## 数学代写|数理逻辑代写Mathematical logic代考|MATH4810

2023年1月6日

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## 数学代写|数理逻辑代写Mathematical logic代考|Substitution

In this section we define how to substitute a term $t$ for a variable $x$ in a formula $\varphi$ at the places where $x$ occurs free, thus obtaining a formula $\psi$. We wish to define the substitution so that $\psi$ expresses the same about $t$ as $\varphi$ does about $x$. We start with an example to illustrate our objective and to show why a certain care is necessary. Let
$$\varphi:=\exists z z+z \equiv x .$$
In $\mathfrak{N}$ the formula $\varphi$ says that $x$ is even; more precisely:
$$(\mathfrak{N}, \beta) \models \varphi \quad \text { iff } \quad \beta(x) \text { is even. }$$
If we replace the variable $x$ by $y$ in $\varphi$, we obtain the formula $\exists z z+z \equiv y$, which states that $y$ is even. But if we replace the variable $x$ by $z$, we obtain the formula $\exists z z+z \equiv z$, which no longer says that $z$ is even; in fact, this formula is valid in $\mathfrak{N}$ regardless of the assignment for $z$ (because $0+0=0$ ). In this case the meaning is altered because at the place where $x$ occurred free, the variable $z$ gets bound. On the other hand, we obtain a formula which expresses the same about $z$ as $\varphi$ does about $x$ if we proceed as follows: First, we introduce a new bound variable $u$ in $\varphi$, and then in the formula $\exists u u+u \equiv x$ thus obtained we replace $x$ by $z$. It is immaterial which variable $u$ (distinct from $x$ and $z$ ) we choose. However, for certain technical purposes it is useful to make a fixed choice.

In the preceding example we replaced only one variable, but in our exact definition we specify the procedure for simultaneously replacing several variables: With a given formula $\varphi$, pairwise distinct variables $x_0, \ldots, x_r$ and arbitrary terms $t_0, \ldots, t_r$, we associate a formula $\varphi \frac{t_0 \ldots t_r}{x_0 \ldots x_r}$, which is said to be obtained from $\varphi$ by simultaneously substituting $t_0, \ldots, t_r$ for $x_0, \ldots, x_r$. The reader should note that $x_i$ has to be replaced by $t_i$ only if
$$x_i \in \operatorname{free}(\varphi) \quad \text { and } \quad x_i \neq t_{i \cdot}$$
In the following inductive definition this is explicitly taken into account in the quantifier step; in the other steps it follows immediately.

## 数学代写|数理逻辑代写Mathematical logic代考|Sequent Rules

A mathematical proof proceeds from one statement to the next until it finally arrives at the assertion of the theorem in question. The individual statements depend on certain hypotheses. These can either be hypotheses of the theorem or additional hypotheses temporarily assumed in the course of the proof. For example, if one wants to prove an intermediate claim $\varphi$ by contradiction, one adds $\neg \varphi$ to the hypotheses; if a contradiction results, then $\varphi$ has been proved, and the additional assumption $\neg \varphi$ is dropped.

This observation leads us to describe a stage in a proof by listing the corresponding assumptions and the respective claim. If we call a nonempty list (sequence) of formulas a sequent, then we can use sequents to describe “stages in a proof”. For instance, the “stage” with assumptions $\varphi_1, \ldots, \varphi_n$ and claim $\varphi$ is rendered by the sequent $\varphi_1 \ldots \varphi_n \varphi$. The sequence $\varphi_1 \ldots \varphi_n$ is called the antecedent and $\varphi$ the succedent of the sequent $\varphi_1 \ldots \varphi_n \varphi$. From Lemma II.4.3 it follows that the formulas which constitute a sequent are uniquely determined. In particular, the antecedent and the succedent are well-defined.

In terms of sequents, the indirect proof sketched above can be represented schematically as follows:
$$\begin{array}{lrr} \varphi_1 \ldots \varphi_n & \neg \varphi & \psi \ \varphi_1 \ldots \varphi_n & \neg \varphi & \neg \psi \ \hline \varphi_1 \ldots \varphi_n & \varphi \end{array}$$
Thus (+) describes the following argument: If under the assumptions $\varphi_1, \ldots, \varphi_n$ and (the additional assumption) $\neg \varphi$ one can obtain both the formula $\psi$ and its negation $\neg \psi$ (that is, a contradiction), then from the assumptions $\varphi_1, \ldots, \varphi_n$ one can infer $\varphi$.

In the following we shall use the letters $\Gamma, \Delta, \ldots$ to denote (possibly empty) sequences of formulas. Then we can write sequents as $\Gamma \varphi \psi, \Delta \psi, \ldots$ and the scheme (+) in the form

As in ( $+)$, we use spaces between elements in a sequent merely for easier reading.

# 数理逻辑代写

## 数学代写|数理逻辑代写Mathematical logic代考|Substitution

$$\varphi:=\exists z z+z \equiv x$$

$(\mathfrak{N}, \beta) \models \varphi \quad$ iff $\quad \beta(x)$ is even.

$$x_i \in \operatorname{free}(\varphi) \text { and } x_i \neq t_i .$$

## 数学代写|数理逻辑代写Mathematical logic代考|Sequent Rules

Ibegin{array}{Irr} Ivarphi_1 \ddots Ivarphi_n \& Ineg Ivar

## 有限元方法代写

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## 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。