# 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

- Rudimentarily Closed & Definability;
- Linear Ordering: Auxilliary Hierarchy;
- Satisfaction -> Skolem term ->Condensation Lemma;
- Acceptability & Consequences;
- $\Sigma_1$-Projectum.

### 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.

- Similar arguments as the Handbook article has mentioned
- (I shall combine similar proofs.)

#### Jensen’s Preprint Book

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

# 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?

**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:

$$f(x_1,…,x_k)<\max_{i\leq k}(rank(x_i))+n.$$

$\square$

## 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 $\alpha$).

*Proof.*

$\square$

**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.* Uses $J^A_\alpha\cap Ord = \alpha$ and the $\Sigma_1$-definability of $S^A_\alpha$.

$\square$

**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 $\mathbf\Sigma^M_1$.

*Proof.*

$\square$

## 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.** $Qv$ 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.(…)

## Part 4: Downward & Upward Extensions

Recall that our aim is to use the uniformly definable $\Sigma_1$-Skolem function to analysis $\Sigma_n$ properties.

**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

$$A_{M}^{p}=\left{\langle i, x\rangle \in \omega \times H_{\rho}^{M} \mid M \vDash \varphi_{i}(x, p)\right}$$

is called the *standard code* determined by $p$. Easy to verify that $A^p_M$ is $\Sigma^M_1{p}$. 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*.

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

## Addendum: Logic embedded in $V$

### Other remarks

## 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*;

Draft of My Fine Structure Notes