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## 数学代写|数理逻辑入门代写Introduction To Mathematical logic代考|The Collection of Axioms Is Recursive

In this section we will exhibit two $\Delta$-formulas that are designed to pick out the axioms of our deductive system.

Proposition 4.11.1. The collection of Gödel numbers of the axioms of $N$ is recursive.

Proof. The formula AxiomOfN is easy to describe. As there are only a finite number of $\mathrm{N}$-axioms, a natural number $a$ is in the set AxiomOFN if and only if it is one of a finite number of Gödel numbers. Thus
$\operatorname{AxiomOfN}(a)$ is:
$$\begin{gathered} a=\overline{\Gamma(\forall x) \neg S x=0\urcorner} \vee \ a=\overline{\Gamma(\forall x)(\forall y)[S x=S y \rightarrow x=y]\urcorner} \vee \ \vdots \ \vee a=\overline{\Gamma(\forall x)(\forall y)[(x<y) \vee(x=y) \vee(y<x)]\urcorner} . \end{gathered}$$
(To be more-than-usually picky, we need to change the $x$ ‘s and $y$ ‘s to $v_1$ ‘s and $v_2$ ‘s, but you can do that.)

Proposition 4.11.2. The collection of Gödel numbers of the logical axioms is recursive.

Proof. The formula that recognizes the logical axioms is more complicated than the formula AxiomOfN for two reasons. The first is that there are infinitely many logical axioms, so we cannot just list them all. The second reason that this group of axioms is more complicated is that the quantifier axioms depend on the notion of substitutability, so we will have to use our results from Section 4.10.

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

It is probably difficult to remember at this point of our journey, but our goal is to prove the Incompleteness Theorem, and to do that we need to write down an $\mathcal{L}{N T}$-sentence that is true in $\mathfrak{N}$, the standard structure, but not provable from the axioms of $N$. Our sentence, $\theta$, will “say” that $\theta$ is not provable from $N$, and in order to “say” that, we will need a formula that will identify the (Gödel numbers of the) formulas that are provable from $N$. To do that we will need to be able to code up deductions from $N$, which makes it necessary to code up sequences of formulas. Thus, our next goal will be to settle on a coding scheme for sequences of $\mathcal{L}{N T}$-formulas.

We have been pretty careful with our coding up to this point. If you check, every Gödel number that we have used has been even, with the exception of 3 , which is the garbage case in Definition 4.7.1. We will now use numbers with smallest prime factor 5 to code sequences of formulas.

Suppose that we have the sequence of formulas
$$D=\left\langle\phi_1, \phi_2, \ldots, \phi_k\right\rangle .$$
We will define the sequence code of $D$ to be the number
$$\left.r D\urcorner=5^{\left.r \phi_1\right\urcorner} 7^{\left.r \phi_2\right\urcorner} \cdots p_{k+2} \phi_k\right\urcorner .$$
So the exponent on the $(i+2)$ nd prime is the Gödel number of the $i$ th element of the sequence. You are asked in the Exercises to produce several useful $\mathcal{L}_{N T}$-formulas relating to sequence codes.

## 数学代写|数理逻辑入门代写Introduction To Mathematical logic代考|The Collection of Axioms Is Recursive

$\operatorname{AxiomOfN}(a)$是:
$$\begin{gathered} a=\overline{\Gamma(\forall x) \neg S x=0\urcorner} \vee \ a=\overline{\Gamma(\forall x)(\forall y)[S x=S y \rightarrow x=y]\urcorner} \vee \ \vdots \ \vee a=\overline{\Gamma(\forall x)(\forall y)[(x<y) \vee(x=y) \vee(y<x)]\urcorner} . \end{gathered}$$
(为了比通常更挑剔，我们需要将$x$和$y$更改为$v_1$和$v_2$，但您可以这样做。)

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

$$D=\left\langle\phi_1, \phi_2, \ldots, \phi_k\right\rangle .$$

$$\left.r D\urcorner=5^{\left.r \phi_1\right\urcorner} 7^{\left.r \phi_2\right\urcorner} \cdots p_{k+2} \phi_k\right\urcorner .$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

Posted on Categories:Mathematical logic, 数学代写, 数理逻辑

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## 数学代写|数理逻辑入门代写Introduction To Mathematical logic代考|The Language, the Structure, and the Axioms of $N$

We work in the language of number theory
$$\mathcal{L}_{N T}={0, S,+, \cdot, E,<},$$
and we will continue to work in this language for the next two chapters. $\mathfrak{N}$ is the standard model of the natural numbers,
$$\mathfrak{N}=\langle\mathbb{N}, 0, S,+, \cdot, E,<\rangle,$$
where the functions and relations are the usual functions and relations that you have known since you were knee high to a grasshopper. $E$ is exponentiation, which will usually be written $x^y$ rather than $E x y$ or $x E y$.

We will now establish a set of nonlogical axioms, $N$. You will notice that the axioms are clearly sentences that are true in the standard structure, and thus if $T$ is any set of axioms such that $T \vdash \sigma$ for all $\sigma$ such that $\mathfrak{N} \vDash \sigma$, then $T \vdash N$. So, as we prove that several sorts of formulas are derivable from $N$, remember that those same formulas are also derivable from any set of axioms that has any hope of providing an axiomatization of the natural numbers.

The axiom system $N$ was introduced in Example 2.8.3 and is reproduced on the next page. These eleven axioms establish some of the basic facts about the successor function, addition, multiplication, exponentiation, and the $<$ ordering on the natural numbers.

## 数学代写|数理逻辑入门代写Introduction To Mathematical logic代考|Recursive Sets and Recursive Functions

For the sake of discussion, suppose that we let $f(x)=x^2$. It will not surprise you to find out that it is the case that $f(4)=16$, so I would like to write $n \models f(4)=16$. Unfortunately, we are not allowed to do this, since the symbol $f$, not to mention 4 and 16, are not part of the language.

What we can do, bowever, is to represent the function $f$ by a formula in $\mathcal{L}{N T}$. To be specific, suppose that $\phi(x, y)$ is $$y=E x S S O$$ Then, if we allow ourselves once again to use the abbreviation $\bar{a}$ for the $\mathcal{C}{N T \text {-term }}^{S S S \cdots S} 0$, we can assert that
$$\boldsymbol{n}=\phi(\overline{4}, \overline{16})$$
which is the same thing as
ๆю= SSSSSSSSSSSSSSSSOESSSSOSSO.
(Boy, aren’t you glad we don’t use the official language very often?) Anyway, the situation is even better than this, for $\phi(4, \overline{16})$ is derivable from $N$ rather than just true in $\mathfrak{n}$. In fact, if you look back at Lemma 2.8.4, you probably won’t have any trouble believing the following statements:

• $N \vdash \phi(\overline{4}, \overline{16})$
• $N \vdash \neg \phi(4,17)$
• $N \vdash \neg \phi(\overline{1}, \overline{714})$

## 数学代写|数理逻辑入门代写Introduction To Mathematical logic代考|The Language, the Structure, and the Axioms of $N$

$$\mathcal{L}_{N T}={0, S,+, \cdot, E,<},$$

$$\mathfrak{N}=\langle\mathbb{N}, 0, S,+, \cdot, E,<\rangle,$$

## 数学代写|数理逻辑入门代写Introduction To Mathematical logic代考|Recursive Sets and Recursive Functions

$$\boldsymbol{n}=\phi(\overline{4}, \overline{16})$$

(天哪，你不高兴我们不经常使用官方语言吗?)无论如何，情况甚至比这更好，因为$\phi(4, \overline{16})$可以从$N$推导出来，而不仅仅是在$\mathfrak{n}$中成立。事实上，如果你回顾引理2.8.4，你可能会毫不费力地相信以下陈述:

$N \vdash \phi(\overline{4}, \overline{16})$

$N \vdash \neg \phi(4,17)$

$N \vdash \neg \phi(\overline{1}, \overline{714})$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

Posted on Categories:Mathematical logic, 数学代写, 数理逻辑

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

There will be a preliminary argument that will show that it is sufficient to prove that if $\Sigma$ is a consistent set of sentences, then $\Sigma$ has a model. Then we will proceed to assume that we are given such a set of sentences, and we will construct a model for $\boldsymbol{\Sigma}$.

The construction of the model will proceed in several steps, but the central idea was introduced in Example 1.6.4. The elements of the model will be variable-free terms of a language. We will construct this model so that the formulas that will be true in the model are precisely the formulas that are in a certain set of formulas, which we will call $\Sigma^{\prime}$. We will make sure that $\Sigma \subseteq \Sigma^{\prime}$, so all of the formulas of $\Sigma$ will be true in this constructed model. In other words, we will have constructed a model of $\Sigma$.

To make the construction work we will take our given set of $\mathcal{L}$ sentences $\Sigma$ and extend it to a bigger set of sentences $\Sigma^{\prime}$ in a bigger language $\mathcal{L}^{\prime}$. We do this extension in two steps. First, we will add in some new axioms, called Henkin Axioms, to get a collection $\hat{\Sigma}$. Then we will extend $\hat{\Sigma}$ to $\Sigma^{\prime}$ in such a way that:

1. $\Sigma^{\prime}$ is consistent.
2. For every $\mathcal{L}^{\prime}$-sentence $\theta$, either $\theta \in \Sigma^{\prime}$ or $(\neg \phi) \in \Sigma^{\prime}$.
Thus we will say that $\Sigma^{\prime}$ is a maximal consistent extension of $\Sigma$, where maximal means that it is impossible to add any sentences to $\Sigma^{\prime}$ without making $\Sigma^{\prime}$ inconsistent.

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

So let us fix our setting for the rest of this proof. We are working in a language $\mathcal{L}$. For the purposes of this proof, we assume that the language is countable, which means that the formulas of $\mathcal{L}$ can be written in an infinite list $\alpha_1, \alpha_2, \ldots, \alpha_n, \ldots$. (An outline of the changes in the proof necessary for the case when $\mathcal{L}$ is not countable can be found in Exercise 6.)

We are given a set of formulas $\Sigma$, and we are assuming that $\Sigma \vDash \phi$. We have to prove that $\Sigma \vdash \phi$.

Note that we can assume that $\phi$ is a sentence: By Lemma 2.7.2, $\Sigma \vdash \phi$ if and only if there is a deduction from $\Sigma$ of the universal closure of $\phi$. Also, by the comments following Lemma 2.7.3, we can also assume that every element of $\Sigma$ is a sentence. So, now all(!) we have to do is prove that if $\Sigma$ is a set of sentences and $\phi$ is a sentence and if $\Sigma \vDash \phi$, then $\Sigma \vdash \phi$.

Now we claim that it suffices to prove the case where $\phi$ is the sentence $\perp$. For suppose we know that if $\Sigma \models \perp$, then $\Sigma \vdash \perp$, and suppose we are given a sentence $\phi$ such that $\Sigma \vDash \phi$. Then $\Sigma U$ $(\neg \phi) \vDash \perp$, as there are no models of $\Sigma \cup(\neg \phi)$, so $\Sigma \cup(\neg \phi) \vdash \perp$. This tells us, by Exercise 4 in Section 2.7.1, that $\Sigma \vdash \phi$, as needed.

So we have reduced what we need to do to proving that if $\Sigma \models \perp$, then $\Sigma \vdash \perp$, for $\Sigma$ a set of $\mathcal{L}$-sentences. This is equivalent to saying that if there is no model of $\Sigma$, then $\Sigma \vdash \perp$. We will work with the contrapositive: If $\Sigma \nvdash \downarrow$, then there is a model of $\Sigma$. In other words, we will prove:

If $\Sigma$ is a consistent set of sentences, then there is a model of $\Sigma$.

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

$\Sigma^{\prime}$ 是一致的。

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

1.2. 从一个循环中移除一个奇异顶点，或者移除一个循环。然后，$S$不减。
1.3. 从链中移除一个内部奇异顶点(即在其两侧有其他奇异顶点)。链的类型不变，从$3 b^$变为$3 b^{\prime}$，或者从$2 a^$变为$2 a^{\prime}$。然后，$P$不会增加，因为在进行此更改时没有元素增加其质量。
1.4. 从链中移除链中唯一的奇异顶点。然后，$S$和$D$不会减少，如果删除一个$b$ -顶点，$K_b$会减少1。
1.5. 在链中不是唯一的挂顶点将从链中移除。如果，从$G$到$o(G), S$传递时不发生变化(与悬边相邻的段为偶数)，则悬端变为非悬端，则链的类型可能发生如下变化:$1 a$变为$3 a, 1 b$变为$3 b, 2$变为$1,2 a$变为$1 a, 2 b$变为$1 b$，或1变为3。在前三种情况下，$D$不变，$P$不变。事实上，当将类型$1 a$从$1 a$更改为$3 a$时，所有包含类型的元素都不会提高其质量;这同样适用于其他两个更改。在最后三种情况下，$D$增加$1, P$或者不变化，或者最多增加$w_a$(考虑反向变化)，而所有其他量都不变化。然后，$T^{\prime}+T^{\prime \prime}-P$最多减少$w_a$。如果$S$增加1(与挂边相邻的段为奇数)，则挂端保持悬挂状态，链条的类型不变。因此，$D$和$P$不变。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

Posted on Categories:Mathematical logic, 数学代写, 数理逻辑

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

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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$\mathbf{L}$。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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

We come back to Theorem 2. Now it is time to specify the value of the L-cardinal $\Omega$, so far left rather arbitrary by Definition 2 on page 13 .
Definition 7 (in $\mathbf{L}$ ). Recall that $1 \leq \mathbb{M}<\omega$ is a number considered in Theorem 2 .

We let $\Omega=\omega_{\mathrm{M}}^{\mathrm{L}}$, and accordingly define $\Omega^{\ominus}=\omega_{\mathrm{M}-1}^{\mathrm{L}}, \Omega^{\oplus}=\omega_{\mathrm{M}+1^{\prime}}^{\mathrm{L}}$
$$\mathbb{H}=\left(\mathbf{H} \Omega^{\oplus}\right)^{\mathbf{L}}=\left(\mathbf{H} \omega_{\mathbb{M}+1}^{\mathrm{L}}\right)^{\mathbf{L}}=\left{x \in \mathbf{L}: \operatorname{card}(\operatorname{TC}(x))<\omega_{\mathbb{M}+1}^{\mathrm{L}} \text { in } \mathbf{L}\right}$$
by Definition 2. Applying Definition 6 with $\Omega=\omega_{\mathrm{M}}^{\mathrm{L}}$, we accordingly fix:

$A \preccurlyeq$-increasing sequence of $\mathbb{R}$-systems $\left{\mathcal{U}z^{\mathbb{R}}\right}{\xi<\mathbb{R}^{\oplus}}$ satisfying (i), (ii), (iii), (iv) of Theorem 6 for the chosen $\mathbf{L}$-cardinal $\Omega=\omega_{\mathrm{M}}^{\mathrm{L}}$,

The limit $\Omega$-system $\mathbb{U}^{\Omega}=\bigvee_{\xi<\Omega} \mathbb{U}_{\xi}^{\Omega}$,

The basic forcing notion $\mathbb{P}^{\Omega}=\mathbf{P}\left[\mathbb{U}^{\Omega}\right]$, and the subforcings $\mathbb{P}\gamma^{\Omega}=\mathbf{P}\left[\mathbb{U}\gamma^{\Omega}\right], \gamma<\Omega^{\oplus}$, and define restrictions $\mathbb{P}^{\Omega}\left|z(z \subseteq \mathcal{I}), \mathbb{P}^{\Omega}\right|^{\geq n}, \mathbb{P}^{\Omega} \uparrow^{<n}, \mathbb{P}^n\lceil\neq\langle n, i\rangle$ etc. as in Section 3.2.

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

To prove Theorem 2 we make use of a certain submodel of a $\left(\mathbb{C} \times \mathbb{P}^{\Omega}\right)$-generic extension of $\mathbf{L}$. First of all, if $g: \omega \rightarrow \mathscr{P}(\omega)$ is any function then we put:
$$\boldsymbol{w}[g]={\langle k, j\rangle: k<\omega \wedge j \in g(k)}$$
Now consider a pair $\langle\zeta, G\rangle,\left(\mathbb{C} \times \mathbb{P}^{\Omega}\right)$-generic over $\mathbf{L}$. Thus $\zeta: \omega \stackrel{\text { onto }}{\longrightarrow} \Xi$ is a generic collapse function, while the set $G \subseteq \mathbb{P}^{\Omega}$ is $\mathbb{P}^{\Omega}$-generic over $\mathbf{L}[\boldsymbol{\zeta}]$. The set:
$$w[\zeta]={\langle k, j\rangle: k<\omega \wedge j \in \zeta(k)} \subseteq \mathcal{I}=\omega \times \omega$$
obviously belongs to the model $\mathbf{L}[\zeta]=\mathbf{L}[\boldsymbol{w}[\zeta]]$, but not to $\mathbf{L}$. Therefore the restrictions $\mathbb{P}^{\Omega} \mid \boldsymbol{w}[\zeta]$, $G\lceil w[\zeta]$ in the next theorem have to be understood in the sense of Definition 5 on page 15 , ignoring Remark 3 since, definitely $\boldsymbol{w}[\zeta] \notin \mathbf{L}$. Thus $\mathbb{P}^{\Omega} \mid \boldsymbol{w}[\boldsymbol{\zeta}]$ is a forcing notion in $\mathbf{L}[\zeta]$, not in $\mathbf{L}$.
The following theorem describes the structure of such generic models.
Theorem 8. Under the assumptions of Definition 7, let a pair $\langle\zeta, G\rangle$ be $\left(\mathbb{C} \times \mathbb{P}^{\Omega}\right)$-generic over $\mathbf{L}$. Then:
(i) $G\left\lceil w[\zeta]\right.$ is a set $\left(\mathbb{P}^n \mid \boldsymbol{w}[\zeta]\right)$-generic over $\mathbf{L}[\zeta]$,
(ii) $\omega_\gamma^{\mathrm{L}[\zeta, G \mid w[\zeta]]}=\omega_{1+\gamma}^{\mathrm{L}}$ for all ordinals $\gamma \geq 1$, in particular, $\mathbb{R}^{\oplus}=\omega_{\mathbb{M}}^{\mathrm{L}[\zeta, G \mid w[\zeta]]}$;
and it is true in the model $\mathbf{L}[\zeta, G\lceil\boldsymbol{w}[\zeta]]$ that
(iii) If $M \geq 2$ then $\Omega=\omega_{M-1}$ and $\Omega^{\oplus}=\Omega^{+}=\omega_M$, whereas if $M=1$ then $\omega<\Omega=\Omega^{\oplus}=\omega_1$;
(iv) GCH holds;
(v) Every constructible real belongs to $\mathbf{D}{1 \mathrm{M}}$, (vi) If $1 \leq m<\omega$ and $m \neq \mathbb{M}$ then $\mathbf{D}{1 m} \notin \mathbf{D}{2 m}$, and (vii) every real in $\mathbf{D}{1 \mathrm{M}}$ is constructible.

## 数学代写|数理逻辑入门代写Introduction To Mathematical logic代考|Further Reformulations and Harrington’s Statement

$$\Delta_3^1, \Delta_4^1, \ldots, \Delta_\omega^1=\text { projective, } \Delta_n^m, 1 \leq n \leq \omega, 2 \leq m \leq \omega$$

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

(ii)如果$n \neq \mathbb{M}$则$\mathbf{D}{1 n} \notin \mathbf{D}{2 n}$;
(iii) GCH持有的一般连续统假设。

(A)一个模型[3]，对于给定的$n \geq 3$，存在一个可数的非空的$\Pi_n^1$实数集，不包含OD元素，而每个可数的$\Sigma_n^1$实数集只包含OD实数;
(B)一个模型[28]，在这个模型中，对于一个给定的$n \geq 2$，有一个$\Pi_n^1$真实的单例，它有效地编码了一个cofinal映射$\omega \rightarrow \omega_1^{\mathbf{L}}$，最小超过$\mathbf{L}$，而每个$\Sigma_n^1$真实是可构造的;
(C)模型[29]，对于给定的$n \geq 2$，存在一个平面的不可均匀化光照面$\Pi_n^1$集，其垂直截面都是可计数的，而所有具有可计数截面的黑体字$\Sigma_n^1$集都是可$\Delta_{n+1}^1$均匀化的;
(D)模型[30]，其中对于给定$n \geq 3$，分离原则对于$\Pi_n^1$失效。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

Posted on Categories:Mathematical logic, 数学代写, 数理逻辑

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## 数学代写|数理逻辑入门代写Introduction To Mathematical logic代考|Further Reformulations and Harrington’s Statement

The problem emerged once again in the early years of forcing, especially in the case $m=1$ corresponding to analytic definability in second-order arithmetic. The early survey [20] by A. R. D. Mathias (the original typescript has been known to set theorists since 1968) contains Problem 3112, that requires finding a model of ZFC in which it is true that:
the set of analytically definable reals is analytically definable
that is, $\mathbf{D}{11} \in \mathbf{D}{21}$. Recall that reals in this context mean subsets of $\omega$. Another problem there, P 3110, suggests a sharper form of this statement, namely; find a model in which it is true that
analytically definable reals are precisely the constructible reals
that is, $\mathbf{D}{11}=\mathscr{P}(\omega) \cap \mathbf{L}$. The set $\mathscr{P}(\omega) \cap \mathbf{L}$ of all constructible reals is (lightface) $\Sigma_2^1$, and hence $\mathbf{D}{21}$, so that the equality $\mathbf{D}{11}=\mathscr{P}(\omega) \cap \mathbf{L}$ implies $\mathbf{D}{11} \in \mathbf{D}_{21}$, that is the case $m=1$ of the sentence (2).
Somewhat later, Problem 87 in Harvey Friedman’s survey One hundred and two problems in mathematical logic [21] requires to prove that for each $n$ in the domain $2<n \leq \omega$ there is a model of:
For $n \leq 2$ this is definitely impossible by the Shoenfield absoluteness theorem. As $\Delta_\omega^1$ is the same as $\mathbf{D}{11}=$ all analytically definable reals, the case $n=\omega$ in (3) is just a reformulation of $\mathbf{D}{11}=\mathscr{P}(\omega) \cap \mathbf{L}$.
At the very end of [21], it is noted that Leo Harrington had solved problem (3) affirmatively. A similar remark, see in [20] (p. 166), a comment to P 3110. And indeed, Harrington’s handwritten notes [22] present the following major result quoted here verbatim:
Theorem 1 (Harrington [22] (p. 1)). There are models of ZFC in which the set of constructible reals is, respectively, exactly the following set of reals:
$$\Delta_3^1, \Delta_4^1, \ldots, \Delta_\omega^1=\text { projective, } \Delta_n^m, 1 \leq n \leq \omega, 2 \leq m \leq \omega$$

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

The goal of this paper is to present a complete proof of the following part of Harrington’s statement in Theorem 1, related to the consistency of the Tarski sentence $\mathbf{D}{1 m} \in \mathbf{D}{2 m}$ and the equality $\mathbf{D}{1 m}=\mathscr{P}(\omega) \cap \mathbf{L}$, strengthened by extra claims (ii) and (iii). This is the main result of this paper. Theorem 2. Let $M \geq 1$. There is a generic extension of $\mathbf{L}$ in which it is true that (i) $\mathbf{D}{1 \mathbb{M}}=\mathscr{P}(\omega) \cap \mathbf{L}$, that is, constructible reals are precisely reals in $\mathbf{D}{1 \mathbb{M}}-$ in particular, $\mathbf{D}{1 \mathbb{M}}$ is a $\Sigma_2^1$ set, hence, $\mathbf{D}{1 M} \in \mathbf{D}{21}$, and even moreso, $\mathbf{D}{1 M} \in \mathbf{D}{2 M}$;
(ii) if $n \neq \mathbb{M}$ then $\mathbf{D}{1 n} \notin \mathbf{D}{2 n}$;
(iii) the general continuum hypothesis GCH holds.
Thus, for every particular $M \geq 1$, there exists a generic extension of $\mathbf{L}$ in which the Tarski sentence $\mathbf{D}{1 M} \in \mathbf{D}{2 M}$ holds whereas $\mathbf{D}{1 n} \notin \mathbf{D}{2 n}$ for all other values $n \neq \mathbb{M}$. We recall that $\mathbf{D}{1 M} \in \mathbf{D}{2 M}$ fails in $\mathbf{L}$ itself for all $M$, see above.

Corollary 1. If $M \geq 1$ then the sentence $\mathbf{D}{1 M} \in \mathbf{D}{2 M}$ is undecidable in $\mathbf{Z F C}$, even in the presence of $\forall n \neq \mathbb{M}\left(\mathbf{D}{1 n} \notin \mathbf{D}{2 n}\right)$.

This paper is dedicated to the proof of Theorem 2. This will be another application of the methods sketched by Harrington and developed in detail in our previous papers [4,5] in this Journal, but here modified and further developed for the purpose of a solution to the Tarski problem.

We may note that problems of construction of models of set theory in which this or another effect is obtained at a certain prescribed definability level (not necessarily the least possible one) are considered in modern set theory, see e.g., Problem 9 in [26] (Section 9) or Problem 11 in [27] (page 209). Some results of this type have recently been obtained in set theory, namely:
(A) a model [3] in which, for a given $n \geq 3$, there exists a countable non-empty $\Pi_n^1$ set of reals, containing no OD element, while every countable $\Sigma_n^1$ set of reals contains only OD reals;
(B) a model [28] in which, for a given $n \geq 2$, there is a $\Pi_n^1$ real singleton that effectively codes a cofinal map $\omega \rightarrow \omega_1^{\mathbf{L}}$, minimal over $\mathbf{L}$, while every $\Sigma_n^1$ real is constructible;
(C) a model [29] in which, for a given $n \geq 2$, there exists a planar non-ROD-uniformizable lightface $\Pi_n^1$ set, all of whose vertical cross-sections are countable, whereas all boldface $\Sigma_n^1$ sets with countable cross-sections are $\Delta_{n+1}^1$-uniformizable;
(D) a model [30] in which, for a given $n \geq 3$, the Separation principle fails for $\Pi_n^1$.

## 数学代写|数理逻辑入门代写Introduction To Mathematical logic代考|Further Reformulations and Harrington’s Statement

$$\Delta_3^1, \Delta_4^1, \ldots, \Delta_\omega^1=\text { projective, } \Delta_n^m, 1 \leq n \leq \omega, 2 \leq m \leq \omega$$

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

(ii)如果$n \neq \mathbb{M}$则$\mathbf{D}{1 n} \notin \mathbf{D}{2 n}$;
(iii) GCH持有的一般连续统假设。

(A)一个模型[3]，对于给定的$n \geq 3$，存在一个可数的非空的$\Pi_n^1$实数集，不包含OD元素，而每个可数的$\Sigma_n^1$实数集只包含OD实数;
(B)一个模型[28]，在这个模型中，对于一个给定的$n \geq 2$，有一个$\Pi_n^1$真实的单例，它有效地编码了一个cofinal映射$\omega \rightarrow \omega_1^{\mathbf{L}}$，最小超过$\mathbf{L}$，而每个$\Sigma_n^1$真实是可构造的;
(C)模型[29]，对于给定的$n \geq 2$，存在一个平面的不可均匀化光照面$\Pi_n^1$集，其垂直截面都是可计数的，而所有具有可计数截面的黑体字$\Sigma_n^1$集都是可$\Delta_{n+1}^1$均匀化的;
(D)模型[30]，其中对于给定$n \geq 3$，分离原则对于$\Pi_n^1$失效。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

Posted on Categories:Mathematical logic, 数学代写, 数理逻辑

## avatest™帮您通过考试

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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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