### 数学代写|数理逻辑代写Mathematical logic代考|MATH318

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|数理逻辑代写Mathematical logic代考|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 he 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”), $\equiv($ “equal”) and variables like $x, y$ and $z$. Formulas of $L$ will be expressions like
$$\forall x x \equiv x, \quad x \equiv y, \quad x \equiv z, \quad \forall x \forall y \forall z((x \equiv y \wedge y \equiv z) \rightarrow x \equiv z) .$$

## 数学代写|数理逻辑代写Mathematical logic代考|The Alphabet of a First-Order Language

We wish to construct formal languages in which we can formulate, for example, the axioms, theorems, and proofs about groups and equivalence relations which we considered in Chapter I. In that context the connectives, the quantifiers, and the equality relation played an important role. Therefore, we shall include the following symbols in the first-order languages: $\neg$ (for “not”), $\wedge$ (for “and”), $\vee$ (for “or”), $\rightarrow$ (for “ifthen”), $\leftrightarrow$ (for “if and only if”), $\forall$ (for “for all”), $\exists$ (for “there exists”), 三 (as symbol for equality). To these we shall add variables (for elements of groups, elements of equivalence structures, etc.) and, finally, parentheses as auxiliary symbols.

To formulate the axioms for groups we also need certain symbols specific to group theory, e.g., a binary function symbol, say $\circ$, to denote the group multiplication, and a symbol, say $e$, to denote the identity element. We call $e$ a constant symbol, or simply a constant. For the axioms of the theory of equivalence relations we need a binary relation symbol, say $R$.

Thus, in addition to the “logical” symbols such as ” $\neg$ ” and ” $\wedge$ “, we need a set $S$ of relation symbols, function symbols, and constants which varies from theory to theory. Each such set $S$ of symbols determines a first-order language. We summarize:

By $\mathbb{A}$ we denote the set of symbols listed in (a) through (e). Let $S$ be the (possibly empty) set of symbols from (f). The symbols listed under (f) must, of course, be distinct from each other and from the symbols in $\mathbb{A}$.

The set $S$ determines a first-order language (cf. Section 3). We call $\mathbb{A}_S:=\mathbb{A} \cup S$ the alphabet of this language and $S$ its symbol set.

We have already become acquainted with some symbol sets: $S_{\mathrm{gr}}:={0, e}$ for group theory and $S_{\mathrm{eq}}:={R}$ for the theory of equivalence relations. For the theory of ordered groups we could use ${0, e, R}$, where the binary relation symbol $R$ is now taken to represent the ordering relation. In certain theoretical investigations we shall use the symbol set $S_{\infty}$, which contains the constants $c_0, c_1, c_2, \ldots$, and for every $n \geq 1$ countably many $n$-ary relation symbols $R_0^n, R_1^n, R_2^n, \ldots$ and $n$-ary function symbols $f_0^n, f_1^n, f_2^n, \ldots$

Henceforth we shall use the letters $P, Q, R, \ldots$ for relation symbols, $f, g, h, \ldots$ for function symbols, $c, c_0, c_1, \ldots$ for constants, and $x, y, z, \ldots$ for variables.

# 数理逻辑代写

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

$\left(^*\right)$ 是否每个合题 $\psi$ 从 $\Phi$ 也可以证明 $\Phi$ ?

(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) 来表达他们的命题。但是，由于日 常语言中的句子通常在含义和结构上并非完全没有歧义，因此无法通过精确的定义来指定它们。为此，我 们将引入一种形式语言 $L$ 反映了数学陈述的特点。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 (“字母表”) 开始，我们获得所谓的公式，作为以标准方式构建的有限符号串。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 $\forall$ (读作”为所有人”)， $\wedge($ “和”)， $\rightarrow($ “如果……那 么”)，三(“等于”) 和变量，如 $x, y$ 和 $z$. 的公式 $L$ 会像这样的表达
$$\forall x x \equiv x, \quad x \equiv y, \quad x \equiv z, \quad \forall x \forall y \forall z((x \equiv y \wedge y \equiv z) \rightarrow x \equiv z)$$

## 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。