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# 数学代写|数理逻辑入门代写Introduction To Mathematical logic代考|. Digression: Definability in HC

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## 数学代写|数理逻辑入门代写Introduction To Mathematical logic代考|Digression: Definability in HC

The next subsection will contain a transfinite construction of a key forcing notion in L relativized to $\mathrm{HC}$. Recall that $\mathrm{HC}$ is the collection of all hereditarily countable sets. In particular, $\mathbf{H C}=\mathbf{L}_{\omega_1}$ in $\mathbf{L}$. In matters of related definability classes, we refer to e.g., Part B, Chapter 5, Section 4 in [20], or Chapter 13 in [21], on the Lévy hierarchy of $\in$-formulas and definability classes $\Sigma_n^X, \Pi_n^X, \Delta_n^X$ for any set $X$, and especially on $\Sigma_n^{\mathrm{HC}}, \Pi_n^{\mathrm{HC}}, \Delta_n^{\mathrm{HC}}$ for $\mathrm{X}=\mathrm{HC}$ in Sections 8 and 9 in [22], or elsewhere. In particular,
$\Sigma_n^{\mathrm{HC}}=$ all sets $\mathrm{X} \subseteq \mathrm{HC}$, definable in $\mathrm{HC}$ by a parameter-free $\Sigma_n$ formula.
$\Sigma_n^{\mathrm{HC}}=$ all sets $\mathrm{X} \subseteq \mathrm{HC}$ definable in $\mathrm{HC}$ by a $\Sigma_n$ formula with sets in HC as parameters.
Something like $\Sigma_n^{\mathrm{HC}}(x), x \in \mathrm{HC}$, means that only $x$ is admitted as a parameter, while $\Sigma_n^{\mathrm{HC}}(M)$, where $M \subseteq \mathrm{HC}$ is a transitive model, means that all $x \in M$ are admitted as parameters. Collections like $\Pi_n^{\mathrm{HC}}, \Pi_n^{\mathrm{HC}}(x), \Pi_n^{\mathrm{HC}}(M)$ are defined similarly, and $\Delta_n^{\mathrm{HC}}=\Sigma_n^{\mathrm{HC}} \cap \Pi_n^{\mathrm{HC}}$, etc.. The boldface classes are defined as follows: $\Sigma_n^{\mathrm{HC}}=\Sigma_n^{\mathrm{HC}}(\mathrm{HC}), \Pi_n^{\mathrm{HC}}=\Pi_n^{\mathrm{HC}}(\mathrm{HC}), \Delta_n^{\mathrm{HC}}=\Delta_n^{\mathrm{HC}}(\mathrm{HC})$.

## 数学代写|数理逻辑入门代写Introduction To Mathematical logic代考|Basic Generic Extension and Regular Subextensions

Recall that an integer $m \geq 2$ and sets $\mathbb{U}{\bar{\zeta}}, \mathbb{M}{\bar{\zeta}}, \mathbb{U}, \mathbb{P}_{\bar{\zeta}}, \mathbb{P}$ are fixed in $\mathbf{L}$ by Definition 16 . These sets are fixed for the remainder.

Suppose that, in $\mathbf{L}, K \subseteq \mathbb{P}$ is a regular subforcing. If $G \subseteq \mathbb{P}$ is a set $\mathbb{P}$-generic over $\mathbf{L}$ then $G \cap K$ is $K$-generic over $\mathbf{L}$ by Lemma 9 (vi), and hence $\mathbf{L}[G \cap K]$ is a $K$-generic extension of $\mathbf{L}$. The following formulas $\mathbb{}i(i \in \mathcal{I})$ will give us a useful coding tool in extensions of this form: $$\mathbb{}_v(S):={ }{\operatorname{def}} v \in \mathcal{I} \wedge S \subseteq \operatorname{Seq} \wedge \forall f \in \operatorname{Fun} \cap \mathbf{L}(f \in \mathbb{U}(v) \Longleftrightarrow \max (S / f)<\omega) .$$
This is based on the next two results. Recall that $|G \cap K|=\bigcup_{p \in G \cap K}|p|$.
Lemma 17. $\mathbb{}v(S)$ as a binary relation belongs to $\Pi{\mathrm{n}-1}^{\mathrm{HC}}$ in any cardinal-preserving generic extension of $\mathbf{L}$.
Proof. The set $W={\langle v, f\rangle: v \in \mathcal{I} \wedge f \in \mathbb{U}(v)}$ is $\Delta_{\mathrm{n}-1}^{\mathrm{HC}}$ in L, by Lemma 16, and hence so is $W^{\prime}={\langle v, f\rangle: v \in \mathcal{I} \wedge f \in$ Fun $\backslash \mathcal{U}(v)}$. Let $\varphi(v, f)$ and $\varphi^{\prime}(v, f)$ be $\Sigma_{n-1}$ formulas that define resp. $W, W^{\prime}$ in HC, in $\mathbf{L}$. Then, in any generic extension of $\mathbf{L}, \mathbb{}v(S)$ is equivalent to $v \in \mathcal{I} \wedge S \subseteq$ Seq $\wedge \forall f \in \operatorname{Fun} \cap \mathbf{L} \Psi(v, f)$, where $\Psi(v, f)$ is the $\Pi{\mathrm{n}-1}$ formula
$$((\mathbf{L} \models \varphi(\nu, f)) \Longrightarrow \max (S / f)<\omega) \wedge\left(\left(\mathbf{L} \models \varphi^{\prime}(v, f)\right) \Longrightarrow \max (S / f)=\omega\right)$$

## 数学代写|数理逻辑入门代写Introduction To Mathematical logic代考|Digression: Definability in HC

$\Sigma_n^{\mathrm{HC}}=$ 全套 $\mathrm{X} \subseteq \mathrm{HC}$，可在 $\mathrm{HC}$ 通过无参数 $\Sigma_n$ 公式。
$\Sigma_n^{\mathrm{HC}}=$ 全套 $\mathrm{X} \subseteq \mathrm{HC}$ 定义于 $\mathrm{HC}$ 由a $\Sigma_n$ 以HC中的集合为参数的公式。

## 数学代写|数理逻辑入门代写Introduction To Mathematical logic代考|Basic Generic Extension and Regular Subextensions

$$((\mathbf{L} \models \varphi(\nu, f)) \Longrightarrow \max (S / f)<\omega) \wedge\left(\left(\mathbf{L} \models \varphi^{\prime}(v, f)\right) \Longrightarrow \max (S / f)=\omega\right)$$

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