Re: Minimal set theory for model theory?

On Apr 26, 11:21 am, LauLuna <laureanol...@xxxxxxxx> wrote:
Ordinarily we understand that the domains
of our models are sets. But
if you introduce proper classes into the picture,
well, then we surely want to interpret set theory
as being about all classes.

No, of course you do NOT want to do that.
Once you introduce proper classes, the set/class
dichotomy is VERY hard and stark, and you STILL
want to be about all SETS. Being about "all" classes
is just sort of fundamentally impossible, INSIDE the
"same" class theory. Bothering with proper classes is
simply ACKNOWLEDING that universal generalization
forces semantic ascent. I will belabor this point to
unreasonable detail in rehashing Rusell's Vicious
Circle Principle below

So, again we would have no model for it,

The class theory ITSELF, JUST as in the old/refuted ZFC
case, is supposed to provide the machinery for constructing
the classes-that-are-going-to-be-models. The point here
is that one can now talk about theories as having sets as
models WITHOUT having a meltdown when either 1) generalizing over all
of them, or 2) considering set
theories. In both of those cases, we now have the
option of using the proper class of all sets as a domain.

unless we admit the existence of a universal class.

In most class theories that is simply impossible, for the
simple reason that one way of distinguishing a proper class
from a set is that proper classes are not members of other
classes. Since No class contains Any proper class, it is
analytic that you are not going to have a class of classes.
BUT YOU CAN STILL have a class of all sets, AND YOU CAN
STILL do most of first-order model theory UNDER THE
RESTRICTION that models HAVE to be sets, or at least
have domains that are sets. Given that everybody was
FORMERLY doing model theory (of everything EXCEPT ZFC
and ITS "supermodels") with domains as ZFC-sets, this CAN'T
POSSIBLY be any worse a state of affairs THAN the one that
prevailed formerly. But you now have the option of creating
a fork in your model-theory between theories whose domains
can be sets those whose domains MUST be proper classes
(you can now assign ZFC to the upper tier). This is analogous
to using 2nd-order logic -- you can ascend from sets to proper
classes as opposed to from 0th-order objects to 1st-order
predicates over them -- a class of sets arguably just IS
a predicate over sets-considered-as-base-objects.


A related difficulty lies in the interpretation of
some FOL theorems like Thomson's theorems:
-Ex Ay Rxy <-> -Ryy.

Calling THIS an FOL theorem is like calling Hurricane
Camille a rainstorm, or WW2 a skirmish!
THIS IS a *2*OL theorem!
The R IS UNIVERSALLY quantified!!

Note that this is closely related to Russell's paradox.

Indeed, it is Rusell's paradox re-phrased as
a 2nd-order validity; it almost doesn't even sink
to the level of "theorem" since it doesn't even
need any premises or axioms! Without even
an axiom to put R into the signature, you can
derive this FROM NOTHING using only *1st*-order
inference rules!

Take R as the 'aboutness' relation among propositions,

But the WHOLE point is that R can be ANYthing.
It NEVER MATTERS what R is.
This "x" is ALWAYS going to come up non-existent.

so defined: p is about q iff there is a
proposition r equivalent to p and the
subject of r either denotes q or a class to which q belongs.
Then we interpret the theorem as:

(1) there is no proposition exactly about
all propositions not about
themselves and only them.

For if there were, would it be about itself or not about itself?
It is for this reason that the object-language/meta-language
distinction was invented. But it is an OLD invention by now.


And consider:

(2) all propositions not about themselves share the property that
there is no proposition exactly about them.

THIS is a MIS-use of "all". THIS is NOT the "all" of the universal
quantifier, the "all" that translates to "each and every",
singularly, individually, conjunctively. THIS is the "all"
that translates into "all COLLECTIVELY", into "the class of all",
considered as ONE thing. You CAN'T use the "individual"
quantifiers of classical logic to talk about "sharing" a property.
THOSE quantifers will ALWAYS be saying about each ONE
thing (or some definitely-finitely-numbered ordered-n-tuple
of things) that it does or doesn't have some property.

Well, (2) is equivalent to (1)

No, it isn't.

and seems to be exactly about all
propositions not about themselves.

To the contrary, it doesn't even seem to be reasonably or
rationally translatable (or even rightly understood,
frankly) by the very people proposing it.

Consequently there is an interpretation of the theorem,
namely (1),
that seems to violate the theorem itself!

That is preposterous. Strawson's "theorem" CAN'T
be violated: IT IS A LOGICAL VALIDITY. A great
many things may ultimately get violated, but NO
*2nd*-order validity IS EVER going to be one of them.
SOME OTHER error is being made; I just identified one.
If you tried to formalize (2), you would see that your natlang
gloss of it is just wrong.

I think we can only escape this by imposing
a restriction on the range
of the quantifiers in the interpreted theorem: the universe of
discourse cannot include the interpreted theorem itself.

DUH! IT HAS ALWAYS BEEN the case that you risk paradox
when you try to produce a summary-statistic over a domain
that WILL THEN contain THAT VERY product!
FOR[classic]EXAMPLE: IF you have a blackboard
with two or more positive natural
numerals written on it, then you canNOT,
on THAT board, write the sum of the numerals-written-on
THAT board (think of that as a vicious circle of diameter 1).
If you have 2 blackboards (again with two or more positive
natural numerals written on them), you canNOT arrange
for EACH of them to have its sum-of-the-numerals-written-on-it
occurring on SOME one of the 2 (on EITHER of the 2) boards:
think of that as a vicious circle of diameter 2. You can expand
this to a diameter of 3: You cannot write two or more positive
natnums on 3 boards such that for each board, the sum of the numbers
written on it occurs on one of THOSE
3 boards. You can expand this to a diameter of 4,
or, inductively, to any natural number n. If you want to ensure
that the sums of the numbers written on all n boards occur
on some boards, then ONE of the boards-on-which-a-sum-
occurs must be OUTSIDE the n boards summed over (putting
the nth&-last one "in" would complete the "circle").

This is Russell's "vicious circle principle". It is completely
general. You could certainly think of Strawson's theorem
as one way of saying it. The left or "x" argument of R ranges
over possible values of the summary-statistic and the right or
"y" argument ranges over the domain being summarized.
Classical quantifers quantify over the SAME domain at all
argument positions, so the range of the summary statistic
winds up forced to be in the right-arg domain ("Rxx") and
paradox ensues.

Equivalently you could say that Russell's Paradox IS
a paradox precisely BECAUSE the correct formal translation
of it is a logical contradiction, and the proof that that
contradiction
IS a contradiction is just to notice that it IS THE DENIAL of
Strawson's theorem, WHICH IS A *2ND*-order validity.
The first-order Russell's paradox is what you get by
instantiating R to epsilon, but the point is, predicates and
their extensions are sufficiently tightly connected that
for a predicate to be true of something and for its(class-
like)extension to contain that something as a member
are arguably merely orthographic variations on the same theme.


So, again, paradoxes can force us
to consider sophisticated issues of
set theory in order to account for the interpretation of FOL
formulas.


The vicious circle principle IS NOT a SOPHISTICATED
issue of set theory! That you cannot have a circle of xK
such that x0 < x1 < x2 < x... < xK < x0 again is a TRIVIAL issue
of linear ordering! To the extent that it is a 2nd-order
validity/contradiction, it is not even specific to set theory
at all, although, again, set theory remains SPECIAL in that
you could insist that extensions ALWAYS exist and that
EVERY instance of a predicate being true of something is
ALSO ALWAYS an instance of that something belonging
to a class. The point is simply that some of these classes
must be AT HIGHER LEVELS than others in order to avoid this
paradox. That is fundamentally what the set-class distinction
is about. That is why it is "irresponsible", HAVING created a
class of all sets, to then ask about the class of all classes.
It goes WITHOUT saying that you can't fit generalizations
about ALL of the framework INTO the framework! Remeber
the blackboards!



.

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