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数学代写|数理逻辑入门代写Introduction To Mathematical logic代考|Transformations and Invariance

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数学代写|数理逻辑入门代写Introduction To Mathematical logic代考|Transformations and Invariance

Here we show that, under certain assumptions, the transformations of the first two groups defined in Section 3.7 preserve forcing approximations forc. This is not an absolutely elementary thing: there is no way to reasonably apply transformations to transitive models $M$ involved in the definition of forc. What we can do is to require that the transformations involved belong to the models involved. This leads to certain complications of different sort.

Family 1: permutations. First of all we have to extend the definition of the action of $\pi$ in Section 3.7 to include formulas. Suppose that $c, c^{\prime} \subseteq \mathcal{I}$. Define the action of any $\pi \in \mathrm{BIJ}_{c^{\prime}}^c$ onto formulas $\varphi$ of $\mathcal{L}\left(\mathbf{P}^*\right)$ such that $|\varphi| \subseteq c$ :

to get $\pi \varphi$ substitute $\pi \cdot \tau$ for any $\tau \in \operatorname{NAM} \varphi$ and $\pi \cdot B$ for any $B \in \operatorname{IND} \varphi$.
Lemma 22. Suppose that $\langle M, U\rangle, K, p, \varphi$ satisfy (F1) of Definition 20, sets $c, c^{\prime} \subseteq \mathcal{I}$ have equal cardinality and are absolute $\Delta_1^{\mathrm{HC}}(M), \pi \in \mathrm{BI}_{c^{\prime}}^c$ is an absolute $\Delta_1^{\mathrm{HC}}(M)$ function, and $|\varphi| \subseteq c,|U| \subseteq c, K \subseteq \mathbf{P}^* \mid c$.

Proof. Under the assumptions of the lemma, in particular, the requirement of $c, c^{\prime}, \pi$ being absolute $\Delta_1^{\mathrm{HC}}(M), \pi$ acts as an isomorphism on all relevant domains and preserves all relevant relations between the objects involved. Thus $\langle M, \pi \cdot U\rangle, \pi \cdot K, \pi \cdot p, \pi \varphi$ still satisfy Definition 20(F1), and $|\pi \varphi| \subseteq c^{\prime},|\pi \cdot U| \subseteq c^{\prime}, \pi \cdot K \subseteq \mathbf{P}^*\left\lceil c^{\prime}\right.$. (For instance, to show that $\pi \cdot U$ still belongs to $M$, note that the set $|U| \subseteq c$ belongs to $M$, thus $\pi|| U \mid \in M$, too, since $\pi$ is an absolute $\Delta_1^{\mathrm{HC}}(M)$ function.) This allows to prove the lemma by induction on the complexity of $\varphi$.

数学代写|数理逻辑入门代写Introduction To Mathematical logic代考|Elementary Equivalence Theorem

This section presents further properties of $\mathbb{P}$-generic extensions of $\mathbf{L}$ and their subextensions, including Theorem 13 and its corollaties on the elementary equivalence of different subextensions.
Assumption 2. We continue to assume $\mathbf{V}=\mathbf{L}$ in the ground universe. Below in this section, a number n $\geq 2$ is fixed, and pairs $\left\langle\mathbb{M}{\bar{\xi}}, \mathbb{U}{\bar{\zeta}}\right\rangle$, the system $\mathbb{U}=V_{\bar{\zeta}<\omega_1} U_{\bar{\zeta}}$, the forcing notions $\mathbb{P}{\bar{\xi}}=\mathbf{P}\left[\mathbb{U}{\bar{\zeta}}\right]$ and $\mathbb{P}=\mathbf{P}[U]=\bigcup_{\bar{\zeta}<\omega_1} \mathbb{P}{\bar{\zeta}}$ are as in Definition 16 for this $\mathrm{n}$. 6.1. Further Properties of Forcing Approximations Coming back to the complete sequence of pairs $\left\langle M{\bar{\zeta}}, \mathbb{U}_{\bar{\xi}}\right\rangle$ introduced by Definition 16, we consider the auxiliary forcing relation forc with respect to those pairs. We begin with the following definition.

Definition 21 (in L). Let $K \subseteq \mathbf{P}^*$ be a regular forcing. Recall that
$$K[U]=K \cap \mathbb{P} \text { and } K\left[U_{\bar{\zeta}}\right]=K \cap P\left[U_{\bar{\zeta}}\right]=K \cap \mathbb{P}{\bar{\zeta}}$$ names in $\mathbb{M}{\xi} \cap \mathbf{S N}\omega^\omega\left(K\left[\mathbb{U}{\tilde{\xi}}\right]\right)$ as parameters, all names $\tau \in \operatorname{NAM} \varphi$ are $K\left[\mathbb{U}{\tilde{\zeta}}\right]$-full below $p$, all indices $B \in \operatorname{IND} \varphi$ belong to $M{\xi}$. The following is an easy consequence of Lemma 18.

数学代写|数理逻辑入门代写Introduction To Mathematical logic代考|Elementary Equivalence Theorem

$$K[U]=K \cap \mathbb{P} \text { and } K\left[U_{\bar{\zeta}}\right]=K \cap P\left[U_{\bar{\zeta}}\right]=K \cap \mathbb{P}{\bar{\zeta}}$$ 姓名 $\mathbb{M}{\xi} \cap \mathbf{S N}\omega^\omega\left(K\left[\mathbb{U}{\tilde{\xi}}\right]\right)$ 作为参数，所有的名称 $\tau \in \operatorname{NAM} \varphi$ 是 $K\left[\mathbb{U}{\tilde{\zeta}}\right]$-下面全部 $p$，所有指标 $B \in \operatorname{IND} \varphi$ 属于 $M{\xi}$． 下面是引理18的一个简单推论。

MATLAB代写

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