# 数学代写|数理逻辑代写Mathematical logic代考|MATH230

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## 数学代写|数理逻辑代写Mathematical logic代考|Num and SUB Are Recursive

To write a $\Delta$-definition $\operatorname{Num}(a, y)$ we will start, once again, with a construction sequence, but this time we will construct the numeral $\bar{a}$. For $a=2$, the construction sequence is
$$\langle 0, S 0, S S 0\rangle,$$
which gives rise to the sequence of Gödel numbers
$$\left\langle 2^9, 2^{11} 3^{2^9}, 2^{11} 3^{2^{11} 3^{\left(2^9\right)}}\right\rangle,$$
which is coded by the number
$$c=2^{\left[{ }^9\right]} 3^{\left[2^{11} 3^{2^9}\right]_5} 5^{113^{11} 2^{\left(2^9\right)}} 1 .$$
Notice that the length of the construction sequence here is 3 , and in general the construction sequence will have length $a+1$ if we seek to code the construction of the Gödel number of the numeral associated with the number $a$. (That is a very long sentence, but it does make sense if you work through it carefully.)
You are asked in Exercise 1 to write down the formula
NumConstructionSequence $(c, a, y)$
as a $\Delta$-formula. The idea is that
$$\mathfrak{N} \models \text { NumConstructionSequence }(c, a, y)$$
if and only if $c$ is the code for a construction sequence of length $a+1$ with last element $y=r \bar{a}$.

## 数学代写|数理逻辑代写Mathematical logic代考|Definitions by Recursion Are Recursive

If you look at the definition of a term (Definition 1.3.1), the definition of formula (Definition 1.3.3), the definition of $u_t^x$ (Definition 1.8.1), and the definition of $\phi_t^x$ (Definition 1.8.2), you will notice that all of these definitions were definitions “by recursion.” For example, in the definition of a term, you see the phrase
$\ldots t$ is $f t_1 t_2 \ldots t_n$, where $f$ in an $n$-ary function symbol of $\mathcal{L}$ and each of the $t_2$ are terms of $\mathcal{L}$
so a term can have constituent parts that are themselves terms. In the last two sections we have used the device of construction sequences to show that the sets TERM, FORMULA, TERMSUB, and SUB are recursive sets. In this section we outline a proof that all such sets of strings, defined “by recursion,” give rise to sets of Gödel · numbers that are recursive. It will be clear from our exposition that a more general statement of our theorem could be proved, but what we present will be sufficient for our needs.

Definition 4.10.1. A string of symbols $s$ from a first-order language $\mathcal{L}$ is called an expression if $s$ is either a term of $\mathcal{L}$ or a formula of $\mathcal{L}$.

# 数理逻辑代写

## 数学代写|数理逻辑代写Mathematical logic代考|Num and SUB Are Recursive

$$\langle 0, S 0, S S 0\rangle,$$

$$\left\langle 2^9, 2^{11} 3^{2^9}, 2^{11} 3^{2^{11} 3^{\left(2^9\right)}}\right\rangle,$$

$$c=2^{\left[{ }^9\right]} 3^{\left[2^{11} 3^{2^9}\right]_5} 5^{113^{11} 2^{\left(2^9\right)}} 1 .$$

NumConstructionSequence $(c, a, y)$

$$\mathfrak{N} \models \text { NumConstructionSequence }(c, a, y)$$

## 数学代写|数理逻辑代写Mathematical logic代考|Definitions by Recursion Are Recursive

$\ldots t$是$f t_1 t_2 \ldots t_n$，其中$f$在$\mathcal{L}$的$n$ – any函数符号中，每个$t_2$都是$\mathcal{L}$的项

4.10.1.定义如果$s$是$\mathcal{L}$的一项或$\mathcal{L}$的公式，则一阶语言$\mathcal{L}$中的符号字符串$s$称为表达式。

## 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。

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