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# 数学代写|数理逻辑入门代写Introduction To Mathematical logic代考|Preservation of the Completeness

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## 数学代写|数理逻辑入门代写Introduction To Mathematical logic代考|Preservation of the Completeness

The next lemma says that the completeness property (iii) of Theorem 6 , of the sequence $\left{\mathbb{U}{\bar{\zeta}}\right}{\bar{\zeta}<\Omega^{\oplus}}$, still holds, to some extent, in rather mild generic extensions of $\mathbf{L}$.

Lemma 19. Under the assumptions and notation of Definition 2, suppose that $\left{\mathbb{U}a\right}{a<\Omega^{\oplus}} \in \mathbf{L}$ is $a \preccurlyeq-$ increasing sequence of $\rrbracket$-systems in $\mathbf{s D S}_{\mathbb{\Omega}}$ satisfying (i)-(iv) of Theorem 6.

Let $Q \in \mathbf{L}$ be a forcing notion with card $Q \leq \mathbb{\Omega}$ in $\mathbf{L}$, e.g., $Q=\mathbb{C}$. Let $F \subseteq Q$ be a set $Q$-generic over $\mathbf{L}$.

Assume that $m<\omega, \delta<\Omega^{\oplus}$, and a set $D \in \mathbf{L}[F], D \subseteq \operatorname{sDS}{\Omega} \mid \geq m$, belongs to $\Sigma{m+3}(H[F])$, and is
Then there is an ordinal $\alpha, \delta \leq \alpha<\mathbb{R}^{\oplus}$, such that $\mathbb{U}\alpha \mid \geq m m$-solves $D$, as in Theorem 6(iii). We recall that $\mathbb{H}=\left(\mathbf{H} \Omega^{\oplus}\right)^{\mathbf{L}}$ and $\mathbb{H}[F]=\left(\mathbf{H} \Omega^{\oplus}\right)^{\mathbf{L}[F]}$ by (5), (6). Proof. As obviously $\left.\operatorname{sDS}{\Omega}\right|^{\geq m} \subseteq H$, we conclude by Theorem 5 (ii) that there is a $\Sigma_{m+3}(H)$ name $t \in \mathbf{L}, t \subseteq Q \times \mathrm{H}$, such that $D=t[F]$.

We argue in $\mathbf{L}$. If $q \in Q, U \in \mathbf{s D S}{\Omega} \Gamma^{\geq m}$, and there is such a condition $h \in Q$ that $h \leqslant q$ (meaning $h$ is stronger) and $\langle h, U\rangle \in t$, then write $A(q, U)$. If $b \in Q$ then we define: $$D(b)=\left{\left.U \in \mathbf{s D S}{\mathbb{R}}\right|^{\geq m}: \exists q \in Q(q \leqslant b \wedge A(q, U))\right} .$$
Each of the sets $D(b) \subseteq \mathbb{H}$ belongs to $\Sigma_{m+3}(H)$ by virtue of Lemma 17 and the choice of $t$. Therefore, by the choice of the sequence of $\Omega$-systems, for every $b \in Q$ there is an ordinal $\alpha(b), \delta<\alpha(b)<\mathbb{\Omega}^{\oplus}$, such that the $\mathbb{R}$-system $\mathbb{U}_{\alpha(b)}||^{\geq m} m$-solves the set $D(b)$.

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

We argue under the assumptions and notation of Definition 2 on page 13 . In particular, a successor L-cardinal $\mathbb{R}>\omega$ is fixed. We make the following arrangements.

Definition 6 (in L). We fix a $\preccurlyeq$-increasing sequence of $\Omega$-systems $\left{\mathbb{U}{\xi}^{\Omega}\right}{\Sigma<\Omega^{\circ}}$ satisfying conditions (i)-(iv) of Theorem 6 for the particular L-cardinal $\Omega$ introduced by Definition 2 .

We define the limit $\Omega$-system $\mathbb{U}^{\Omega}=\bigvee_{\xi<n \oplus \mathbb{U}{\xi}^{\Omega}}$, the basic forcing notion $\mathbb{P}^{\Omega}=\mathbf{P}\left[\mathbb{U}^{\Omega}\right]$, and the subforcings $\mathbb{P}\gamma^{\Omega}=\mathbf{P}\left[\mathbb{U}\gamma^{\curvearrowleft}\right], \gamma<\Omega^{\oplus}$. Define restrictions $\mathbb{P}^n|z, G| z\left(z \subseteq \mathcal{I}, G \subseteq \mathbb{P}^n\right),\left.\mathbb{P}^n\right|{\neq\langle n, i\rangle}$ etc. as in Section 3.2.
Thus by construction $\mathbb{P}^{\Omega} \in \mathbf{L}$ is the L-product of sets $\mathbb{P}^n(n, i)=P\left[\mathbb{U}^{\Omega}(n, i)\right], n, i \in \omega$. Lemma 14 implies some cardinal characterictics of $\mathbb{P}^{\Omega}$, namely:
(I) $\operatorname{card} \mathbb{P}^n=\Omega^{\oplus}$ in $\mathbf{L}$,
(II) $\mathbb{P}^{\Omega}$ satisfies $\mathbb{R}^{\oplus}-\mathrm{CC}$ in $\mathbf{L}$,
(III) $\mathbb{P}^{\Omega}$ is $\Omega^{\ominus}$-closed and $\Omega^{\ominus}$-distributive in $\mathbf{L}$.
Corollary 2. $\mathbb{P}^{\cap}$ does not adjoin new reals to $\mathbf{L}$.
Proof. The result follows from (III) because $\mathbb{R}^{\ominus} \geq \omega$ by Definition 2 .

As for definability, the set $\mathbb{U}^{\Omega}$ is not parameter free definable in $\mathbb{H}=\left(\mathbf{H} \Omega^{\oplus}\right)^{\mathbf{L}}$, yet its slices are:
Lemma 20 (in L). Let $n<\omega$. Then the set $\left.\mathbb{U}^{\Omega}\right|^n=\left{\langle i, f\rangle: f \in \mathbb{U}^{\Omega}(n, i)\right}$ belongs to $\Sigma_{n+4}^{\mathbb{H}}$. In addition there is a recursive sequence of parameter free $\in$-formulas $u_n(i, f)$ such that, for any $n<\omega$, if $i<\omega$ and $f \in \mathbf{F u n}_{\Omega}$ then $f \in \mathbb{U}^{\Omega}(n, i)$ iff $\mathbb{H} \models \boldsymbol{u}_n(i, f)$.

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

(一)$\mathbf{L}$中的$\operatorname{card} \mathbb{P}^n=\Omega^{\oplus}$;
(二)$\mathbb{P}^{\Omega}$满足$\mathbf{L}$中的$\mathbb{R}^{\oplus}-\mathrm{CC}$;
(三)$\mathbb{P}^{\Omega}$为$\Omega^{\ominus}$封闭，$\Omega^{\ominus}$分布于$\mathbf{L}$。

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