As I indicated in my comment, there will be many silly
examples if we interpret your notion of $Phi$ is
preserved by an iteration $P_delta$'' to mean that
whenever all the proper initial segments of the iteration
$P_alpha$ for $alpha<delta$ have $Phi$, then $P_delta$
has $Phi$. The reason is that if $delta=beta+1$ is a
successor ordinal, then this way of stating the property
doesn't place any requirement on the stage $beta$ forcing.
For example, let $Phi$ hold of a forcing notion $P$ if and
only if $P$ is equivalent to countably closed forcing. This
is preserved by countable support limit ordinal iterations,
since if every initial segment $P_alpha$ for
$alpha<delta$ is countably closed, then so is the whole
iteration $P_delta$, provided $delta$ is a limit. But if
$Q$ is any non-countably forcing notion, consider the
iteration of length $1$, forcing with $Q$ at stage $0$.
Since $P_0$ is trivial, it is countably closed, but the
whole iteration $P_1$ is not, so it violates your property
in the ignoring-the-last-stage manner that you have stated
it.
Another silly example: let $Phi$ hold of proper forcing.
This is preserved by countable support limits, by Shelah's
theorem, but it is not preserved by finite iterations in
the ignoring-the-last-stage sense that you describe.
So I think you don't really want to ignore the last stage
like that. Perhaps you are interested in something like
this: a property $Phi$ of forcing notions (respecting the
equivalence of forcing) is preserved by an iteration of
length $delta$, if whenever each stage $Q_beta$ of an
iteration is forced over $P_beta$ to have the property,
then $P_delta$ also has the property. For example, proper
forcing is preserved by countable support iterations. The
countable chain condition is preserved by finite support
iterations. Countably closed forcing is preserved by
countable support iterations.
Now, you want to ask whether there is a property that is
preserved by all limit iterations, but not by some finite
iterations.
If trivial forcing has property $Phi$, which is the
typical situation (e.g. trivial forcing is c.c.c., proper,
closed, cardinal-preserving, GCH-preserving, etc. etc.),
then the answer is no, again for a silly reason. Suppose
that $P_n$ is a finite iteration that witnesses that $Phi$
is not preserved, so that each stage $Q_m$ of $P_n$ has
$Phi$, but the iteration $P_n$ itself does not. Now let
$P_omega$ simply continue the forcing with trivial forcing
out to stage $omega$, and use countable support. Thus,
every stage of $P_omega$ has property $Phi$, but the
iteration altogether is forcing equivalent to $P_n$, which
does not have property $Phi$.
You can tweak this example to make an iteration $R_omega$
of length $omega$, which is nontrivial at every stage in
the weak sense that no $R_m$ forces with $1$ that $Q_m$ is
trivial, but such that anyway $R_omega$ is forcing
equivalent to the original $P_n$. For example, let the
first stage of forcing generically choose a natural number
$k$, which is interpreted as the place where the actual
iteration $P_n$ will begin. Every stage $m$ has a nonzero
Boolean possibility of being in the nontrivial part of the
iteration, and so no stage of this forcing is forced by $1$
to be trivial.
Perhaps a more interesting question would be to inquire:
Question. Is there a first order property that holds
in all the forcing extensions by limit length countable
support iterations, but not in all forcing extensions by
finite length forcing iterations?
That is, we inquire not about properties of the forcing,
but rather about properties of the universe to which the
forcing leads. In this case, to avoid the silly kind of
example above, let us insist that every stage of the
iteration forces that the next stage of forcing is
nontrivial below every condition.
In this case, the answer is Yes. One easy example is to let
$Phi$ be the assertion: ``the universe is not a forcing
extension of $L$ by adding one Sacks real.'' If I have an
iteration $P_delta$ of any limit length $delta$,
nontrivial at every stage, then it cannot be that
$P_delta$ is equivalent to adding a Sacks real, since that
forcing admits no intermediate models, such as the forcing
extensions arising during the iteration. But the iteration
over $L$ of length $1$, adding a Sacks real at that stage,
fails property $Phi$.
More generally, however, for any class $Gamma$ of forcing
notions, the statement $Psi =$ ``the universe is
obtained by forcing over $L$ with forcing of a certain type
$Gamma$'' is first order expressible. So if you have any
class $Gamma$, such as the class of forcing that is
equivalent to countable support limit length iterations,
the statement $Psi$ will be true exactly in the forcing
extensions of $L$ you are inquiring about, and only those.
This idea generalizes to forcing over $V$, if you allow
parameters, although this is a bit more subtle.
No comments:
Post a Comment