fom
Posts:
1,037
Registered:
12/4/12
|
|
Re: fom - 01 - preface
Posted:
Dec 9, 2012 9:36 PM
|
|
On 12/9/2012 2:05 PM, Ross A. Finlayson wrote:
> On Dec 8, 11:21 pm, fom <fomJ...@nyms.net> wrote:
>> On 12/8/2012 10:50 PM, Ross A. Finlayson wrote:
<snip>
> http://en.wikipedia.org/wiki/Forcing_(mathematics)
Thank you for the web pages. I have not had
an internet account for some time and never
thought to see what was written on wikipedia
So, this is from that webpage,
CONSISTENCY
The above can be summarized by saying the
fundamental consistency result is that given a
forcing poset P, we may assume that there exists
a generic filter G, not in the universe V, such
that V[G] is again a set theoretic universe,
modelling ZFC. Furthermore, all truths in V[G] can
be reduced to truths in V regarding the
forcing relation.
====
This is completely legitimate. Relative to the deductive
calculus, the consistency spoken about here justifies
investigating the independence of the axioms from one
another. These questions are wrapped in the completeness
and soundness of the deductive system.
Just before Jech makes precisely the same observation,
he writes:
"To conclude this section, I shall briefly
touch the question of existence of generic
sets. Of course, if M is countable, then
every PeM has an M-generic filter. On the
other hand, in general one can at least prove
something like: Given PeM, then it is consistent
that P has an M-generic filter. Even when
formulated correctly, one has to be careful
to exclude the obvious false cases (like
when M=the universal class). In fact, it
seems necessary to formulate the theorem only
for M having an absolute definition (like
M=the consructible universe.)
In other words, if V=L is true, generic class
model that take L as the ground model are
illegitimate candidates for consideration
of foundational questions.
Suppose one asks, why might it be
consistent to assume the existence of a
filter converging outside of the model?
My answer was:
Definition of proper part:
AxAy(xcy <-> (Az(ycz -> xcz) /\ Ez(xcz /\ -ycz)))
Definition of membership:
AxAy(xey <-> (Az(ycz -> xez) /\ Ez(xez /\ -ycz)))
In other words, membership is essential in that
its own occurrence is necessary to its own
definition, but, membership is not the primitive
of the language. The proper subset relation
can be characterized with a self-defining
syntax. Thus, forcing methods are possible
because the part relation can be manipulated.
This is exactly how Jech gives his simplest
description:
"... to introduce "imaginary" sets and
adjoin an imaginary set G to the universe
to obtain an extension V[G] that is a
model of ZFC."
As for those definitions, while Russell's mistake
was becoming the ontology of mathematical logic,
there had been others thinking differently. In
"Cantorian Set Theory and Limitation of Size"
Michael Hallet discusses the 1906 position taken
by Young and Young:
"After giving Cantor's 1882 definition
of a set the Youngs [1906], p.146,
comment: 'Here Cantor emphasizes the
fact that in any logical and more
especially mathematical thinking, we must
confine our ideas to some particular
field ...' They then underline (though
they do not refer explicitly to the
paradoxes) the danger associated with
arbitrary 'conceptual spheres' or
'fundamental regions' as they call
them: namely, that there is a temptation
to allow the fundamental region itself
to be a set. Certainly, this would be
logically convenient and many of the
natural 'fundamental regions' are treated
as sets in other contexts. But, say
the Youngs:
'... it should be postulated that the
fundamental region itself is not to be
regarded as a set unless there is
another fundamental region containing
the first fundamental region together
with other elements.'
This certainly characterizes the
situation for any hierarchy of set
models. But, with regard to the universe,
it means something like this:
Definition of top:
Ax(x=V() <-> Ay(-(ycx <-> y=x)))
Assumption of top:
ExAy(-(ycx <-> y=x))
where the fundamental region has proper
parts, and, only proper parts can be
sets. One can still speak of classes
as grammatical forms associated with
formulas (which is the formal way of
speaking as "a ZFCer" on Sunday when
every mathematician is a "formalist").
But, the universe is not defined in
terms of membership and is not technically
a class in terms of second-order identity
by extensions.
The real problem with forcing is that
it is somewhat dishonest. The partial
order in the explanation given in the
web page,
A forcing poset is an ordered triple
(P, ?, 1)
where "?" is a preorder on P that satisfies following
splitting condition:
For all p ? P, there are q, r ? P such that q, r ? p with no s ? P such
that s ? q, r
and 1 is a largest element, that is:
p ? 1 for all p ? P,.
Members of P are called conditions. One reads
p ? q
as
p is stronger than q.
seems innocent. But, look at how it is applied,
The simplest nontrivial forcing poset is ( Fin(?,2), ?, 0 ),
the finite partial functions from ? to 2={0,1} under
reverse inclusion
Now, suppose I assume (for definiteness) -(V=L).
Next, I take a partial order bounded below. It either
has a bottom or it is like a forest of trees. That is,
when I reverse the inclusion relation, I have a
direction that is guaranteed to converge. Moreover,
I have stipulated a further condition matching the
definition of convergent filterbases that
characterize the neighborhood base definitions
of a topology. In other words, I have taken the
Cantor nested set theorem with vanishing diameter
and pointed it to heaven.
Is it any surprise that I suddenly find something
outside of what I have assumed to be partial?
You cannot do that with
Definition of top:
Ax(x=V() <-> Ay(-(ycx <-> y=x)))
Assumption of top:
ExAy(-(ycx <-> y=x))
Just like with the coatoms one element away from the
whole set in the ordering of its power set, all
notions of convergence end at the universe.
As far as the consistency aspects of forcing, the
best explanation I have found is in Rosser. He
compares it to something that has nothing to do
with partial orders. He compares it with Cayley's
demonstration that the axiom of parallels is
independent:
"... That is, he began by assuming
the truth of postulate 5. As we
mentioned in Section B, we begin
similarly by assuming AC and GCH,
preparatory to proving their independence.
This is not self-defeating, because
we then modify somewhat the notions
of 'set,' 'well-ordering,' 'cardinal
number,' etc. just as Cayley introduced
modified 'points,' 'lines,' 'distances,'
etc. Indeed, we need the presence of
powerful principles such as AC and
GCH to be able to prove that our
unorthodox 'sets,' etc. have the
desired properties, just as Cayley needed
Postulate 5 to prove the properties of
his 'points,' 'lines,' 'distances,' etc.
"Note further that Cayley did not
modify his 'points,' 'lines,' etc. very
much. After all, he wished to invalidate
only Postulate 5. Thus his 'points' were
actual points; he just didn't use all
possible points."
The last sentence strongly suggests that
partiality is a fundamental requirement
for independence proofs in order to manipulate
an existing system of definitions to show that
the new definitions yield a system satisfying
the assumptions that are held invariant.
|
|
