# $A^\#$ and inner models

For a set of ordinals $A$ we say that $A^\#$ exists if there exists a closed and unbounded class of indiscernibles, $I\subseteq\operatorname{Ord}$, for $L[A]$. Formally, if such class exists we define $A^\#=\{\varphi\mid L_{i_\omega}[A]\models\varphi(i_1,\ldots,i_n)\}$, where $\varphi$ is a formula in the language $\{\in,A\}$. If we consider the Godel encoding of $\varphi$ we can think of $A^\#$ as a subset of $\omega$ or as a real number.

Suppose $V$ is a model of $ZFC$, $A$ a set of ordinals and $M$ is a transitive inner model such that $A\in M$ and $M\models A^\#\text{ exists}$.

Does that imply $V\models A^\#\text{ exists}$? What about the other direction (i.e. $A^\#$ exists in $V$, and $A\in M$)?

My intuition says the answer is yes in both cases, but my intuition was completely obliterated and needs to be rebuilt.

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Asaf: $A^\sharp$ is more complicated than your description, which only captures $0^\sharp$. You want $L_{i_\omega}[A]$ rather than $L_{i_\omega}$, for example, and you want to specify your language so that you have access to $A$ and its elements. – Andrés E. Caicedo Jul 27 '11 at 18:55
@Andres: Yes. You are correct. I did not specify that I meant $L_{i_\omega}[A]$ and that $\varphi$ is in the language of $\{\in, A\}$. I will fix that right away. – Asaf Karagila Jul 27 '11 at 19:06
I think by the other direction you mean that if $A^\sharp\in M$ then $M$ thinks that $A^\sharp$ exists, but perhaps you mean that if $A\in M$ and $A^\sharp$ exists in $V$ then it does in $M$. These questions have different answers. I tried to address both below. – Andrés E. Caicedo Jul 27 '11 at 19:23

One can use the class of indiscernibles to generate elementary embeddings of $L[A]$ to itself with critical point above the supremum of $A$, in much the same way as the indiscernibles of $L$ give us embeddings of $L$ into itself.

By results of Kunen (for $L$, but they generalize straightforwardly), the existence of an embedding $j:L[A]\to L[A]$ that has critical point above the supremum of $A$ implies the existence of $A^\sharp$, so both statements are equivalent.

So, suppose $M$ is an inner model and $M$ thinks that $A^\sharp$ exists. Then $M$ sees an elementary embedding of $L[A]$ into itself. This is clearly seen in $V$ as well. So $A^\sharp$ exists in $V$. The converse does not hold. After all, $L[A]$ sees $A$ but never $A^\sharp$. (However, if $M$ is an inner model that thinks that $A^\sharp$ exists, then $A^\sharp$ exists and coincides with $M$'s idea of it. This can be seen descriptive set theoretically (see below), or using the Kunen characterization. Also, if $A^\sharp$ belongs to $M$, then $M$ thinks that $A^\sharp$ exists, because from the sharp one can easily produce an $L[A]$-normal measure on the indiscernible $i_0$, and one can use this to recover the elementary embeddings.)

There is a better way of thinking about sharps, though, namely in terms of mice. This is probably folklore by now, but a nice reference is Schimmerling's "The ABC's of mice", Bulletin of Symbolic Logic 7 (2001) 485-503.

A sharp is then a model of the form $M=(L_\alpha[A],\in,A,U)$ where $U$ is an (external) measure on some $L_\alpha[A]$ cardinal $\kappa$. We add certain requirements that depend on your fine structural taste (say, $\alpha=\kappa^{++}$ in the sense of $L_\alpha[A]$), $M$ has the same size as $A$ and is sound and iterable. Soundness is a technical condition which essentially ensures $M$ is as small as possible.

Iterability is more complicated but it means that (as you imagine) iterating the process of forming ultrapowers by $U$ and its images only produces well-founded models. All the conditions describing the mouse $A^\sharp$ except for iterability are clearly absolute between inner models and $V$.

Iterability is as well, but this requires an argument. But then absoluteness gives us a positive answer to your question. (This means that if an inner model thinks that the mouse $A^\sharp$ exists, then it does in $V$, and if $A^\sharp$ exists and belongs to an inner model, then the inner model knows that it is $A^\sharp$.)

Consider first the case where $A$ is a subset of $\omega$. Then if one iterates $M$ some countable ordinal number of times, the whole iteration is codable by reals, and we can see that the statement that a real $x$ coding a model is iterable is $\Pi^1_2(x)$ (you basically have to say: every real coding an iteration of $x$ gives a well-founded model). The point is that if some iterate of $x$ is ill-founded, then some countable iterate is ill-founded, so you only need to describe iterations "accessible" by reals.

Finally, if $A$ is not a real, we can pretend that it is by working on a collapse of a sufficiently large initial segment of the universe to verify absoluteness. Or we can form the tree of attempts to build $A^\sharp$ and check that if $M$ thinks that $A^\sharp$ exists, then the tree is ill-founded, so $A^\sharp$ exists in $V$, and if $A^\sharp$ is in $M$, hen $M$ has a witness to the ill-foundedness of the tree. You can see a more detailed sketch of that idea in my paper with Schindler, "Projective well-orderings of the reals", Archive for Mathematical Logic 45 (7) (2006), 783-793, available at my page.

(Of course, one can also go as Jech does and verify the absoluteness of sharps in terms of blueprints, but I prefer the two other approaches I sketched.)

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Many thanks, Andres. I do have two questions regarding the answer. First is what does "is probably folklore" mean? I think I have seen that twice today, in the same context. The other is the first sentence of the seventh paragraph is unclear, it is as though it refers to a previous sentence, but there is none. Lastly, I have been skimming through inner model introductory texts for this assignment (this question is a byproduct of a byproduct of a question, and you did not reveal any actual part of the solution, so no worries there), I have made plans with a friend to study inner models soon :-) – Asaf Karagila Jul 27 '11 at 20:17
Hi Asaf. "Folklore" typically refers to the body of knowledge in the field that is mostly oral tradition and whose origins seem lost in the fog of time; it is hard to find precise references in the literature, but people know of it. – Andrés E. Caicedo Jul 27 '11 at 20:23
Oh, seventh paragraph: Iterability of a mouse $m$ is (absolute) as well (between $V$ and inner models $M$ with $m\in M$). One word of caution is that for larger mice the appropriate versions of iterability are no longer absolute. – Andrés E. Caicedo Jul 27 '11 at 20:25
Nice to hear about inner model theory. It takes some work to go through the preliminaries, but I think it is worth the effort. Let me know by email if you need suggestions regarding references. – Andrés E. Caicedo Jul 27 '11 at 20:27
The folklore thing sounds like the proof of transitivity for the Mitchell order. It seems that no one ever published one :-) as for the suggestions about reading references I would be delighted to have some. I'll let you know when I'm planning to start, and ask for your opinion. Thanks! – Asaf Karagila Jul 27 '11 at 20:32