |
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Bill Taylor says...
[quote]In the case of reals, presumably
if every real is definable at some level,
Yes.
then the collection of all such levels has a supremum, alpha.
No; as I said, w_1^CK doesn't really exist in the defineability sense.
[/quote]
To me, that's a *completely* meaningless claim. w_1^CK is perfectly
definable: It's defined to be the supremum of all ordinals alpha such
that there exists a recursive well-ordering of the naturals of order type
alpha.
You can certainly talk about cutting off the cumulative hierarchy at
w_1^CK, so that it doesn't exist in your model, but to say it doesn't
exist, period, is nonsensical to me. We can talk about, reason about it
consistently. It exists in the same sense that any other abstract
mathematical object exists: the square-root of 2, imaginary numbers,
etc.
[quote]You are saying that AC
is bad because it produces undefinable sets of reals. But it
doesn't *unless* there are already undefinable reals.
This is clearly wrong, and I suspect not what you meant to say.
[/quote]
No, it's clearly true, and it is exactly what I meant to say.
If X is a set of nonempty sets of definable reals, then there
will *always* be a choice function on X.
[quote]Compare - all naturals are defineable, but (you said above)
there may be undefineable collections of them (i.e. reals).
[/quote]
Sure, but that has nothing to do with choice. AC does not produce
any undefinable sets of reals unless there are *already* undefinable
reals.
[quote]AC would
not be the *source* of the undefinability. It's the innocent bystander.
I would rather put it, the executoror enforcer of an immoral law:
powerset.
[/quote]
I have trouble making any sense of concern about the power set. If A
is a set, then the collection of all subsets of A exists as a *concept*.
It exists as a proper class (since proper classes are basically just
formulas with free variables ranging over some set). To deny power set
is to say that this collection doesn't exists *AS* *A* *SET*. But what
does that even mean? It certainly makes sense to say *relative* to a
model---certain collections appear in the model and other collections
do not. But I don't see how it makes sense to talk about P(A) not existing
as a set in any absolute sense.
Unless you want to say that there is a *standard*, God-given (or Bill-given,
since you're an atheist) model for set theory, and it doesn't happen to
contain P(A). I still can't grasp what that could mean. If you have in
mind a model M that has no P(A) for some particular A, then I can certainly
imagine another model M' that is obtained by extending M to a new model
that includes P(A).
I have a hard time knowing what in the world you could mean by saying
that P(A) does not exist, for some A. Do you have some Platonic universe
of sets to appeal to, or what?
--
Daryl McCullough
Ithaca, NY |
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| Bill Taylor... |
| Posted: Thu Nov 05, 2009 5:40 pm |
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stevendaryl3... at (no spam) yahoo.com (Daryl McCullough) wrote:
[quote]stevendaryl3... at (no spam) yahoo.com (Daryl McCullough) wrote:
That isn't true. Real numbers did not have names before Cantor.
Oh heavens, of course they did!
No, they did not. *Some* reals had names.
[/quote]
Oh. It would have been clearer if you'd prefaced the above with
"some" or "all" . No matter. Go ahead with your evidence that
(pre-Cantor) there were any reals that did not have names.
By "name", OC, I mean a definition.
[quote]I'll ignore the trivial fact that all Euclid's diagrams had letters
You don't understand the difference between a *variable* and a
unique name? When someone says "Let A be a point, and let R be
[/quote]
OK OK, it was just a flippancy. I won't make any more!
[quote]No-one doubts he [Descartes] was doing Euclidean geometry, and
extending it, and all his points were real number pairs, see just above.
Not every real has a name, then not every pair of reals has a name.
[/quote]
So, we agree as regard geometry - that it comes down to naming reals.
So we cannot proceed, until we have your answer to the above request.
[quote]That is completely ridiculous. Of course, if a real comes up
in an application, then you *give* it a name. That's where "e"
came from. That's where "pi" came from.
[/quote]
OC. So where does any real come up that is NOT nameable,
in this sense? This is still what I'm wondering about.
-- Wondering William |
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| Daryl McCullough... |
| Posted: Fri Nov 06, 2009 6:54 am |
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Bill Taylor says...
[quote]
stevendaryl3... at (no spam) yahoo.com (Daryl McCullough) wrote:
stevendaryl3... at (no spam) yahoo.com (Daryl McCullough) wrote:
That isn't true. Real numbers did not have names before Cantor.
Oh heavens, of course they did!
No, they did not. *Some* reals had names.
Oh. It would have been clearer if you'd prefaced the above with
"some" or "all" . No matter. Go ahead with your evidence that
(pre-Cantor) there were any reals that did not have names.
By "name", OC, I mean a definition.
[/quote]
Cantor proved this fact. But it didn't *become* true when Cantor
proved it. So you are the one who is making a claim
that is provably false. Your argument is there were no reals that
people had reason to talk about before Cantor that had no definition.
That's certainly true, and almost tautological---if you have reason to
talk about a specific real, then you are motivated to give it a
name.
Prior to Cantor there was no naming *scheme* that assigned a name
to each real. Your counter-argument is that you can name all reals
by a hierarchy of schemes (perhaps indexed by computable ordinals).
Well there was no such infinite hierarchies of schemes prior to
Cantor, either. So your claims make no sense at all.
Unless you are just saying the trivial fact that prior to Cantor,
people never studied any *particular* real unless that real had
a finite description. Of course, that's tautologically true. How
could it be otherwise? If I tried to get people to be interested
in studying the properties of the real 2.126745..., it would take
an infinite length of time to tell people which real I was talking
about, and then there would be no time to study it.
The fact that humans communicate using finite noun phrases is
certainly true. Cantor didn't change that.
--
Daryl McCullough
Ithaca, NY |
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| Peter Webb... |
| Posted: Sat Nov 07, 2009 12:08 pm |
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"Bill Taylor" <w.taylor at (no spam) math.canterbury.ac.nz> wrote in message
news:accf8622-d756-425f-9ccf-b34ea8bef1c6 at (no spam) u36g2000prn.googlegroups.com...
[quote]stevendaryl3... at (no spam) yahoo.com (Daryl McCullough) wrote:
stevendaryl3... at (no spam) yahoo.com (Daryl McCullough) wrote:
That isn't true. Real numbers did not have names before Cantor.
Oh heavens, of course they did!
No, they did not. *Some* reals had names.
Oh. It would have been clearer if you'd prefaced the above with
"some" or "all" . No matter. Go ahead with your evidence that
(pre-Cantor) there were any reals that did not have names.
By "name", OC, I mean a definition.
I'll ignore the trivial fact that all Euclid's diagrams had letters
You don't understand the difference between a *variable* and a
unique name? When someone says "Let A be a point, and let R be
OK OK, it was just a flippancy. I won't make any more!
No-one doubts he [Descartes] was doing Euclidean geometry, and
extending it, and all his points were real number pairs, see just above.
Not every real has a name, then not every pair of reals has a name.
So, we agree as regard geometry - that it comes down to naming reals.
[/quote]
Point of order. All Reals constructible with straightedge and compass are
name-able; they are named by their construction. Indeed they are a tiny
subset of the algebraic numbers. |
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| Daryl McCullough... |
| Posted: Sat Nov 07, 2009 12:36 pm |
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Peter Webb says...
[quote]Point of order. All Reals constructible with straightedge and compass are
name-able; they are named by their construction. Indeed they are a tiny
subset of the algebraic numbers.
[/quote]
Yes, that's right. But a lot of Euclidean constructed started with
an *arbitrary* initial state: Given a line and a point not on that
line, it's possible to draw a second line through that point parallel
to the first line (or whatever). Was there an assumption that such
starting points were constructible? Was there an assumption that
for an arbitrary pair of line segments, the ratio between their
lengths was a constructible real? Maybe the ancient Greek
geometers did believe that at some point.
--
Daryl McCullough
Ithaca, NY |
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| Bill Taylor... |
| Posted: Mon Nov 09, 2009 5:04 pm |
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Isn't it remarkable how threads can wander!?
This thread started out with a (perhaps disingenuous) enquiry as to
the alleged non-obviousness of well-ordering. After a brief detour
through the anecdote about AC WOT and Zorn, (walking into a bar?),
it then transferred its attention to AD, the axiom of determinacy.
After lingering there for some time, it has now devolved
into a debate about Powerset, let's call it PS, and what
it could possibly mean for it to be false.
Now I am in a quandary. I might defend its possibility
of being false, but this would necessitate going OUTSIDE
Cantorian set theory. And as almost everyone else here feels
inescapably INSIDE Cantorian set theory, pre-Z, it might be called,
I cannot defend it! All the objections and incredulity expressed
about this are from confirmed Cantorians, which is most mathies.
Including myself (!), at least for operational purposes.
What *is* pre-Z ? It is informal set theory, Cantor style,
which means, effectively, the basic set theory of Boole
and perhaps earlier, and perhaps up to Peano; and Cantor's
own unique original ideas. And what were these? NOT (not chiefly)
AC or well-ordering, as much as he was determined that universal
well-orderability should turn out to be true; not that, IMHO.
The essential Cantorian departure was POWER SET. PS. (IMHO).
Most of the slick uniformity of C20 math is derived from this.
And a little more is derived from AC.
It was Zermelo who turned pre-Z into Z in 1908(?), or rather
into ZC, as Zermelo regarded AC as "obvious", and sufficient
to prove wellorder, as it obviously is. Truly, as one book title
has it, we ought to speak of "Zermelo's Axiom of Choice".
Z is formalized Z, that is, axiomatized pre-Z.
ZC is Zermelo's set theory, not Cantor's. Cantor's set theory
is basic (Boolean) set theory plus PS. ZFC later introduced F,
a further mild addition, in 1920 or so. But it is intriguing
that mathematicians and mathematical logicians have always
slightly reversed history and regarded ZF (which never arose
historically!) as an important stepping-stone on the way to ZFC.
And such was the utter slick and encompassing nature of ZF(C)
that mathematicians from 1920 onwards almost uniformly adopted
the new ideas wholesale. Occasionally with doubts expressed
about AC, but NEVER with any doubts about PS. PS was "obviously"
true, and needed to ensure lots of other stuff like the reals, R,
general Cartesian products etc, and students were introduced
to it so smoothly (usually without even being aware of its
axiomatic nature), that no-one ever noticed what a huge dead
fish they were swallowing! Only in recent decades, has it
become more suspicious again. And even then only to
math loggies, rather than mathies proper.
As Halmos(?) said, "The Axiom of Choice is unique in its ability
to trouble the conscience of the working mathematician.
This conscience-troubling, as I noted, is only apparently due
to AC, but really, the blame lies with PS, and such is the intimacy
with which mathies enjoy with Z(F(C)) that they never even notice.
And indeed, react with utter incredulity to the suggestion
that it be in some way false, or at least dangerous and dark.
Mathematics, now, runs on ZFC the way Freecell runs on Windows.
So what would math be with a non-Cantorian set theory as its
platform or operating system? Most people might say, hugely
different, but I think they would be wrong. Obviously some
stuff that is intimately entwined with set theory would have
to change, but not greatly. We could still have Cartesian
Products, ordered multiplets, sequences, continuous functions,
and the whole panoply of C19 math - differential equations,
optimization, analysis, tensors, etc etc virtually unchanged,
without PS. No-one has ever tried this (outside constructivism),
mostly because, "Why bother?" - a comment often levelled at
constructivists as well, intriguingly. But it could be done,
without too much trouble, as well.
And finally, what about the reals, R ? They would certainly
look a lot different, or rather, the way we standardly handle
them would be somewhat different. Essentially, we would have
to abandon our habits of theft over honest toil, a la Russell,
and take a lot more constructive, or rather *definitional*
approach to them. They would have to be built up slowly
from basic unexceptionable ones, through increasingly "artificial"
(but necessary for the handling of l.u.b etc) levels of posterior
definability, exactly mirroring the building up of recursive
ordinals to any level below but NOT including w_1^CK
(which need not, does not, exist outside Cantorian Z).
Such a study has not yet been done, though small steps
along the way have been made here and there.
If this system were ever worked out in full - WHEN this system is
worked out in full - it will be seen as a much more ontologically
reliable system than Z, though admittedly far more cumbersome,
and thus always likely to be ignored by working mathematicans.
And it is on this (as yet incomplete) basis, that I say with
arrogant confidence, that the more egregious absurdities
following from AC are simply *false*, though this
would take a lot of proving, or even explicating.
And even more radically, I declare that Powersets of
infinite sets, at least in their Cantorian conception,
simply *will not exist* - and will not be needed.
Like everyone else, I adore ZF because of its uniformizing
slickness, but must reluctantly concede that it sometimes produces
falsities/absurdities; but these are almost entirely due to
the tacking on of AC - that uniquely conscience-troubling axiom!
I have endeavoured to indicate why it might be so unwittingly
troublesome.
-- Basic Bill |
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| Herman Jurjus... |
| Posted: Tue Nov 10, 2009 2:22 am |
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Bill Taylor wrote:
[quote]... utter incredulity to the suggestion
that it be in some way false, or at least dangerous and dark.
[/quote]
Just in case this wasn't clear before: I agree with your assessment of
the power set axiom.
[quote]Mathematics, now, runs on ZFC the way Freecell runs on Windows.
So what would math be with a non-Cantorian set theory as its
platform or operating system? Most people might say, hugely
different, but I think they would be wrong. Obviously some
stuff that is intimately entwined with set theory would have
to change, but not greatly. We could still have Cartesian
Products, ordered multiplets, sequences, continuous functions,
and the whole panoply of C19 math - differential equations,
optimization, analysis, tensors, etc etc virtually unchanged,
without PS. No-one has ever tried this (outside constructivism),
mostly because, "Why bother?" - a comment often levelled at
constructivists as well, intriguingly. But it could be done,
without too much trouble, as well.
[/quote]
What do you think about the separation axiom schema?
Should it be restricted to formulas in which all the quantifiers are
bounded (i.e. 'for all x in V' instead of 'for all x' sec)?
If you allow the current 'full' version, you'd still allow (a.o.)
quantifying over all subsets of N. Does that make sense, when throwing
out PS?
--
Cheers,
Herman Jurjus |
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| Aatu Koskensilta... |
| Posted: Tue Nov 10, 2009 4:42 am |
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Bill Taylor <w.taylor at (no spam) math.canterbury.ac.nz> writes:
[quote]This thread started out with a (perhaps disingenuous) enquiry as to
the alleged non-obviousness of well-ordering.
[/quote]
Bill, I will get back to you on your mumblings about the powerset
axiom, definability, what not, later. Here I'd just like to note this
thread started out with a (perfectly ingenuous though possibly not
very serious-minded) enquiry as to the alleged /counter-intuitiveness/
of the well-ordering theorem, this enquiry prompted by my (possibly
erroneous and arbitrary) notion that people's statements about
counter-intuitiveness, evidence, etc. often are almost totally
arbitrary -- that is, it is often impossible to get any explanation
whatever of e.g. what intuitions are contradicted by this or that.(I
don't recall if it's been mentioned already, but Sol Feferman's paper
_Mathematical Intuition vs. Mathematical Monsters_ is an enjoyable and
relevant read in this context.)
--
Aatu Koskensilta (aatu.koskensilta at (no spam) uta.fi)
"Wovon man nicht sprechen kann, darüber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus |
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| Bill Taylor... |
| Posted: Wed Nov 11, 2009 6:19 pm |
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Aatu Koskensilta <aatu.koskensi... at (no spam) uta.fi> wrote:
[quote]Bill, I will get back to you onyourmumblingsabout the powerset
axiom, definability, what not, later.
[/quote]
Excellent. I look forward to some helpful mutterings from you.
[quote]Here I'd just like to note this
thread started out with a (perfectly ingenuous though possibly not
very serious-minded) enquiry
[/quote]
Possibly not. And it goes along with your notable article here
about what kinds of joy you (and I) get from Usenet debates.
[quote]as to the alleged /counter-intuitiveness/
of the well-ordering theorem, this enquiry prompted by my (possibly
erroneous and arbitrary) notion that people's statements about
[/quote]
I was surprised by the initial inquiry, coming from *you*, as it did.
It's more the sort of thing that John Jones would ask. Surprising,
especially in view of the (I think obvious) fact that large numbers
of people DO find it unintuitive, as the well-known and
here-repeated anecdote attest.
[quote]counter-intuitiveness, evidence, etc. often are almost totally
arbitrary -- that is, it is often impossible to get any explanation
whatever of e.g. what intuitions are contradicted by this or that.
[/quote]
Again, I am surprised, as I would have thought the elucidations
were common enough, even if not up to rigorous logical standards.
It seemed to me, almost, as if your original enquiry were provocative.
But then, so what if it was!
[quote]I don't recall if it's been mentioned already, but Sol Feferman's paper
_Mathematical Intuition vs. Mathematical Monsters_ is an enjoyable and
relevant read in this context.)
[/quote]
Thanks for the ref! It sounds fun, indeed! Will read asap.
-- Breathless Bill |
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| Bill Taylor... |
| Posted: Wed Nov 11, 2009 7:08 pm |
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Herman Jurjus <hjm... at (no spam) hetnet.nl> wrote:
[quote]Just in case this wasn't clear before:
I agree with your assessment of the power set axiom.
[/quote]
Excellent! We are now TWO voices alone in the wilderness. :)
[quote]What do you think about the separation axiom schema?
Should it be restricted to formulas in which all the quantifiers
are bounded (i.e. 'for all x in V' instead of 'for all x' sec)?
[/quote]
I'm not fully sure I follow the question,
(e.g. what was that "sec" in the last line?)
However, noting that quantifying over sets of naturals
is (encyptically) quantifying over reals, there is still
a considerable difference, seemingly, between quantifying
over reals and quantifying over SETS of reals.
Quantifying over reals doesn't seem to hold many terrors,
(though I could be wrong); but quantifying over sets of them
is a whole nother matter. It is in this latter that LUB makes
its appearance. Typically, the LUB of a set of reals occurs
at a greater definability level than any of its boundees,
and (presumably) of the set of them.
In a similar way, Cantor's uncountability theorem has its
content subtly altered, though not if its original statement
is kept in its proper form - that for any list of reals there is
a real not on the list. This now is seen as a rather simple
corollary of the fact that reals occur in distinct levels,
which are indefinitely extensible.
It seems that any statement involving sets of reals, must
inevitably be interpreted as thinking of *some* set of reals
at *some* level. So no, not all reals. And the consequent
of the statement must allow the possibility of going beyond
that original level. But nevertheless the theory itself
need not, cannot, make any mention of the levels, much as
ordinary ZF never mentions the class V, its intended model.
[quote]Does that make sense, when throwing out PS?
[/quote]
I hope it makes *some* sense.
Any further comments would be most welcome.
-- Baffled Bill |
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| Rupert... |
| Posted: Thu Nov 12, 2009 1:01 pm |
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Guest
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On Nov 13, 5:23 am, stevendaryl3... at (no spam) yahoo.com (Daryl McCullough)
wrote:
[quote]Bill Taylor says...
Herman Jurjus <hjm... at (no spam) hetnet.nl> wrote:
Just in case this wasn't clear before:
I agree with your assessment of the power set axiom.
Excellent! We are now TWO voices alone in the wilderness. :)
People skeptical of the power set usually balk at the very first
application that gets you something really new: P(omega), or the
reals. So abandoning the power set axiom means, in this case,
considering the collection of all reals to be a proper class,
rather than a set.
But what does that really mean? What does it mean to say that
the collection of all reals (which exists in the sense that it
is a definable class) does not exist *as* *a* *set*? It seems
to me that we only accept the set/class distinction because
it is *forced* on us by consistency. In the case of P(omega),
it's *not* forced on us by consistency considerations, so
why consider it a proper class, rather than a class? I can't
understand the motivation for that.
--
Daryl McCullough
Ithaca, NY
[/quote]
Unrestricted comprehension gets you into trouble, right? Most set
theorist types want as much comprehension as you can get without
inconsistency. But there's also a case for saying we should be
conservative. We know unrestricted comprehension gets you into
trouble, so maybe we should have as little comprehension as is
indispensible for scientifically applicable mathematics. That is the
point of view of the predicativist such as Feferman. |
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