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# 数学代写|数理逻辑代考Mathematical logic代写|HPH203 A Preliminary Analysis

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## 数学代写|数理逻辑代考Mathematical logic代写|A Preliminary Analysis

We now sketch some aspects which the two examples just given have in common.
In each case one starts from a system $\Phi$ of propositions which is taken to be a system of axioms for the theory in question (group theory, theory of equivalence relations). The mathematician is interested in finding the propositions which follow from $\Phi$, where the proposition $\psi$ is said to follow from $\Phi$ if $\psi$ holds in every structure which satisfies all propositions in $\Phi$. A proof of $\psi$ from a system $\Phi$ of axioms shows that $\psi$ follows from $\Phi$.

When we think about the scope of methods of mathematical proof, we are led to ask about the converse:
$(*)$ Is every proposition $\psi$ which follows from $\Phi$ also provable from $\Phi$ ?
For example, is every proposition which holds in all groups also provable from the group axioms (G1), (G2), and (G3)?

The material developed in Chapters II through V and in Chapter VII yields an essentially positive answer to $()$. Clearly it is necessary to make the concepts “proposition”, “follows from”, and “provable”, which occur in $()$, more precise. We sketch briefly how we shall do this.
(1) The Concept “Proposition.” Usually mathematicians use their everyday language (e.g., English or German) to formulate their propositions. But since sentences in everyday language are not, in general, completely unambiguous in their meaning and structure, one cannot specify them by precise definitions. For this reason we shall introduce a formal language $L$ which reflects features of mathematical statements. Like programming languages used today, $L$ will be formed according to fixed rules: Starting with a set of symbols (an “alphabet”), we obtain so-called formulas as finite symbol strings built up in a standard way. These formulas correspond to propositions expressed in everyday language. For example, the symbols of $L$ will include $\forall$ (to be read “for all”), $\wedge$ (“and”), $\rightarrow$ (“if . . then”), 三 (“equal”) and variables like $x, y$ and $z$. Formulas of $L$ will be expressions like
$$\forall x x \equiv x, \quad x \equiv y, \quad x \equiv z, \quad \forall x \forall y \forall z((x \equiv y \wedge y \equiv z) \rightarrow x \equiv z) .$$
Although the expressive power of $L$ may at first appear to be limited, we shall later see that many mathematical propositions can be formulated in $L$. We shall even see that $L$ is, in principle, sufficient for all of mathematics. The definition of $L$ will be given in Chapter II.
(2) The Concept “Follows From” (the Consequence Relation). Axioms (G1), (G2), and (G3) of group theory obtain a meaning when interpreted in structures of the form $\left(G, \circ^{G}, e^{G}\right)$. In an analogous way we can define the general notion of an $L$-formula holding in a structure. This enables us (in Chapter III) to define the consequence relation: $\psi$ follows from (is a consequence of) $\Phi$ if and only if $\psi$ holds in every structure where all formulas of $\Phi$ hold.
(3) The Concept “Proof.” A mathematical proof of a proposition $\psi$ from a system $\Phi$ of axioms consists of a series of inferences which proceed from axioms of $\Phi$ or propositions that have already been proved, to new propositions, and which finally ends with $\psi$. At each step of a proof mathematicians write something like “From … and $\ldots .$ one obtains directly that $\sim \sim \sim$,” and they expect it to be clear to anyone that the validity of $\ldots$ and of $\ldots-$ entails the validity of $\sim \sim \sim$.

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

By an alphabet $\mathbb{A}$ we mean a nonempty set of symbols. Examples of alphabets are the sets $\mathbb{A}{1}={0,1,2, \ldots, 9}, \mathbb{A}{2}={a, b, c, \ldots, z}$ (the alphabet of lower-case letters), $\mathbb{A}{3}=\left{0, \int, a, d, x, f,\right),(}$, and $\mathbb{A}{4}=\left{c_{0}, c_{1}, c_{2}, \ldots\right}$.

We call finite sequences of symbols from an alphabet $\mathbb{A}$ strings or words over $\mathbb{A}$. By $\mathbb{A}^{}$ we denote the set of all strings over $\mathbb{A}$. The length of a string $\zeta \in \mathbb{A}^{}$ is the number of symbols, counting repetitions, occurring in $\zeta$. The empty string is also considered to be a word over $\mathbb{A}$. It is denoted by $\square$, and its length is zero.
Examples of strings over $\mathbb{A}{2}$ are $$\text { softly, xdbxaz. }$$ Examples of strings over $\mathbb{A}{3}$ are
$$\int f(x) d x, \quad x \circ \iint a .$$
Suppose $\mathbb{A}={\mid, |}$, that is, $\mathbb{A}$ consists of the symbols $a_{1}:=\left.\right|^{1}$ and $a_{2}:=|$. Then the string || $\mid$ over $\mathbb{A}$ can be read in three ways: as $a_{1} a_{1} a_{1}$, as $a_{1} a_{2}$, and as $a_{2} a_{1}$. In the sequel we allow only those alphabets $\mathbb{A}$ where any string over $\mathbb{A}$ can be read in exactly one way. The alphabets $\mathbb{A}{1}, \ldots, \mathbb{A}{4}$ given above satisfy this condition.

# 数理逻辑代写

## 数学代写数理逻辑代考Mathematical logic代写|A Preliminary Analysis

$\Phi$ ，其中命题 $\psi$ 据兑从 $\Phi$ 如果 $\psi$ 在满足所有命题的每一个结构中都成立 $\Phi$.个证明 $\psi$ 从一个系统 $\Phi$ 的公理表明 $\psi$ 从 $\Phi .$

$(*)$ 是每一个命题 $\psi$ 綮随其后的是 $\Phi$ 也可从 $\Phi$ ?

(1) 概念”命题”。通常数学家使用他们的日常语言 (例如，英语或德语) 来制定他们的命题。但是，由于日常语言中的句子通常在 含义和咭构上并不是完全明确的，因此无法通过精确的定义来具体说明它们。出于这个原因，我们将介绍一种形式语言 $L$ 它反映了 数学炼述的特点。就像今天使用的编程语言一样，L将根据固定规则形成: 从一组符号 (“字母”) 开始，我们获得所谊的公式，即 以标准方式构建的有限符号字符串。这些公式对应于用日常语言表达的命题。例如，符号 $L$ 会包括 $V$ (读作”为所有人”)， $\wedge$
$$\forall x x \equiv x, \quad x \equiv y, \quad x \equiv z, \quad \forall x \forall y \forall z((x \equiv y \wedge y \equiv z) \rightarrow x \equiv z) .$$

(2) “源于”的概念 (后果关系)。群的公理 $(G 1) 、(G 2)$ 和 (G3) 在以下列形解释时获得意义 $\left(G, o^{G}, e^{G}\right)$ ). 以类似的方式， 我们可以定义一个般概念 $L$ – 保存在结构中的公式。这使我们 (在第三章) 能哆定义结果关系: $\psi$ 来自 (是的结果) $\Phi$ 当且仅当 $\psi$ 适用于所有公式的每个结构 $\Phi$ 抓住。
(3) 概今证明”。命题的数学证明 $\psi$ 从一个䒺统 $\Phi$ 公理由一系列从公理得出的推论组成 $\Phi$ 或已经被证明的命题, 到新的命题, 最后 $\sim \sim \sim$.

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

\left 的分隔符缺失或无法识别
，和 \eft 的分隔符矩失或无法识别

softly, xdbxaz.

$$\int f(x) d x, \quad x \circ \iint a .$$

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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。