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# 数学代写|数理逻辑入门代写Introduction To Mathematical logic代考|Forcing Approximation

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## 数学代写|数理逻辑入门代写Introduction To Mathematical logic代考|Formulas

Here we introduce a language that will help us to study analytic definability in $\mathbf{Q}[U]$-generic extensions, for different systems $U$, and their submodels.

Let $\mathcal{L}$ be the 2 nd order Peano language, with variables of type 1 over $\omega^\omega$. If $K \subseteq \mathbf{Q}^*$ then an $\mathcal{L}(K)$ formula is any formula of $\mathcal{L}$, with some free variables of types 0,1 replaced by resp. numbers in $\omega$ and names in $\mathbf{S N}_\omega^\omega(K)$, and some type 1 quantifiers are allowed to have bounding indices $B$ (i.e., $\exists^B$, $\forall^B$ ) such that $B \subseteq \mathcal{I}^{+}$satisfies either $\operatorname{card} B \leq \omega_1$ or $\operatorname{card}(\mathcal{I} \backslash B) \leq \omega_1$ (in L). In particular, $\mathcal{I}^{+}$itself can serve as an index, and the absence If $\varphi$ is a $\mathcal{L}\left(\mathbf{Q}^\right)$ formula, then let \begin{aligned} \text { NAM } \varphi & =\text { the set of all names } \tau \text { that occur in } \varphi ; \ \text { IND } \varphi & =\text { the set of all quantifier indices } B \text { which occur in } \varphi ; \ |\varphi|^{+} & =\bigcup_{\tau \in \text { NAM } \varphi}|\tau|^{+} \text {(a set of } \omega_1 \text {-size); } \ |\varphi| & =|\varphi|^{+} \cup(\bigcup \operatorname{UND} \varphi)-\text { so that }|\varphi|^{+} \subseteq|\varphi| \subseteq \mathcal{I}^{+} . \end{aligned} If a set $G \subseteq \mathbf{Q}^$ is minimally $\varphi$-generic (that is, minimally $\tau$-generic w.r.t. every name $\tau \in$ NAM $\varphi$, in the sense of Section 3.5), then the valuation $\varphi[G]$ is the result of substitution of $\tau[G]$ for any name $\tau \in \operatorname{NAM} \varphi$, and changing each quantifier $\exists^B x, \forall^B x$ to resp. $\exists(\forall) x \in \omega^\omega \cap \mathbf{L}[G \mid B]$, while index-free type 1 quantifiers are relativized to $\omega^\omega ; \varphi[G]$ is a formula of $\mathcal{L}$ with real parameters, and some quantifiers of type 1 relativized to certain submodels of $\mathbf{L}[G]$.

An arithmetic formula in $\mathcal{L}(K)$ is a formula with no quantifiers of type 1 (names in $\mathbf{S N}\omega^\omega(K)$ are allowed). If $n<\omega$ then let a $\mathcal{L} \Sigma_n^1(K)$, resp., $\mathcal{L} \Pi_n^1(K)$ formula be a formula of the form $$\exists^{\circ} x_1 \forall^{\circ} x_2 \ldots \forall^{\circ}\left(\exists^{\circ}\right) x{n-1} \exists(\forall) x_n \psi, \quad \forall^{\circ} x_1 \exists^{\circ} x_2 \ldots \exists^{\circ}\left(\forall^{\circ}\right) x_{n-1} \forall(\exists) x_n \psi$$
respectively, where $\psi$ is an arithmetic formula in $\mathcal{L}(K)$, all variables $x_i$ are of type 1 (over $\omega^\omega$ ), the sign – means that this quantifier can have a bounding index as above, and it is required that the rightmost (closest to the kernel $\psi$ ) quantifier does not have a bounding index.

## 数学代写|数理逻辑入门代写Introduction To Mathematical logic代考|Forcing Approximation

We introduce a convenient forcing-type relation $p \operatorname{forc}U^M \varphi$ for pairs $\langle M, U\rangle$ in sJS and formulas $\varphi$ in $\mathcal{L}(K)$, associated with the truth in $K$-generic extensions of $\mathbf{L}$, where $K=\mathbf{Q}[U] \subseteq \mathbf{Q}^*$ and $U \in \mathbf{L}$ is a system. (F1) First, writing $p \operatorname{forc}_U^M \varphi$, it is assumed that: (a) $\langle M, U\rangle \in \mathbf{s J S}$ and $p$ belongs to $\mathbf{Q}[U]$, (b) $\varphi$ is a closed formula in $\mathcal{L} \Pi_k^1(\mathbf{Q}[U], M) \cup \mathcal{L} \Sigma{k+1}^1(\mathbf{Q}[U], M)$ for some $k \geq 1$, and each name $\tau \in \operatorname{NAM} \varphi$ is $\mathbf{Q}[U]$-full below $p$.
Under these assumptions, the sets $U, \mathbf{Q}[U], p$, NAM $\varphi$ belong to $M$.
The definition of forc goes on by induction on the complexity of formulas.
(F2) If $\langle M, U\rangle \in \mathbf{s J S}, p \in \mathbf{Q}[U]$, and $\varphi$ is a closed formula in $\mathcal{L} \Pi_1^1(\mathbf{Q}[U], M)$ (then by definition it has no quantifier indices), then: $p$ forc $_U^M \varphi$ iff (F1) holds and $p \mathbf{Q}[U]$-forces $\varphi[\underline{G}]$ over $M$ in the usual sense. Please note that the forcing notion $\mathbf{Q}[U]$ belongs to $M$ in this case by (F1).
(F3) If $\varphi(x) \in \mathcal{L} \Pi_k^1(\mathbf{Q}[U], M), k \geq 1$, then:
(a) $p$ forc $_U^M \exists^B x \varphi(x)$ iff there is a name $\tau \in M \cap \mathbf{S N}\omega^\omega(\mathbf{Q}[U]) \mid B, \mathbf{Q}[U]$-full below $p$ (by (F1)b) and such that $p \operatorname{forc}_U^M \varphi(\tau)$. (b) $\quad p$ forc $_U^M \exists x \varphi(x)$ iff there is a name $\tau \in M \cap \mathbf{S N}\omega^\omega(\mathbf{Q}[U]), \mathbf{Q}[U]$-full below $p$ (by (F1)b) and such that $p \operatorname{forc}_U^M \varphi(\tau)$.

## 数学代写|数理逻辑入门代写Introduction To Mathematical logic代考|Formulas

$\mathcal{L}(K)$中的算术公式是没有类型1的量词的公式($\mathbf{S N}\omega^\omega(K)$中的名称是允许的)。如果$n<\omega$那么让一个$\mathcal{L} \Sigma_n^1(K)$，请回复。， $\mathcal{L} \Pi_n^1(K)$公式为$$\exists^{\circ} x_1 \forall^{\circ} x_2 \ldots \forall^{\circ}\left(\exists^{\circ}\right) x{n-1} \exists(\forall) x_n \psi, \quad \forall^{\circ} x_1 \exists^{\circ} x_2 \ldots \exists^{\circ}\left(\forall^{\circ}\right) x_{n-1} \forall(\exists) x_n \psi$$形式的公式

## 数学代写|数理逻辑入门代写Introduction To Mathematical logic代考|Forcing Approximation

(F2)如果$\langle M, U\rangle \in \mathbf{s J S}, p \in \mathbf{Q}[U]$，并且$\varphi$是$\mathcal{L} \Pi_1^1(\mathbf{Q}[U], M)$中的封闭公式(那么根据定义，它没有量词指标)，那么:$p$ force $_U^M \varphi$ iff (F1)在通常意义上成立并且$p \mathbf{Q}[U]$ -force $\varphi[\underline{G}]$ over $M$。请注意，在这种情况下，强制概念$\mathbf{Q}[U]$属于$M$ (F1)。
(F3)若$\varphi(x) \in \mathcal{L} \Pi_k^1(\mathbf{Q}[U], M), k \geq 1$，则:
(a) $p$ forc $_U^M \exists^B x \varphi(x)$如果在$p$(由(F1)b)下面有一个名称$\tau \in M \cap \mathbf{S N}\omega^\omega(\mathbf{Q}[U]) \mid B, \mathbf{Q}[U]$ -full，并且$p \operatorname{forc}_U^M \varphi(\tau)$。(b) $\quad p$ forc $_U^M \exists x \varphi(x)$如果在$p$(由(F1)b)下面有一个名称$\tau \in M \cap \mathbf{S N}\omega^\omega(\mathbf{Q}[U]), \mathbf{Q}[U]$ -full，并且$p \operatorname{forc}_U^M \varphi(\tau)$。

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