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

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

avatest™

## 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

•200+ 英语母语导师

•70分以下全额退款

## 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。