Draft of My Fine Structure Notes

Notice: This is a Reading note of the Handbook article written by Zeman & Schindler, about basics of fine structure theory. The notation may vary but the enumeration of theorems, lemmas and corollaries are the same.

Things I have written so far

• Part I:
  • Rudimentarily Closed & Definability;
  • $S^A_\gamma$ and $<^A_\gamma$;
• Part II:
  • $\Sigma_1$-definability of $S^A_\gamma$ and $<^A_\gamma$;
  • $\Sigma_1$-Satisfaction and the Skolem function;
  • Condensation Lemma;
• Part III:
  • Acceptability & Consequences;
  • $\Sigma_1$-Projectum.
• Part IV:
  • Reducts & Good Parameters;
  • Downward & Upward Extension.

Things I am planning to add in

Constructibility

• $\mathcal L_V$ Theory
  • $Fml(x)$. Mind the absoluteness.
  • Basic Set Theory. $\Sigma_0$ Comprehension.
  • Definability & Rudimentary Closed.
• Applications: $\square_\kappa$ holds in $L$;

Jensen’s Preprint Book

• Admissibility;
• Recursion Theory, similar argument like rudimentarily closedness

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Aim: To give a new way of describing the constructible hierarchy by levels and to apply the uniform $\Sigma_1$-definable $\Sigma$-Skolem function in $\Sigma_n$-Skolem functions, which are not uniformly definable.

Part 1: Rudimentarily Closed & Definability

Rudimentarily Closed: A structure $M = \langle |M|,\in,A\rangle$ that is closed under following function schemas:

• $f(\vec{x}) = \vec x(i)$;
• (Admissibility)$f(\vec x) = \vec x\cap A$;
• $f(\vec x) = {\vec x(i),\vec x(j)}$;
• $f(\vec x) = \vec x(i) - \vec x(j)$;
• $f(\vec x) = \bigcup_{y\in \vec x(0)} g(\langle y,x_2,…,x_n\rangle)$;
• $f(\vec x) = h(\vec g(\vec x))$;

The collection(as a proper class) of all these functions (called the $rud_A$ functions)is defined independent with the structure. A relation is called $rud_A$ if its characteristic function is $rud_A$. $rud_\emptyset$ is simplified as $rud$.

A variety of functions and relations like $\not\in$ and $x’’y$ can be regarded as $rud_A$ or $rud$(See Handbook). The closure of $rud_A$ functions is denoted as $rud_A(M)$. Notice that here $M$ itself cannot be referred as a $\vec x(i)$.

The $J$-Hierarchy is defined very similar as the $L$-Hierarchy, but only use the limit ordinals. $J^A_0 = \emptyset$ and for limit ordinal $\alpha$, $J^A_{\alpha\omega} = \bigcup_{\gamma<\alpha}J^A_{\gamma\omega}$. For successor ordinals, $J^A_{\alpha+\omega} = rud_A(J^A_{\alpha}\cup{J^A_{\alpha}})$. Actually, Lemma 1.4 shows by induction that $L_\alpha = J_\alpha$ if $\alpha$ is a limit ordinal.

One may argue that the collection of $rud_A$ functions could be very complicated due to the function generating schemas(the last two). Thus we introduce the idea of auxillary hierarchy, denoted as $S^A_\alpha$. It is defined via the closure of 15 defined functions $V^2\to V$, and it actually uses all the ordinals.

Fact. $S^A_\alpha = J^A_\alpha (=L_\alpha[A])$ if $\alpha$ is a limit ordinal.

After introducing the preliminaries, we can now discuss the relation between constructibility and $rud_A$ closedness. Remember that $L_\alpha \subseteq \mathcal P(L_\alpha)$ due to its definition.

Lemma 1.4 Let $U$ be a transitive structure with $A\cap V_{rank(U)+\omega}\subset U$. Then $\mathcal P(U)\cap rud_A(U\cup{U}) = \mathcal P(U)\cap \mathbf\Sigma_\omega^{\langle U,\in,A\rangle}$. In particular, $J^A_{\alpha+\omega}\cap \mathcal P(J_\alpha) = def_A(J_\alpha)$.

Remark. As we shall show later, $rud_A$ functions can only add a finite amount of rank to its variables. Thus to restrict $A\cap V_{rank(U)+\omega}\subset U$, we actually let the $rud_A$ functions be $rud_{A\cap U}$ functions.

$$\mathcal{P}(U) \cap \Sigma_{\omega}^{\langle U, \in, A\rangle}=\mathcal{P}(U) \cap \Sigma_{0}^{\langle U \cup{U}, \in, A \cap U\rangle},$$

since now all the quantifiers are bounded. Now we only need to show that
$$\mathcal P(U)\cap rud_A(U\cup{U}) = \mathcal P(U)\cap \mathbf\Sigma_0^{\langle U \cup{U}, \in, A \cap U\rangle}.$$

“$\supseteq$” By induction we can say that every $\Sigma_0$-definable relation would be $rud_A$. Thus for every $x\in\mathcal{P}(U) \cap \Sigma_{0}^{\langle U \cup{U}, \in, A \cap U\rangle}$, $y\in x$ iff $y\in U$ and for some $\vec x\in U^k$, $R(y,\vec x)$ is true for some $rud_A$ relation $R$. To get $U\cap {y:R(y,\vec x)}\in rud_A(U\cup{U})$ we need to use the fact that $f(y,\vec x) := y\cap{z:R(z,\vec x)}$ is $rud_A$ if $R$ is $rud_A$.

“$\subseteq$” Prove by induction that every $rud_A$ function $f$ satisfies the following property: $v_0\in f(v_1,…,v_m)$ is equivalent with a $\Sigma_0$ formula. (Here the assumption in the Remark is required.) Hence, all sets defined like ${y\in U:y \in f(x_1,…,x_n)}$ is in $\Sigma_0^{\langle U\cup{U},\in,A}({x_1,…,x_n})$.

$\square$

Question. What is the maximal/minima ordinal that is/isn’t $\Sigma_0$-definable without parameter? $\omega$ surely satisfies the condition. Moreover, what about the $\Sigma_{n/\omega}$-definable ordinals?

Answer. For $\Sigma_0$-definable ordinals, they are referred as the recursive ordinals. $\omega_1^{CK}$ is the supremum of them, but not recursive itself.

Remark. The core idea of this proof is that $\Sigma_0$-definable relations are also $rud_A$, and the converse is also true. It is quite difficult to see in the direct defintion of $rud_A$, but if one refers to the auxillary hierarchy, then it becomes clear.

Example. Not all $\Sigma_0$ functions (in $ZF$) are rudimentary: $f:x\mapsto \omega$ gives an example of $\Sigma_0$-definable functions which are not rudimentary. The point is, it is easy to verify that every rudimentary function has the so-called finite rank property:

Fact. If $f:V^k\to V$ is $rud$, then there is a $n\in\omega$ such that:
$$f(x_1,…,x_k)<\max_{i\leq k}(rank(x_i))+n.$$
$\square$

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Part 2: Problems arised in $L$-Hierarchy

Being another construction of $L[A]$, the $J$-Hierarchy has many properties that is very similar with the $L$-Hierarchy. Actually many theorems and lemmata are made in comparision with the $L$-Hierarchy.

In the Handbook article, the authors introduced many interesting properties of $J$-Hierarchy. Many similar properties (as well as their proofs) are also included in Devlin’s book Constructibility. Our next goal is to show that in the $J$-Structure, there is a uniformly $\Sigma_1$-definable $\Sigma_1$ Skolem function. We shall remark the usefulness of this statement at the end of this part.

The proofs and explanations are quite tedious, so I just listed all relevant results below.

• “$x=J_\beta^A$” is uniformly $\Sigma_1$-definable over $J_\alpha^A$ where $\beta<\alpha$.
• Similarly, a well-ordering $<^A_\beta$ is uniformly $\Sigma_1$-definable over $J_\alpha^A$ where $\beta<\alpha$.
• The satisfaction $\vDash^{\Sigma_0}_M$ is $\Delta_1$-definable. Thus $\vDash^{\Sigma_1}_M$ is $\Sigma_1$-definable.
• (Condensation) Any transitive structure that is $\Sigma_1$-embeddable into some $J$-structure is itself a $J$-structure.
• If $\alpha$ is closed under Godel’s Pairing funcion (in particular, $\alpha$ is a cardinal), then there is a $\Sigma_1$-definable function $f:\alpha\to J_\alpha^A$ without parameter. For arbitrary $\alpha$ the function requires parameters.

Another usage of the auxillary hierarchy is the $\Sigma_1$-definition of itself and a well-ordering:

Lemma. Both $\langle S^A_\gamma:\gamma<\alpha\rangle$ and $\langle <^A_\gamma:\gamma<\alpha\rangle$ are $\Sigma_1^{J^A_\alpha}$ uniformely definable(the definition does not depend on $\alpha$).

Remark. It is easy to show by induction which is discussed in Jensen’s notes and the Handbook article in detail. We only need to notice that the $\exists$ quantifier is used to declare the level.

Definition. Denote $#(\phi(a))$ as the Godel number of the $\Sigma_0$-sentence $\phi(a)$, with $a$ being the free variable. We call the function
$$f:\omega\to M, #(\phi(a))\mapsto x$$
the $\Sigma_1$ Skolem function iff $M\vDash \phi(x)$. Thus, $M$ satisfies the $\Sigma_1$-sentence $\exists a(\phi(a))$.

Remark. The Skolem function, in short, is a function that witnesses the truthfulness for the $\exists$ quantifier. As Brouwer would agree, a existence sentence couldn’t be viewed as proved if the corresponding elements are not constructed.

In Proposition 1.12, the defined function $f$ maps $i<n$, the Godel number of some $\Sigma_0$-formula(with some free variables), to the collection of all $v(i)$-tuple which satisfies the formula. Notice that $f$ is only a finite initial segment.

Proof. The proof uses finite induction and for each regular step, uses the Lemma 1.1. That is, the ordered pair $\langle n,T\rangle$ we want to add is definable.

We let $\Theta(f,N,n)$ be the $\Sigma_0$-formula which says that “$N$ is transitive, $f = f^N_n$ is defined like above”. Thus the predicate $\vDash^{\Sigma_0}_M$ would be $\Delta_1$, and thus absolute.

Thus for the $\Sigma_1$-definable $\Sigma_1$ Skolem function $h_M$, we only need to “pick out” some elements that is $\Sigma_1$-definable in $f(i)$. The strategy is to pick the $<^A_\beta$-minimal element in $S^A_\beta$, which we proved erlier that these two concepts are both $\Sigma_1$.

Question. I think the reason why he only uses the “first component” is to avoid further tedious problem of definable well-ordering in $n$-tuples of $S^A_\beta$. However I am not quite sure why defining $\Sigma_2$-Skolem function will need $\Sigma_3$ definition.

As an application, we shall now prove the condensation lemma for $J$-Structures.

Theorem. Any transitive structure that is $\Sigma_1$-embeddable into some $J$-structure is itself a $J$-structure.

Proof. Let $\pi:\bar M\xrightarrow[\Sigma_1]{}M$ be such an emebdding, together with $M=\langle J^A_\alpha,\in,B\rangle$. Let $\bar A=\pi^{-1}A$, $\bar B=\pi^{-1}B$ and $\bar \alpha = \bar M\cap Ord$. Now we check $\bar M = \langle J^{\bar A}{\bar \alpha},\in,\bar B\rangle$. Clearly, by the $\Sigma_1$-definability of $S^A_\alpha$, for all $\beta<\bar\alpha$, $S^A{\beta}\in \bar M$, thus $\bar M=J^{\bar A}_{\bar \alpha}$. For the other direction, only to notice that every $x\in\bar M$ is mapped to some $S^A_\beta$ for $\beta<\alpha$.
$\square$

Question. What is the usage of $B$ here?? Is $B$ really affecting the structure of $M$? I don’t think so.

Remark. The below “Convention” is definitely misleading. I think $h_M[X]$ would be much better.

Lemma. If $\alpha$ is closed under Godel paring function, then tere is a surjection $g:\alpha\to J^A_\alpha$ which is $\Sigma^M_1$. For arbitrary ordinal, this function would be $\mathbf\Sigma^M_1$.

Remark. I am not planning to include this proof right now, since we neither introduce the notion of Godel pairing function, nor it has anymore use for us at all. Anyhow, this lemma is interesting and useful.

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Part 3: Acceptability and the $\Sigma_1$ Projectum

In this part we start to construct some of the basic analysis of the $J$-Hierarchy. It turns out that under suitable assumption, $J^A_\rho = H^M_\rho$ and satisfies $ZFC^-$. For the $\Sigma_1$-Projectum, we shall talk about some of the settings for our next note, about upward & downward extension of embeddings.

Definition. A $J$-Structure $M = \langle J^A_\alpha,B\rangle$ is acceptable iff:

• Whenever $\xi<\alpha$ is a limit ordinal and $\tau<\xi$, if $\mathcal P(\tau)\cap J^A_{\xi+\omega}\not\subseteq J^A_\xi$, then there is a surjectve $f: \tau\to\xi$ in $J^A_{\xi+\omega}$(or, $J^A_{\xi+\omega}\vDash |\tau|\geq \xi$.)

Question. The authors wrote that the acceptability can be viewed as a strong version of $GCH$. What does it mean? According to the fact that $L[A]$(or $L_\alpha[A]$) are not generally satisfying $GCH$, $A$ has to satisfy more properties.

Acceptability is a $Q$-property with respect to the following definition.

Definition. $Qv$(called “there are cofinally many $v$”) is the abbriviation of $\forall u\exists v\supset u$. If $\Phi$ is a (fixed) $Q$-sentence, we say a structure $M$ satisfy the $Q$-property defined by $\Phi$ iff $M\vDash \Phi$.

By a cofinal map $\sigma:U\to V$ we declare that $\sigma$ possess the following property:
$$\forall y\in V\exists x\in U(y\subseteq \sigma(x)).$$
Clearly, $\Sigma_1$-elementary embedding is $Q$-preserving downwards. Moreover, if the map is also cofinal, then it is also $Q$-preserving upwards. Thus,

Corollary 1.22 Acceptability is downward absolute with respect to $\Sigma_1$-elementary embeddings. Furthermore, it is (upward) absolute with respect to $Q$-elementary embeddings.

Acceptable structures canbe more well-behaved. For example, Lemma 1.23 says that acceptable structures add no new subsets of sets that are already possessed by lower level with cardinal height.

Proof. While using Lemma 1.17, note that every infinite cardinal is closed under Godel Pairing Function. Consider the surjection $g:\tau\to u$ and the preimage of some $a\subset u$. Towards a contradiction, let the preimage not in $J^A_\rho$. By definition of acceptability, suppose the preimage(a set of ordinals $<\tau$ and $\Sigma_1$-definable) is contained in some $J^A_{\xi+\omega}-J^A_{\xi}$, we have a surjection $f:\tau\to \xi$ in $J^A_{\xi+\omega}$. Thus $\tau\geq\xi\geq\rho$, which is impossible since $\tau<\rho$. Thus the preimage is in $J^A_{\rho}$ and $a\in J^A_{\rho}$. $\square$

Corollary. $|J^A_\rho| = H^M_{\rho}$ provides that $J^A_\rho$ is acceptable and $\rho$ is a cardinal.

Remark. The cardinal property make sure that its size is preserved in lower level.(The comparision of size in lower level is “true”.)

Question. So let us now consider what makes an acceptable $J$-Structure: the set $A$ used to constrct the hierarchy and some ordinal $\alpha$. Is Corollary 1.22 proves that all of the $J$-structure constructed from $A$ is acceptable? Should we refer to the Condensation Lemma?

Lemma 1.24 Let $\langle M,\in, A\rangle$ be an acceptable $J$-structure. Then for any $\rho,\alpha\in Ord^M$, if $\alpha\subset J^A_\rho$ and $Card(a)<\rho$ in $M$, then $\rho$ being an infinite successor cardinal in $M$ gives $\alpha\in J^A_\rho$.

Proof.
$\square$

The proof of Lemma 1.23 gives us the intuition to find the $\Sigma_1$ Projectum in some acceptable $J$-Structure.

Definition. The $\Sigma_1$ Projectum $\rho(M)$ is defined as the least ordinal which satisfies
$$\mathcal P(\rho(M))\cap\mathbf \Sigma_1^M\not\subseteq M.$$

Remark. Equivalently, $\rho = \rho(M)$ is the $\Sigma_1$-Projectum iff there exists a $\mathbf{\Sigma}_1$-definable function $f$ in $M = J^A_\alpha$ such that
$$f’’J^A_{\rho} = J^A_\alpha.$$

In our proof above, we actually shows that $\Sigma_1$-Projectum can only be a $\Sigma_1$-cardinal, which means there are no $\Sigma_1$-definable surjection $f:\tau\to \rho(M)$ for some $\tau<\rho(M)$. In particular, $\rho(M)$ would be a cardinal if $\rho(M)\in M$.

Proof.
$\square$

Remark. The above theorem shows that the two definitions are indeed equivalent.(…)

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Part 4: Downward & Upward Extensions

Recall that our aim is to use the uniformly definable $\Sigma_1$-Skolem function to analysis $\Sigma_n$ properties. In this part we shall introduce some…

Definition. Let $M = \langle J^B_\alpha,D\rangle$ be an acceptable $J$-structure, $\rho = \rho(M)$ is the $\Sigma_1$ Projectum and $p\in M$. Then
$$A_{M}^{p}=\left{\langle i, x\rangle \in \omega \times J_{\rho}^{A} \mid M \vDash \varphi_{i}(x, p)\right}$$

is called the standard code determined by $p$. The strucure $M^p = \langle J^B_\rho,A^p_M\rangle$ is called the reduct determined by $p$. Moreover, the set
$$P_M = {p \in[\rho(M), M \cap \mathrm{On})^{<\omega}:\exists B\in\Sigma^M_1({p})[B\cap \rho\not\in M]}$$

is called the set of good parameters; the set
$$R_M = {r:p \in[\rho(M), M \cap \mathrm{On})^{<\omega}:h_M(\rho\cup{r}) = M}$$

is called the set of very good parameters.

It is easy to see that $R_M\subset P_M$, and both of them are non-empty. Now we would like to show that $\Sigma_0$-elementary embeddings between reducts defined by very good parameters can be extended to the whole structure, which is also $\Sigma_1$-preserving.

Lemma 3.1 Let $\pi: \bar M^{\bar p}\xrightarrow[\Sigma_0]{}M^p$, where $\bar p\in R_{\bar M}$. Then there exists a unique extension $\tilde \pi:\bar M\xrightarrow[\Sigma_0]{} M$ with $\tilde\pi(\bar p) = p$. Moreover, $\tilde \pi$ is $\Sigma_1$-preserving.

Proof.
$\square$

Remark. The above lemma actually shows that every $\Sigma_n$-elementary embedding between reducts can be extended to a unique $\Sigma_{n+1}$-elementary embedding between the original structures, given that $p\in R_M$ also.

Proof.
$\square$

Now we can give a generalized Condensation Lemma for reducts defined by a very good parameter.

Lemma 3.3. Let $\pi:N\xrightarrow[\Sigma_0]{} M^p$, where $N$ is a $J$-strcture and $p\in R_M$. Then there exists a unique $\bar M$ and $\bar p\in R_{\bar M}$ such that $N = \bar M^{\bar p}$.

Proof.
$\square$

The above lemma, together with Lemma 3.1 and 3.2, is called the “downward extension of embeddings”. To give a dual lemma, we first notice that such a extension of the codomain may not exist. Thus, we need to strengthen the power of the embedding.

Definition. A $\Sigma_1$-elementary embedding $\pi$ is called strong iff the well-foundedness of rudimentary functions with the same definition is preserved under $\pi$.

Lemma 4.1. Let $\pi:\bar M^{\bar p}\to N$ is strong, where $N$ is acceptable and $\bar p\in R_{\bar M}$. Then there are unique $M$ and $p\in R_M$ such that $N = M^p$. Moreover, the extended embedding $\tilde \pi:\bar M\xrightarrow[\Sigma_1]{} M$ satisfies $\tilde \pi(\bar p)=p$.

Proof.
$\square$

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Addendum: Application of Fine Structure: $L\vDash \square_\kappa$

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Addendum: Logic embedded in $V$

Other remarks

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Reference

[1] Handbook article of Zeman & Schindler;

[2] Devlin’s book Constructibility;

[3] Jensen’s unpublished notes Manuscript on fine structure, inner model theory, and the core model below one Woodin cardinal. It is on his personal website.

[4] Sy D. Friedman’s book Fine Structure and Class Forcing;

[5] Jensen’s original paper of $J$-Hierarchy The fine structure of the constructible hierarchy;