Godel's NBG and realism
exama...@gmail.com
A few days ago, I've had the chance to briefly review a set theory
textbook founded on Godel's NBG system. I had only heard informal
expositions of the ideas in the theory before. From what I could
gather, the theory seems to allow nothing but a realist interpretation
of mathematics. Is this a right view of the theory, or am I pushing
things too far?
Regards,
--
Eray Ozkural
rupertm...@yahoo.com
There's no reason why you couldn't use the theory if you were a
formalist. The theory is just a system for proving theorems in
mathematics and is not committed to any particular mathematical
philosophy. This is like how you can be a utilitarian in normative
ethics but a relativist in meta-ethics.
Jesse F. Hughes
How could any axiomatic theory require a realist interpretation? Why
couldn't a formalist use NBG? A structuralist? (Okay, maybe not an
intuitionist as I understand the term.)
--
Jesse F. Hughes
"I am the barbarian at the gates. I am a revolutionary, a discoverer,
a guy who didn't just try, but did, who didn't just wonder, but
accomplished." -- James S. Harris gives Hollywood its tagline
David C. Ullrich
>Hello there,
>
>A few days ago, I've had the chance to briefly review a set theory
>textbook founded on Godel's NBG system. I had only heard informal
>expositions of the ideas in the theory before. From what I could
>gather, the theory seems to allow nothing but a realist interpretation
>of mathematics.
A _formal_ _axiomatic_ theory that only allows a realist
interpretation????????
>Is this a right view of the theory, or am I pushing
>things too far?
No, you're not "pushing things too far" - there's nothing
in NBG that even tends to hint at pointing in that direction,
that you might be pushing. What you're doing is inventing
an interpretation of something you read that has no
justification whatever.
>Regards,
************************
David C. Ullrich
exama...@gmail.com
Thank you for the explanation.
Regards,
--
Eray
Nath Rao
realism?
[It might seem extravagant to posit a separate constant for each set. I
recall a paper by P. Maddy on a theory of classes that did just that.]
exama...@gmail.com
Others say it does not... Your mileage may vary.
Cheers,
--
Eray
David C. Ullrich
wrote:
>If a theory requires one constant for each 'set', does that presuppose
>realism?
If it's requiring one constant for each "real" set then yes.
(If it's assuming that there's a big model of set theory where
everything's happening and uses one constant for each element
of that model then no.)
>[It might seem extravagant to posit a separate constant for each set. I
>recall a paper by P. Maddy on a theory of classes that did just that.]
Are you sure it was one constant for each "real" set, as opposed to
each element of some model of set theory?
************************
David C. Ullrich
David C. Ullrich
>
>Nath Rao wrote:
>> If a theory requires one constant for each 'set', does that
>presuppose
>> realism?
>>
>> [It might seem extravagant to posit a separate constant for each set.
>I
>> recall a paper by P. Maddy on a theory of classes that did just
>that.]
>
>Others say it does not...
I haven't seen anyone say this is false. Are you claiming that
NBG uses a constant for each set? I don't believe that...
The first definition of NBG I find on Google,
http://planetmath.org/encyclopedia/VonNeumannBernausGodelSetTheory.html
doesn't appear to me to say anything about there being a constant
for each set. Maybe it does and I missed it.
>Your mileage may vary.
>
>Cheers,
************************
David C. Ullrich
Chris Menzel
> If a theory requires one constant for each 'set',
How, exactly, would a theory do that? A theory would explicitly have to
axiomatize its language (to quantify over its own constants) and a
denotation relation (to be able to say that every set is denoted by a
constant), which is atypical to say the least.
> does that presuppose realism?
I don't see that it would have any more implications for realism than
does set theory as it stands right now.
> [It might seem extravagant to posit a separate constant for each set. I
> recall a paper by P. Maddy on a theory of classes that did just that.]
Are you referring to her paper "Proper Classes"? I don't recall
any such thing in that theory (though it's admittedly been a long time
since I looked at it).
Chris Menzel
exama...@gmail.com
Aatu Koskensilta
If Nath is speaking about Maddy's article Proper classes in Journal of
Symbolic Logic in 1983, then Maddy indeed consideres a a *language* with
a constant for every set. There is nothing particularly problematic
about such languages, as they are in most cases definable (without
parameters) in the language of set theory, as is the case with Maddy's
construction.
I have outlined this construction in a news article here:
<bm9odp$2pq$1...@phys-news1.kolumbus.fi>.
--
Aatu Koskensilta (aatu.kos...@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
exama...@gmail.com
Jesse F. Hughes wrote:
> exama...@gmail.com writes:
>
> > Hello there,
> >
> > A few days ago, I've had the chance to briefly review a set theory
> > textbook founded on Godel's NBG system. I had only heard informal
> > expositions of the ideas in the theory before. From what I could
> > gather, the theory seems to allow nothing but a realist
interpretation
> > of mathematics. Is this a right view of the theory, or am I pushing
> > things too far?
>
> How could any axiomatic theory require a realist interpretation? Why
> couldn't a formalist use NBG? A structuralist? (Okay, maybe not an
> intuitionist as I understand the term.)
I wondered your opinion, and thanks for telling it.
--
Eray
Nath Rao
No. 'Proper Classes' (according to MR) is too old for what I remember
(not being a logician, I did not make notes or a copy). It is probably
'A theory of sets and classes' in "Between logic and Intuition".
To respond to other replies in this thread: I was going off on a tangent
and wasn't talking about vNBG itself.
Nath Rao
David C. Ullrich
<aatu.kos...@xortec.fi> wrote:
>David C. Ullrich wrote:
>
>> On Mon, 07 Feb 2005 13:22:28 -0500, Nath Rao <RnNaDt...@yahoo.com>
>> wrote:
>>
>>
>>>[It might seem extravagant to posit a separate constant for each set. I
>>>recall a paper by P. Maddy on a theory of classes that did just that.]
>>
>>
>> Are you sure it was one constant for each "real" set, as opposed to
>> each element of some model of set theory?
>
>If Nath is speaking about Maddy's article Proper classes in Journal of
>Symbolic Logic in 1983, then Maddy indeed consideres a a *language* with
>a constant for every set. There is nothing particularly problematic
>about such languages, as they are in most cases definable (without
>parameters) in the language of set theory, as is the case with Maddy's
>construction.
I didn't mean to suggest there was anything problematic about such
a thing. Whether or not this has anything to do with "presupposing
realism" does seem to me to depend on whether they're talking
about a constant for each "real" set or for each set in some
(explicit or implicit) model of set theory.
Now, Chris Menzel did see something problematic here. I was
puzzled by that. My conjecture is that he was taking the original
"If a theory requires one constant for each 'set', does that
presuppose realism?"
more literally than I did. A language that contains a constant
for each set, no problem - a language that contains a constant
for each set is also exactly what I assumed we were talking
about. Otoh a _theory_ that _requires_ a constant for each
set is a little different (how would a theory "require" that?)
(Ah, looking back I see you emphasized the word "*language*",
presumably for the sake of pointing out this distinction.)
>I have outlined this construction in a news article here:
><bm9odp$2pq$1...@phys-news1.kolumbus.fi>.
************************
David C. Ullrich
Chris Menzel
<ull...@math.okstate.edu> said:
> On Thu, 10 Feb 2005 19:07:30 +0200, Aatu Koskensilta wrote:
>>David C. Ullrich wrote:
Yes, this was my worry. Obviously, *given* some (possibly class-size)
model, we can, in the metalanguage, tailor a (possibly class-size)
language and its semantics, relative to that model, so that every
element of the model is denoted by some constant. (IIRC, this is what
happens in the Maddy construction, which is vaguely emerging from the
mists of my memory -- I should be ashamed; I was her student when she
wrote that paper.) My concern was with the idea of a theory *itself*
"requiring" that every set be denoted by some constant. I don't know
how (NOT to be read: there is no way) a theory could do that without
quantifying over its own language and having axioms like, well, "Every
set is denoted by some constant."
Then again maybe I'm just not getting it.
Chris Menzel
David C. Ullrich
<cme...@remove-this.tamu.edu> wrote:
My conjecture is that "If a theory requires one constant for each
'set'" really wasn't intended that way, was supposed to refer
just to a language that happens to have such a colllection of
constants.
>Chris Menzel
************************
David C. Ullrich