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In the following question a property (of a forcing notion) is preserved by a CS-iteration if the following implication holds: If Pa has this property (for every ordinal a< d, for d being the iteration's length) then so does Pd.

  1. Is there a non-trivial property Phi such that Phi is preserved under CS-iterations whose length is a limit ordinal, but is not preserved under finite iterations?
  2. Is there any reasonable way to characterize such properties?
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I don't think you're asking the question that you intend to ask. If by Pa you mean the forcing iteration $P_\alpha$ of length $\alpha\lt\delta$, where $P_\delta$ is the whole iteration, then when $\delta=\beta+1$ is a successor ordinal you have not placed any restriction on the stage $\beta$ forcing, so there are many silly examples. But I think you probably intend to. Also, do you want your property to respect the equivalence of forcing? And probably you need to clarify quantifiers: I think you mean that $\Phi$ is preserved by all limit iterations, but not by all finite iterations. – Joel David Hamkins Aug 22 '10 at 2:36
Please allow me to clarify (I apology for the extremely inelegant notation): 1. In the successor case, given (Pa, Qa*: a< d+1) my assumption is that each Pa (for a smaller than d+1) satisfies Phi (but it's not necessarily forced that each Qa* satisfies Phi). In case that such assumption makes trivial counterexamples possible, we may also assume that it's forced by Pd that Qd* satisfies Phi. 2. The answer to your remaining questions is 'Yes'. – Haim Aug 22 '10 at 7:22

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.

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First of all, I'd like to thank you for this well-written answer. Regarding my initial assumption, I'd like to clarify that my assumption was motivated by the following result of Judah and Repicky: Given a CS-iteration of proper forcing notions of limit length such that each Pa does not add a dominating real, it holds that Pd does not add a dominating real. So by "non-trivial properties" I was looking for examples in the spirit of "P does not add a dominating real". – Haim Aug 22 '10 at 16:17

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