2nd order replacement and non-standard models of ZF?
MoeBlee
more about it:
If one takes a second-order axiom of replacement, one can prove that
ZF has non-standard interpretations, as mentioned in Stewart Shapiro's
book on second order logic.
Can anyone supply an outline on this?
Thanks,
MoeBlee
george
> Can anyone supply an outline on this?
I can't, but I think you need to re-phrase the question.
You don't NEED 2nd-order anything to prove the existence of
non-standard models of ZF.
> If one takes a second-order axiom of replacement, one can prove that
> ZF has non-standard interpretations, as mentioned in Stewart Shapiro's
> book on second order logic.
The Lowenheim-Skolem theorem, for example, proves that there is
a countable model of EVERY recursively axiomatizable FIRST-order
theory, including first-order ZFC. Obviously any countable model
of the powerset axiom and the axiom of infinity is non-standard.
george
> ZF has non-standard interpretations, as mentioned in Stewart Shapiro's
> book on second order logic.
I think this is an oversimplification of the book.
If you are using the standard semantics for 2nd-order logic
then that would give you FEWER models than you have in 1st-
order, NOT MORE. When you use a 2nd-order axiom of induction
(in a simpler axiom-set, PA, which is basically ZF minus Infinity),
you LOSE all the non-standard models.
MoeBlee
> The Lowenheim-Skolem theorem, for example, proves that there is
> a countable model of EVERY recursively axiomatizable FIRST-order
> theory, including first-order ZFC. Obviously any countable model
> of the powerset axiom and the axiom of infinity is non-standard.
I don't very much doubt that the orginal question might not have been
specific enough; I was only relaying what someone else asked me about.
I think he knows about Lowenheim-Skolem, so I think he must have
something more in mind.
I shouldn't have offered to present the question for him. It seems
like it will be too cumbersome trying to communicate by relay. I'll
just tell him about this thread and let him speak for himself if he
wants to.
MoeBlee
hurb...@live.fr
hurb...@live.fr
taught me that result was telling me baloney and seeling me snake oil.
If you take second order replacement, you rule out non-standard models
of ZF and as a consequence no models will have € not wellfounded and
infinte descending chains. For some reason I became convinced it was
in Shapiro's book. What I've been told is simply false!
check this out (George Hellman book on google):
http://books.google.fr/books?id=WbzLMruh8vEC&pg=P...
Sorry for the inconvenience. No wonder I couldn't prove or understand
that result! It was simply false...
You can now relax George...
hurb...@live.fr
sorry forgot to mention the information in on pages 66-68
Aatu Koskensilta
> Sorry for the inconvenience. No wonder I couldn't prove or
> understand that result! It was simply false...
Right. All models of second-order set theory are of the form V_kappa
with kappa inaccessible. (Or, without replacement, of the form
V_lambda with lambda a limit ordinal greater than omega.)
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon man nicht sprechen kann, darüber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
hurb...@live.fr
> hurbur...@live.fr writes:
> > Sorry for the inconvenience. No wonder I couldn't prove or
> > understand that result! It was simply false...
>
> Right. All models of second-order set theory are of the form V_kappa
> with kappa inaccessible. (Or, without replacement, of the form
> V_lambda with lambda a limit ordinal greater than omega.)
>
> --
>
> "Wovon man nicht sprechen kann, darüber muss man schweigen"
> - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Thank you Aatu Koskensilta!
Aatu Koskensilta
> Thank you Aatu Koskensilta!
It occurs to me that there is a way of interpreting what you were told
so it isn't trivially false: it is provable in second-order set theory
that ZFC has a non-standard model, provided we have full impredicative
comprehension, that is, if we take "second-order set theory" to refer
to Morse-Kelley set theory.
--
Aatu Koskensilta (aatu.kos...@uta.fi)
gre...@email.unc.edu
> it is provable in second-order set theory
> that ZFC has a non-standard model, provided we have full impredicative
> comprehension, that is, if we take "second-order set theory" to refer
> to Morse-Kelley set theory.
But obviously, you don't need AS MUCH AS all that.
All you NEED is the Lowenheim-Skolem theorem.
Obviously any countable model of the axiom of infinity
and the powerset axiom is non-standard.
Chris Menzel
The L-S theorem alone entails only that ZFC has a countable model IF it
has a model at all, on which point the theorem is silent. You need the
firepower Aatu notes to prove that ZFC in fact has a model and to define
what it means for such a model to be non-standard.
Aatu Koskensilta
> The L-S theorem alone entails only that ZFC has a countable model IF
> it has a model at all, on which point the theorem is silent. You
> need the firepower Aatu notes to prove that ZFC in fact has a model
> and to define what it means for such a model to be non-standard.
Why do you think impredicative comprehension is needed to define what
it means for a model of ZFC to be non-standard? (Full impredicative
comprehension is of course an overkill for the existence of a model of
ZFC; Delta-1-1 comprehension will do.)
Frederick Williams
> Right. All models of second-order set theory ...
I hope that this isn't a silly question but what _is_ second-order set
theory?
--
Science is a differential equation.
Religion is a boundary condition.
--Alan Turing
Aatu Koskensilta
> I hope that this isn't a silly question but what _is_ second-order set
> theory?
ZFC with replacement and separation in their second-order form instead
of as schemata.
Chris Menzel
<aatu.kos...@uta.fi> said:
> Chris Menzel <cme...@remove-this.tamu.edu> writes:
>
>> The L-S theorem alone entails only that ZFC has a countable model IF
>> it has a model at all, on which point the theorem is silent. You
>> need the firepower Aatu notes to prove that ZFC in fact has a model
>> and to define what it means for such a model to be non-standard.
>
> Why do you think impredicative comprehension is needed to define what
> it means for a model of ZFC to be non-standard? (Full impredicative
> comprehension is of course an overkill for the existence of a model of
> ZFC; Delta-1-1 comprehension will do.)
Ever the stickler, Aatu! I wasn't really thinking precise details here.
The point of my response to George was simply that L-S alone wasn't
sufficient to prove the existence of nonstandard models of ZFC, contrary
to what he had asserted. I suppose "more firepower" rather than "the
firepower Aatu notes" would have been a more appropriate wording.
MoeBlee
> Frederick Williams <frederick.willia...@tesco.net> writes:
> > I hope that this isn't a silly question but what _is_ second-order set
> > theory?
>
> ZFC with replacement and separation in their second-order form instead
> of as schemata.
But is there consensus as to what the logical axioms are?
MoeBlee
Fjodor
> Frederick Williams <frederick.willia...@tesco.net> writes:
> > I hope that this isn't a silly question but what _is_ second-order set
> > theory?
>
> ZFC with replacement and separation in their second-order form instead
> of as schemata.
>
> --
>
> "Wovon man nicht sprechen kann, darüber muss man schweigen"
> - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
I am still left wondering about how it may be that Z +PI-n replacement
(separation) can be finitely axiomatized.l I am interested in this,
and do not have any colloeagues nd soin the know to talk about it with
this semester, References and/or explanations anyone? (Yes, I know,
this may be in elementary text-books. But I am a philosopher, so there
are some holes in my education and shortcomings in my acument in these
matters.)
Aatu Koskensilta
> I am still left wondering about how it may be that Z +PI-n
> replacement (separation) can be finitely axiomatized.
What, bald unsubstantiated assertions aren't enough for you? To see
that Pi-n separation can be finitely axiomatised, observe that it can
be expressed as
(*) (p)(x)(if p is a code of a Pi-n formula with one free variable
v1, then (Ez)(y)(y in z iff True-Pi-n(p[y/v1])))
where the formula True-Pi-n is given by induction as follows:
True-atomic(p) <--> True-atomic(p) <--> if p = "a in b" and a in b or
if p = "a = b" and a = b
True-Sigma-0(p) <--> True-Pi-0(p) <--> p is a Boolean combination of
atomic sentences that evaluates as true by the truth tables
True-Pi-n+1(p) <--> p = "(v)q" with q Sigma-n and for all x,
True-Sigma-n(q[x/v])
True-Sigma-n+1(p) <--> p = "(Ev)q" with q Pi-n and there is an x,
such that True-Pi-n(q[x/v])
Here p and q are codes for formulas in the language of set theory with
set parameters. That is, the language has a constant symbol a' for
each set a. Also, p[x/v] denotes the result of substituting the name
of the set x for the variable v in p. (There are obviously a proper
class of formulas in this extended language, but being a Pi-n or a
Sigma-n formula is definable in the language of set theory without
parameters.)
--
Aatu Koskensilta (aatu.kos...@uta.fi)
Aatu Koskensilta
> What, bald unsubstantiated assertions aren't enough for you? To see
> that Pi-n separation can be finitely axiomatised, observe that it can
> be expressed as
>
> (*) (p)(x)(if p is a code of a Pi-n formula with one free variable
> v1, then (Ez)(y)(y in z iff True-Pi-n(p[y/v1])))
Ahem. That should be
(*) (p)(x)(if p is a code of a Pi-n formula with one free variable
v1, then (Ez)(y)(y in z iff y in x and True-Pi-n(p[y/v1])))
Aatu Koskensilta
> Ever the stickler, Aatu!
Always!
> I wasn't really thinking precise details here. The point of my
> response to George was simply that L-S alone wasn't sufficient to
> prove the existence of nonstandard models of ZFC, contrary to what
> he had asserted.
Sure. I was just wondering whether you had something interesting in
mind with the on the face of it obviously false claim, that you need
the firepower I note to define what it means for a model of ZFC to be
non-standard -- that is, given that in the usual sense the definition
can be stated perfectly well in the first-order language of set
theory, I'm not just quibbling over the exact amount of second-order
comprehension required or some other piece of tedious technical
arcana.
Chris Menzel
<aatu.kos...@uta.fi> said:
> Chris Menzel <cme...@remove-this.tamu.edu> writes:
>
>> Ever the stickler, Aatu!
>
> Always!
>
>> I wasn't really thinking precise details here. The point of my
>> response to George was simply that L-S alone wasn't sufficient to
>> prove the existence of nonstandard models of ZFC, contrary to what he
>> had asserted.
>
> Sure. I was just wondering whether you had something interesting in
> mind
Nah, my technical knowledge in mathematical logic is pretty much
confined to just good solid basics. If I ever appear to be alluding to
something Deep and Arcane, take a closer look. :-)
> with the on the face of it obviously false claim, that you need the
> firepower I note to define what it means for a model of ZFC to be
> non-standard -- that is, given that in the usual sense the definition
> can be stated perfectly well in the first-order language of set
> theory, I'm not just quibbling over the exact amount of second-order
> comprehension required or some other piece of tedious technical
> arcana.
Understood!
gre...@email.unc.edu
> > But obviously, you don't need AS MUCH AS all that.
> > All you NEED is the Lowenheim-Skolem theorem.
> > Obviously any countable model of the axiom of infinity and the
> > powerset axiom is non-standard.
On Mar 1, 8:31 pm, Chris Menzel <cmen...@remove-this.tamu.edu> wrote:
> The L-S theorem alone entails only that ZFC has a countable model IF it
> has a model at all, on which point the theorem is silent. You need the
> firepower Aatu notes to prove that ZFC in fact has a model and to define
> what it means for such a model to be non-standard.
Well stroked.
But this is still weird.
The LST is a META-theorem.
It is not even clear what axioms you are supposed to be proving it
FROM.
It's a FIRST-order theorem and it is provable in first-order ZFC,
AMONG OTHER
things. By default, around here, for first-order theories, first-
order
ZFC *IS THE* standard/default model-construction language.
It is what you use to prove ANY OR EVERY thing consistent, by proving
that a model of it exists. Of course, by Godel, you can't use it to
prove that a model of ZFC exists. But people USUALLY get around that
just by adding THAT prior model-existence AS A HYPOTHESIS WHILE
REMAINING in first-order ZFC.
OK, so I was wrong, about THAT part -- it is precisely as you say
(and I didn't) NOT true (as I falsely said, and you corrected) that
"All you NEED is the Lowenheim-Skolem theorem".
But MY point CONTINUES to be that you STILL don't need
a second-order set-theory. ANY OLD UNDECIDABLE SENTENCE
WILL DO, to prove the existence of a non-standard model.
If there are models both ways then one of them HAS to be non-standard.
You don't NEED to be able to "completely" define "non-standard"
in order to prove that non-standard models exist. Any two
models giving opposite truth-values to any sentence show
that at least one of them (even if you don't know which) HAS
to be non-standard.
By G2, the undecidable sentence that would probably be best to choose
is "ZFC has a model", i.e., is consistent. Which is (tragically) a
loose use
of "i.e." since Con(ZFC) under the usual Godelian encoding does NOT
appear to mean "ZFC has a model" -- the usual encoding of "ZFC has a
model" is something altogether different from the usual Godelian
encoding of unprovability of something, even though the two have to be
equivalent.
My main point here is simply that it is common practice
to use FIRST-order ZFC *and not* second- to prove the existence
of sets that could be domains -- and related sets that could be
functions
and predicates over those domains) for models of VARIOUS first-
order theories. That you can't use this technique for first-order ZFC
*itself* seems almost like a Godelian or Russellian ACCIDENT -- it
seems that the SELF-reference is what is getting in the way.
Or is it the "power" instead??
Fjodor
> Aatu Koskensilta <aatu.koskensi...@uta.fi> writes:
> > What, bald unsubstantiated assertions aren't enough for you? To see
> > that Pi-n separation can be finitely axiomatised, observe that it can
> > be expressed as
>
> > (*) (p)(x)(if p is a code of a Pi-n formula with one free variable
> > v1, then (Ez)(y)(y in z iff True-Pi-n(p[y/v1])))
>
> Ahem. That should be
>
> (*) (p)(x)(if p is a code of a Pi-n formula with one free variable
> v1, then (Ez)(y)(y in z iff y in x and True-Pi-n(p[y/v1])))
>
> --
>
> "Wovon man nicht sprechen kann, darüber muss man schweigen"
> - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
As you, I am on occasions prone to make bold, bald and unsubstantiated
assertions. Amongst the things we may agree upon, is that we are both
as fallible as many.
Your construction is fairly much as I expected. I believed I would see
a satisfaction predicate in it, but this may serve its purpose. (Your
"delta-0 condition" came as a surprise, but, again, this will work in
the context.) There are at least two concerns which make me harbour
doubt as to whether this may qualify as part of a finite xiomatization
of Z(F) with PI-n separation(replacement): (1) In order to define the
used PI-n truth-predicate it seems to me that you need something
stronger than Bounded Zermelo set theory, which is finitely
axiomatizable. So there is the concern that a set theory able to
define its own PI-n truth predicate will not itself be a finitely
axiomatizable theory. (2) If you import the truth predicate from
outside, or concern (1) may be put aside, there is still the problem
that in order to put your (*) to any use, one is required to unravel
the meaning of the Gödel codes used via the inductive definition of
the truth predicate. But this seems like being schematic in
camouflage. And indeed, the camouflage is not, it seems to me, very
effective.
Am I missing something?
Aatu Koskensilta
> But people USUALLY get around that just by adding THAT prior
> model-existence AS A HYPOTHESIS WHILE REMAINING in first-order ZFC.
Sure. What I was pointing out was that the existence of a model, and
hence by compactness the existence of a non-standard model, of ZFC is
provable in second-order set theory.
> But MY point CONTINUES to be that you STILL don't need
> a second-order set-theory. ANY OLD UNDECIDABLE SENTENCE
> WILL DO, to prove the existence of a non-standard model.
Nope. In this context "standard model" is technical jargon, and means
essentially a well-founded model. The existence of undecidable
sentences does not in itself give us the existence of non-standard
models (though the existence of an undecidable sentence of specific
logical form does). The easiest way to go from the existence of a
model for ZFC to the existence of a non-standard model is to invoke
compactness.
> Which is (tragically) a loose use of "i.e." since Con(ZFC) under the
> usual Godelian encoding does NOT appear to mean "ZFC has a model" --
> the usual encoding of "ZFC has a model" is something altogether
> different from the usual Godelian encoding of unprovability of
> something, even though the two have to be equivalent.
Why should the formalisations of two distinct but mathematically
equivalent sentences be the same? On the face the tragedy you're
lamenting is as horrible as the distinctness of the axiom of choice
and the well-ordering principle.
Aatu Koskensilta
> I believed I would see a satisfaction predicate in it, but this may
> serve its purpose.
I have a strong irrational dislike of the satisfaction predicate and
much prefer to work with a truth predicate for a language with lots of
additional constants instead.
> There are at least two concerns which make me harbour doubt as to
> whether this may qualify as part of a finite xiomatization of Z(F)
> with PI-n separation(replacement): (1) In order to define the used
> PI-n truth-predicate it seems to me that you need something stronger
> than Bounded Zermelo set theory, which is finitely axiomatizable. So
> there is the concern that a set theory able to define its own PI-n
> truth predicate will not itself be a finitely axiomatizable
> theory.
I am not sure I understand this worry. In order to make use of the
Pi-n truth predicate, we need just make sure there are codes for all
formulas in the language of set theory with set parameters. This can
certainly be done with a finite number of axioms.
> (2) If you import the truth predicate from outside, or concern (1)
> may be put aside, there is still the problem that in order to put
> your (*) to any use, one is required to unravel the meaning of the
> Gödel codes used via the inductive definition of the truth
> predicate. But this seems like being schematic in camouflage. And
> indeed, the camouflage is not, it seems to me, very effective.
Well, to prove that the separation axiom expressed using a Pi-n truth
predicate yields all instances of the schematic form we obviously need
to use "informal induction", that is, reason about the theory by using
induction. If this is not what you have in mind I altogether fail to
see what you mean by "importing the truth predicate from outside". The
truth predicate in question is, after all, expressed by a single
formula in the language of set theory.
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon man nicht sprechen kann, darüber muss man schweigen"
Aatu Koskensilta
> But is there consensus as to what the logical axioms are?
In this context it doesn't matter, since we were discussing semantical
matters. Second-order set theory, when taken to be a deductive system,
usually has full impredicative comprehension and possibly the axiom of
choice (i.e. global choice) as logical axioms.
ross.fi...@gmail.com
> MoeBlee <jazzm...@hotmail.com> writes:
> > But is there consensus as to what the logical axioms are?
>
> In this context it doesn't matter, since we were discussing semantical
> matters. Second-order set theory, when taken to be a deductive system,
> usually has full impredicative comprehension and possibly the axiom of
> choice (i.e. global choice) as logical axioms.
>
> --
>
> "Wovon man nicht sprechen kann, darüber muss man schweigen"
> - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Hi Aatu,
With impredicativity there is often also the use of the naive set
theory's expectations (model theoretic). Defining parallels then
symmetries, instead of the disc being unitized, the disc unitizes.
Definition of transform notation arrays in the natural linear and
spiral model form complete symmetry groups of all container theoretic
operations, yet surely it takes a smaller area to make the program
than the specific program. Thus in the noodle universe, mice follow
the noodles. Consider the heap, or the noodle. Also that means mice.
(Ramblings to Aatu omitted.)
Then, I am to convince people that you are one of the few people who
can explain it.
Regards,
Ross F.
MoeBlee
> With impredicativity there is often also the use of the naive set
> theory's expectations (model theoretic). Defining parallels then
> symmetries, instead of the disc being unitized, the disc unitizes.
> Definition of transform notation arrays in the natural linear and
> spiral model form complete symmetry groups of all container theoretic
> operations, yet surely it takes a smaller area to make the program
> than the specific program. Thus in the noodle universe, mice follow
> the noodles. Consider the heap, or the noodle. Also that means mice.
Self parody! Ross, I didn't know you had it in you!
MoeBlee
Chris Menzel
said:
Are you certain he does...?
Alan Smaill
gre...@email.unc.edu
> Nope.
Wrong.
> In this context "standard model" is technical jargon,
Not really.
> and means essentially a well-founded model.
No, it doesn't. Good grief. The standard models at a bare minimum
have
to at least all be elementarily equivalent to each other, even if they
are not
isomorphic.
> The existence of undecidable
> sentences does not in itself give us the existence of non-standard
> models
Of course it does.
The standard model decides every sentence in exactly one way.
Every model that decides that sentence another way therefore must
be non-standard, QED.
You can object to "the" standard model on the grounds that
standardness may be loosely-enough defined that there could
be different (non-isomorphic) standard models. But I repeat,
at a BARE MINIMUM, they have to form an elementary-equivalence class.
Somebody is no doubt getting ready to tell me that I am far from
qualified to re-direct the well-established parlance of an entire
community
of people who understand the subject far better than I do.
They'd be wrong AGAIN.
Aatu Koskensilta
> On Mar 4, 9:25 am, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:
>> Nope.
>
> Wrong.
Your bizarre imaginings and the unique way you express your
terminological confusion are certainly interesting from a certain
point of view but not really of any logical relevance. If you wish to
learn about set theory, standard models of ZFC, etc. I suggest you
consult a text on the subject instead of ranting incoherently in the
news.
> Somebody is no doubt getting ready to tell me that I am far from
> qualified to re-direct the well-established parlance of an entire
> community of people who understand the subject far better than I do.
> They'd be wrong AGAIN.
A fine piece of Greenery! There are many instances of unfortunate
terminology in set theory, but "standard model" really isn't one of
them. That is, it is not liable to lead to confusion in general,
though individuals with a foundational agenda do manage to befuddle
themselves from time to time, as you have demonstrated. (Your
confusion is of course easily explained -- you seem to think "standard
model of ZFC" is analogous to "standard model of arithmetic".)
gre...@email.unc.edu
> Your bizarre imaginings
I am not imagining anything.
> and the unique way you express your
> terminological confusion
The fact that I do not know how the community uses certain terms
does NOT mean that *I* am the one who is confused!
IF YOU HAVE SENSE ENOUGH to go back to THE BEGINNING OF THIS
thread to see WHAT IT is about, YOU WILL SEE the phrase "the
standard model of set theory". YOU DID NOT ACCUSE the person who
said THAT of imagining things or being terminologically confused!
YET YOU DO accuse ME!
My point is that you are coming from a place where "the" standard
model is simply WRONG (as OPPOSED to merely "confused")
because there is more than one "standard" model, ACCORDING TO YOU.
Yet when you suddenly need to refer to "THE" Cumulative Hierarchy,
suddenly, IT'S UNIQUE again. I repeat, *I* am *NOT* the one who is
confused.
> . If you wish to
> learn about set theory, standard models of ZFC, etc. I suggest
Please, don't bother.
> > Somebody is no doubt getting ready to tell me that I am far from
> > qualified to re-direct the well-established parlance of an entire
> > community of people who understand the subject far better than I do.
> > They'd be wrong AGAIN.
>
> A fine piece of Greenery! There are many instances of unfortunate
> terminology in set theory, but "standard model" really isn't one of
> them. That is, it is not liable to lead to confusion in general,
> though individuals with a foundational agenda do manage to befuddle
> themselves from time to time, as you have demonstrated. (Your
> confusion is of course easily explained -- you seem to think "standard
> model of ZFC" is analogous to "standard model of arithmetic".)
HOW did you get to be SUCH an ASSHOLE???
We have been reading each other for 20 years and
it was NOT ever thus!
I repeat, *I* am not confused!
The confusion is IN THE COMMUNITY and this terminological
confusion IS OBJECTIVELY present!
OF COURSE, IF ANYone says *THE* standard model of set
theory, then I am going to assume that the ZFC case is analogous
to the PA case insofar as there DOES IN FACT ACTUALLY EXIST
A REFERENT of *THE*!! I am going to assume that "the" is
terminologically
AND LOGICALLY *justifiable* in this locution!
AND IF YOU GOOGLE
"THE standard model of set theory", YOU WILL GET 167 hits,
of which only about 67 will acknowledge that this is problematic
BECAUSE THERE IS NO "the" there, and the other 100 will
blithely continue on as though this were legitimate!
Worse, if you just google "Standard model of set theory" WITHOUT
the "the", YOU ONLY GET 200 hits! MOST of the time, if people who
have managed to get something onto the web have used this phrase,
THEY WERE USING the terminology AS THOUGH "the" model were
unique!
http://www.google.com/search?q=%22standard+model+of+set+theory%22&ie=utf-8&oe=utf-8&aq=t&rls=org.mozilla:en-US:official&client=firefox-a
As you have been at pains to explain, IT ISN'T, THEREFORE YOUR
QUARREL IS WITH THE BRUTE WEIGHT OF ESTABLISHED USAGE AND NOT
with me! I by ACCIDENT wound up CONFORMING to that while seeking to
attack it!
I realize that google isn't the absolute authority on what's typical
and that
what's typical in the myriad journal articles that are NOT present on
the
web would probably (necessarily I would hope!) tip this scale (of
weight of
usage) in the opposite direction.
But the point is, YOU DON'T GET to ABUSE ME AS
terminologically "confused" or "bizarre" WHEN THE WHOLE GOOGLE
DATABASE OF USES OF THIS TERM
swings MY WAY AND NOT YOURS! At least not unless you (far more
embarrassingly to you than
to anyone else) wind up AGREEING WITH MY PERSPECTIVE that the typical
usage IS OFTEN
not only confusing, BUT CONFUSED, ITself!
MoeBlee
> But the point is, YOU DON'T GET to ABUSE ME AS
> terminologically "confused" or "bizarre" WHEN THE WHOLE GOOGLE
> DATABASE OF USES OF THIS TERM
> swings MY WAY AND NOT YOURS! At least not unless you (far more
> embarrassingly to you than
> to anyone else) wind up AGREEING WITH MY PERSPECTIVE that the typical
> usage IS OFTEN
> not only confusing, BUT CONFUSED, ITself!
Your ending would be effective were it not "ITself!" but rather
"itSELF!".
MoeBlee
gre...@email.unc.edu
> terminology in set theory, but "standard model" really isn't one of
> them.
You are just factually refutable on this. This isn't even a matter of
anything as subjective as "unfortuante".
> you seem to think "standard
> model of ZFC" is analogous to "standard model of arithmetic".)
The point is, YOU DON'T think that.
So the one of us who is confused is the one whose parlance
about this most DIFFERS from what's TYPICALLY SAID in
the community. In PA, the standard model is (up to isomorphism)
unique -- THERE IS a "the" standard model.
If you google "the standard model" with "model of set theory", you
will get about 7000 hits. If you google "a standard model" with
"model of set theory", you will only get about 2000
(I had smaller numbers for more exact phrases in the previous
message; I am trying something looser here in the hope of getting
a bigger and more generally relevant sample).
My point is, I AM NOT OUTNUMBERED in referring to, and expecting,
THE standard model to be ONE standard model!
That is obviously unreasonable and unfortunate (yet I was doing it
anyway) purely from the standpoint of the ordinal HEIGHT of the
cumulative hierarchy (every limit ordinal is a candidate height,
so surely the most one could ask for is one model per height),
but even there, you could, pace the Burali-Forti paradox, just
insist that THE model was the one encompassing ALL those ordinals.
Nobody really needs to do that, but that is not the point.
The point is simply that
the fact that
"standard" isn't a strong enough critierion in this context
to get us down to one model, IS GENERALLY OVERLOOKED
IN JARGON ABOUT this topic. *THE* standard model of set theory
IS NOT my bizarre imagining or MY terminological confusion:
LOTS OF OTHER people are talking THAT way. MORE, in fact,
that are (perhaps more reasonably or more correctly) talking about
*A* standard model (one of many) of set theory.
Alan Smaill
> On Mar 5, 6:44 pm, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:
> >There are many instances of unfortunate
> > terminology in set theory, but "standard model" really isn't one of
> > them.
>
> You are just factually refutable on this. This isn't even a matter of
> anything as subjective as "unfortuante".
>
> > you seem to think "standard
> > model of ZFC" is analogous to "standard model of arithmetic".)
>
>
> The point is, YOU DON'T think that.
> So the one of us who is confused is the one whose parlance
> about this most DIFFERS from what's TYPICALLY SAID in
> the community. In PA, the standard model is (up to isomorphism)
> unique -- THERE IS a "the" standard model.
>
> If you google "the standard model" with "model of set theory", you
> will get about 7000 hits. If you google "a standard model" with
> "model of set theory", you will only get about 2000
> (I had smaller numbers for more exact phrases in the previous
> message; I am trying something looser here in the hope of getting
> a bigger and more generally relevant sample).
>
> My point is, I AM NOT OUTNUMBERED in referring to, and expecting,
> THE standard model to be ONE standard model!
I did a small experiment, bu googling on "standard model", and "set theory";
the highlighted phrases involving "standard model" in the first ten results
were:
* The assumption that there exists a standard model of ZFC
* In set theory, a minimal model is a minimal standard model of ZFC
* In fact, if there is a standard model, then there is a smallest
standard model called the minimal modelMinimal model (set theory) .
* Adjoin, to a countable standard model $M$ of Zermelo-Fraenkel set theory
(ZF), a countable set $A$ of independent Cohen generic reals.
* i.e. the class of constructible sets of a given standard model
* We have proved that K2(W) proves the existence of a standard model of ZF
* Some basic definition questions of set theory Set Theory, Logic, ...
String theory my definition, alien609, Beyond the Standard Model,
[the standard model here is from physics, not set theory)
* The latter means that all formulas that are true in the standard
model are theorems.
* Every world in our possible worlds semantics is the standard model of
arithmetic
* However in the case of the relation between the standard model, var epsilon
[ the standard model here is to do with the "BeltramiPoincare
projection of the compactified Klein modular curve ]
This is a small sample, of course, but it says that out of the 10 mentions,
2 are not about standard models of set theory at all,
1 talks about about *the* standard model of set theory, and 7 are about a
notion of standard model where there is no unique standard model of set theory.
--
Alan Smaill
Aatu Koskensilta
> IF YOU HAVE SENSE ENOUGH to go back to THE BEGINNING OF THIS thread
> to see WHAT IT is about, YOU WILL SEE the phrase "the standard model
> of set theory".
No I won't.
> I repeat, *I* am *NOT* the one who is confused.
If you weren't confused about what a standard model of set theory is
your remarks in this thread are very puzzling.
> OF COURSE, IF ANYone says *THE* standard model of set theory, then I
> am going to assume that the ZFC case is analogous to the PA case
> insofar as there DOES IN FACT ACTUALLY EXIST A REFERENT of *THE*!!
> I am going to assume that "the" is terminologically AND LOGICALLY
> *justifiable* in this locution!
Well, who in this context, apart from you, said anything about "the
standard model of set theory"? Since people use confused terminology
and language all the time, it is a good idea to pause for a moment and
reflect on what they're saying before jumping to absurd conclusions.
Aatu Koskensilta
> On Mar 5, 6:44 pm, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:
>
>> you seem to think "standard model of ZFC" is analogous to "standard
>> model of arithmetic".)
>
> The point is, YOU DON'T think that. So the one of us who is
> confused is the one whose parlance about this most DIFFERS from
> what's TYPICALLY SAID in the community. In PA, the standard model
> is (up to isomorphism) unique -- THERE IS a "the" standard model.
Right, and in case of set theory there is no standard model in the
analogous sense. Rather, as already explained, "standard model of set
theory" means "well-founded model". This is just a piece of jargon.