# Blog

## 我所收藏的硬币

1：澳大利亚

1.1 两元硬币

1.1.1 形态

1.1.2 背后的一些东东

1.2 50分硬币

1.2.1 形态

50分硬币正面是澳大利亚的国徽，左边袋鼠，右边鸸鹋，都是澳大利亚特有的动物。最上面有一颗七角星，中心是一块盾牌，上面刻制了六个州的徽章，表明六个州联合组建成了澳大利亚联邦，背景是澳大利亚的国花金合欢，下面是大大的“50”。背面和两元硬币一样。需要注意的是：这个硬币是十二边形的。

1.2.2 背后的一些东东

1.3 20分硬币

1.3.1 形态

20分硬币正面是鸭嘴兽在水里游泳，泛起一阵一阵的水波，右上角有个大大的“20”。背面和其他硬币一样。

1.3.2 背后的一些东东

1.4 10分硬币

1.4.1 形态

1.4.2 背后的一些东东

1.5 5分硬币

1.5.1 形态

5分硬币正面的图案是澳大利亚特有的珍稀动物针鼹。这只针鼹团作一团，样子很可爱。下方有一个“5”的字样。背面和之前的硬币一样。

1.5.2 背后的一些东东

1.6 一元硬币（我木有）

1.6.1 形态

1.6.2 背后的一些东东

1.7 一个奇怪的变异了的一元硬币（求大神指教）

1.7.1 形态

1.7.2 背后的一些东东

2：美国

2.1 一元硬币

2.1.1 形态

12.5 50分（五毫）硬币

12.5.1 形态

12.5.2 背后的一些东东

12.6 20分（贰毫）硬币

12.6.1 形态

12.6.2 背后的一些东东

12.7 10分（壹毫）硬币

12.7.1 形态

12.7.2 背后的一些东东

13：人民币

13.1 1元硬币

13.1.1 形态

13.1.2 背后的一些东东

13.2 5角硬币

13.2.1 形态

13.2.2 背后的一些东东

13.3 1角硬币

13.3.1 形态

13.3.2 背后的一些东东

14：土耳其里拉

14.1 1里拉硬币

14.1.1 形态

14.1.2 背后的一些东东

14.2 50库鲁硬币

14.2.1 形态

14.2.2 背后的一些东东

14.3 25库鲁硬币

14.3.1 形态

14.3.2 背后的一些东东

14.4 10库鲁硬币

14.4.1 形态

14.4.2 背后的一些东东

14.5 5库鲁硬币

14.5.1 形态

14.5.2 背后的一些东东

15：以色列谢克尔

15.1 1/2谢克尔

15.1.1 形态

15.1.2 背后的一些东东

15.2 10阿哥拉硬币

15.2.1 形态

15.2.2 背后的一些东东

15.3 10新阿哥拉硬币

15.3.1 形态

15.3.2 背后的一些东东

15.4 5新阿哥拉硬币

15.4.1 形态

15.4.2 背后的一些东东

15.5 2新阿哥拉硬币

15.5.1 形态

15.5.2 背后的一些东东

15.6 1新阿哥拉硬币

15.6.1 形态

15.6.2 背后的一些东东

16：捷克克朗

16.1 概述

16.2 介绍

17：俄罗斯卢布与戈比

17.1 概述

17.2 介绍

18：杂币

18.1.1 挪威10克朗硬币

18.1.2 挪威国王哈拉尔五世

18.2.1 100韩元硬币

18.2.2 李舜臣

18.3.1 500印度尼西亚卢比

18.3.2 印度尼西亚国徽

18.4.1 澳门5元硬币

18.4.2 大三巴牌坊

18.5.1 罗马尼亚巴尼们

50巴尼，正面是字样“50 BANI”，10巴尼，正面是字样“10 BANI”，5巴尼，正面是字样“5 BANI”，反面都是“ROMANIA”和年份加上国徽。

18.5.2 罗马尼亚国徽

18.6.1 10菲律宾比索硬币

18.6.2 俩人物

Apolinario Mabini和Maranan（1864年7月23日 – 1903年5月13日）是菲律宾革命领袖，教育家，律师和政治家，他首先担任革命政府的法律和宪法顾问，然后担任第一任总理。菲律宾成立第一个菲律宾共和国。他被视为“utak ng himagsikan”或“革命的大脑”。他的两部作品，El Verdadero Decalogo（真正的十诫，1898年6月24日）和Programa Constitucional dela Republica Filipina（菲律宾共和国的宪法纲领，1898年）在起草最终被称为马洛洛斯的作品方面发挥了重要作用。尽管已经失去了使用他的双腿都小儿麻痹症马比尼执行的所有他的革命和政府的活1896年菲律宾革命前不久。马比尼在菲律宾历史上的角色使他在菲律宾革命的开放日期面对西班牙的第一次殖民统治，然后在菲律宾 – 美国战争时期进行美国殖民统治。后者看到马比尼被美国殖民当局俘虏并流放到关岛，只能在1903年5月最终死亡前两个月返回

18.7.1 阿联酋迪拉姆

18.7.2 阿拉伯联合酋长国

18.8.1 马来西亚仙们

18.8.2 朱瑾

18.9.1 10德国芬尼硬币

18.9.2 德国

18.10.1 印度卢比们

18.10.2 印度

18.11.1 伊朗硬币，无法描述

2018年9月2日于家中

## How to define membership relation from subset relation and power set operation

In my talk at 2018 Chinese Mathematical Logic Conference, I asked if $$(V,\subset,P)$$ is epsilon-complete, namely if the membership relation can be recovered in the reduct. Professor Joseph S. Miller approached to me during the dinner and pointed out that it is epsilon-complete. Let me explain how.

Theorem

Let $$(V,\in)$$ be a structure of set theory, $$(V,\subset,P)$$ is the structure of the inclusion relation and the power set operation, which are defined in $$(V,\in)$$ as usual. Then $$\in$$ is definable in $$(V,\subset,P)$$.

Proof.

Fix a set $$x$$. Define $$y$$ to be the $$\subset$$-least such that

$\forall z \big((z\subset x\wedge z\neq x)\rightarrow P(z)\subset y\big).$

Actually, $$y=P(x)-\{x\}$$, so $$\{x\}= P(x) – y$$. Since set difference can be defined from subset relation and $$(V,\subset,\{x\})$$ can define $$\in$$, we are done.

$$\Box$$

Here is another argument figured out by Jialiang He and me after we heard Professor Miller’s Claim.

Proof.
Since $$\in$$ can be defined in $$(V,\subset,\bigcup)$$ (see the slides). Fix a set $$A$$, it suffices to show that we can define $$\bigcup A$$ from $$\subset$$ and $$P$$.

Let $$B$$ be the $$\subset$$-least set such that there is $$c$$, $$B=P(c)$$ and $$A\subset B$$. Note that
$\bigcap\big\{P(d)\bigm|A\subset P(d)\big\}= P\big(\bigcap\big\{d\bigm|A\subset P(d)\big\}\big).$
Therefore, $$B$$ is well-defined. Next, we show that
$\bigcap\big\{d\bigm|A\subset P(d)\big\}=\bigcup A.$
Clearly, $$A\subset P(\bigcup A)$$. This proves the direction from left to right. For the other direction, if $$x$$ is in an element of $$A$$, then it is in an element of $$P(d)$$ given $$A\subset P(d)$$, i.e. it is an element of such $$d$$.

Therefore $$\bigcup A$$ is the unique set whose power set is $$B$$.

$$\Box$$

## Set Theory II (Forcing) 2018

Lecture: HGW2403, T 18:30-21
Section: HGW2403, R 18:30-20

Syllabus

#### Instruction for the final paper

You can choose one of the following topics.

1. An introduction to one specific forcing notion from this list (for Cohen forcing you can introduce an application not discussed in our course) and explain how it works.
2. On one or more issues concerning the meta-theory of forcing. For example, why and how we can talk about an “outer model” or a “object” not in our universe, why we can and why we need to assume the existence of a transitive model, how to account forcing arguments as purely constructive methods, etc.

#### Problem set 01

1. Let $$\pi$$ be the canonical interpretation of PA into ZF. Can we prove “for each arithmetic formula $$\varphi$$, if ZFC $$\vdash\pi(\varphi)$$, then PA $$\vdash\varphi$$”? Prove it if we can, explain it if we cannot.
2. Assume ZF $$\vdash\varphi^L$$ for each formula $$\varphi\in\Sigma$$, and $$\Sigma\vdash\psi$$. Show that ZF $$\vdash\psi^L$$.
3. Why Con(ZF) does not imply there is a countable transitive model of ZF?

#### Problem set 02

Kunen’s set theory (2013) Exercise I.16.6 – I.16.10, I.16.17.

#### Problem set 03

Kunen’s set theory (2013) Exercise II.4.6, 4.8.

Jech’s set theory (2002) Exercise 7.1, 7.3 – 7.5, 7.13, 7.16, 7.18 – 7.20, 7.22 – 7.33.

#### Problem set 04

1. Let $$M^\mathbb{B}$$ be a Boolean valued model. Prove the following statements are valid in $$M^\mathbb{B}$$.
• $$\forall y\big(\forall x\varphi(x)\rightarrow\varphi(y)\big)$$.
• $$\forall x(\varphi\rightarrow\psi)\rightarrow\forall x\varphi\rightarrow\forall x\psi$$.
• $$\alpha\rightarrow\forall x\alpha$$, $$x$$ does not occur in $$\alpha$$ freely.

Jech’s set theory (2002) Exercise 14.12.

#### Problem set 05

1. Let $$\sigma$$ be a $$\mathbb{B}$$-name. Show that $|\!|\exists x\in\sigma~\varphi(x)|\!| = \sum_{\xi\in\textrm{dom}\sigma}\sigma(\xi)\cdot|\!|\varphi(\xi)|\!|.$
2. For any partial order $$\mathbb{P}$$, there is a separative partial order $$\mathbb{Q}$$ and a surjection $$h:\mathbb{P}\to\mathbb{Q}$$ such that
• $$x\leq y$$ implies $$h(x)\leq h(y)$$;
• $$x$$ and $$y$$ are compatible in $$\mathbb{P}$$ if and only if $$h(x)$$ and $$h(y)$$ are compatible in $$\mathbb{Q}$$.

Such $$\mathbb{Q}$$ is unique up to isomorphism. We call it the separative quotient of $$\mathbb{P}$$.

Jech’s set theory (2002) Exercise 14.1, 14.9, 14.14, 14.16. Lemma 14.13.

## Modal Logic 2018

Lecture: HGX205, M 18:30-21
Section: HGW2403, F 18:30-20

Syllabus

#### Exercise 01

1. Prove that $$\neg\Box(\Diamond\varphi\wedge\Diamond\neg\varphi)$$ is equivalent to $$\Box\Diamond\varphi\rightarrow\Diamond\Box\varphi$$. What you have assumed?
2. Define strategy and winning strategy for modal evaluation games. Prove Key Lemma: $$M,s\vDash\varphi$$ iff V has a winning strategy in $$G(M,s,\varphi)$$. Prove that modal evaluation games are determined, i.e. either V or F has a winning strategy.

And all exercises for Chapter 2 (see page 23, open minds)

#### Exercise 02

1. Let $$T$$ with root $$r$$ be the tree unraveling of some possible world model, and $$T’$$ be the tree unraveling of $$T,r$$. Show that $$T$$ and $$T’$$ are isomorphic.
2. Prove that the union of a set of bisimulations between $$M$$ and $$N$$ is a bisimulation between the two models.
3. We define the bisimulation contraction of a possible world model $$M$$ to be the “quotient model”. Prove that the relation links every world $$x$$ in $$M$$ to the equivalent class $$[x]$$ is a bisimulation between the original model and its bisimulation contraction.

And exercises for Chapter 3 (see page 35, open minds): 1 (a) (b), 2.

#### Exercise 03

1. Prove that modal formulas (under possible world semantics) have ‘Finite Depth Property’.

And exercises for Chapter 4 (see page 47, open minds): 1 – 3.

#### Exercise 04

1. Prove the principle of Replacement by Provable Equivalents: if $$\vdash\alpha\leftrightarrow\beta$$, then $$\vdash\varphi[\alpha]\leftrightarrow\varphi[\beta]$$.
2. Prove the following statements.
• “For each formula $$\varphi$$, $$\vdash\varphi$$ is equivalent to $$\vDash\varphi$$” is equivalent to “for each formula $$\varphi$$, $$\varphi$$ being consistent is equivalent to $$\varphi$$ being satisfiable”.
• “For every set of formulas $$\Sigma$$ and formula $$\varphi$$, $$\Sigma\vdash\varphi$$ is equivalent to $$\Sigma\vDash\varphi$$” is equivalent to “for every set of formulas $$\Sigma$$, $$\Sigma$$ being consistent is equivalent to $$\Sigma$$ being satisfiable”.
3. Prove that “for each formula $$\varphi$$, $$\varphi$$ being consistent is equivalent to $$\varphi$$ being satisfiable” using the finite version of Henkin model.

And exercises for Chapter 5 (see page 60, open minds): 1 – 5.

#### Exercise 05

Exercises for Chapter 6 (see page 69, open minds): 1 – 3.

#### Exercise 06

1. Show that “being equivalent to a modal formula” is not decidable for arbitrary first-order formulas.

Exercises for Chapter 7 (see page 88, open minds): 1 – 6. For exercise 2 (a) – (d), replace the existential modality E with the difference modality D. In the clause (b) of exercise 4, “completeness” should be “correctness”.

#### Exercise 07

1. Show that there are infinitely many non-equivalent modalities under T.
2. Show that GL + Id is inconsistent and Un proves GL.
3. Give a complete proof of the fact: In S5, Every formula is equivalent to one of modal depth $$\leq 1$$.

Exercises for Chapter 8 (see page 99, open minds): 1, 2, 4 – 6.

#### Exercise 08

1. Let $$\Sigma$$ be a set of modal formulas closed under substitution. Show that $(W,R,V),w\vDash\Sigma~\Leftrightarrow~ (W,R,V’),w\vDash\Sigma$ hold for any valuation $$V$$ and $$V’$$. Define a $$p$$-morphism between $$(W,R),w$$ and $$(W’,R’),w’$$ as a “functional bisimulation”, namely bisimulation regardless of valuation. Show that if there is a $$p$$-morphism between $$(W,R),w$$ and $$(W’,R’),w’$$, then for any valuation $$V$$ and $$V’$$, we have $(W,R,V),w\vDash\Sigma~\Leftrightarrow~ (W’,R’,V’),w\vDash\Sigma.$

Exercises for Chapter 9 (see page 99, open minds).

#### Exercise the last

Exercises for Chapter 10 and 11 (see page 117 and 125, open minds).