Original version:

(第一段)

“As truth seekers,” that’s how I view mathematicians. The research of the foundations of mathematics, then, is an investigation of the essence of truth. With extreme zeal to discover truth deep in my soul for so long, I dream to be an excellent mathematical logician with great contributions to this field.

//xination: 開頭是還可以。但是用字上有些太誇張, 像是 extreme zeal.

(第二段)

My math journey started from the elegant Euler’s identity, e^(iπ) + 1= 0.

Just like any great wisdom, this formula conveys a significant amountof knowledge in only a handful of symbols: the five most important constants0, 1, π, i and e (roughly for the different fields of arithmetics, geometry, algebra and analysis, respectively) connected by equality using the three most fundamental operations addition, multiplication and exponentiation.

//xination: 盡量不要使用(),不重要的資訊就不要加來，重要的資訊就不要用括號。這邊以經是第二段了，但沒有切入更深的討論作者的經歷，反而再說general fact, 這是不必要的資訊，串場的篇幅太大。

(第三段)

Inspiration by Euler’s identity induced my appreciation of the beautyof math. In my opinion, math is the purest form of human knowledge. What makes math so unique and different from other disciplines such as natural science, I believe, is essentially mathematicians’ confirmation of a statement as true by means of a proof, i.e. a process of step-by-step sound reasoning starting from trivial basic principles called axioms.

//xination:變成傳教師在傳教的感覺了 其中what makes....那段非常的不順。

(第四段)

To better understand the axioms or basic properties commonly used in such fields as analysisor algebra, I consulted E. Landau’s Foundations of Analysis. I was not only amazed by the book’s remarkable style of recurrent pattern‘definition, theorem, proof ’, but also delighted to see how elegantly familiar number systems N, Z, Q, R and C (or actually the systems N, Q+, R+ and C) were constructed; it was then did I totally comprehend the crucial importance of the principle of induction to N.

//xination:還是太多general fact.

(第五段)

The following paragraph “Accordingly, the following are trueon purely logical grounds:

(1) x = x for every x.

(2) If x = y then y = x.

(3) If x = y, y = z then x = z.”,

quoted from the very beginning of Section §1.1, though, prompted me to take yet a further step in my journey: Logic.

//xination:我不認為這是對申請很必要的資訊.

Investigation deep into the level of logic soon brought forth to me the big picture of how the mechanism behind a proof works, accompanied by my firm conviction that any mathematical statement can be either proved or refuted in a clearly defined framework. This conviction was even strengthened by my practice of doing a lot of proofs (an example is given in chapter 1 of my Research Document). It did not, however, last long before my encounter with the book — Godel: A Life of Logic by J. L. Casti and Werner DePauli— that brought about a great shock to my belief.

//xination:討論書的方式走偏了

(第六段)

My junior year was hard due to pressure of strict course demands in addition to recurring pain of xxxx. That book was the only source of my happiness. From it I came to know the monumental Hilbert’s Program, the 23 problems proposed by Hilbert, the Halting Problem, and of course the famous Godel’s Incompleteness Theorems, among other topics. In particular, Hilbert’s Program, whose aim is to reduce infinistic rgument that had long been employed in classical mathematics to finitistic means to be followed by justification of these methods and a consistency proof, only to be rendered impossible by Godel’s Incompleteness Theorems, significantly aroused my interest in mathematicallogic.

//xination:that book 會讓人搞不懂是哪本書

// In particular ... 那句太長，會讓讀者不知道重點在哪。且太多general fact.

(第七段)

After many times of consulting Professor xxxx in my graduate school, an expert in logic, I wrote my master thesis concerning that topic and graph theory (see chapter 2 of my Research Document) under the guidance of my thesis advisor Professor xxxx, a specialist in graph algorithms. On the other hand, the topics of Halting Problem and Godel’s First Incompleteness Theorem and the interrelationship between them are treated fairly thoroughly in C. H. Papadimitriou’s Computational Complexity, which was the textbook for the course Computing Theory in my department when in graduate school. I did pretty well in that course, scoring among the top in the whole class, due largely to my strong enthusiasm for the study of logic and computation.

//xination:master thesis 很重要，為什麼只寫一句而已？

(第八段)

My knowledge of logic has even been enriched by the experience of work at Academia Sinica with Dr. xxx, an expert in functional programming as well as type theory and logic, through the research using automatic theorem provers like Agda and Coq; there I learned logic from an intuitionistic point of view, writing programs for proofs by Curry-Howard Correspondence of logic to computation.

//xination:當RA感覺是個很可以發揮的點為什麼只寫到這邊就停了？

(第九段)

To provide myself with a solid background for the study of logic, I spent a whole year literally reading through (i.e. every single character in) what later became my favorite logic text--- Mathematical Logic by H.-D. Ebbinghaus, J. Flum and W. Thomas --- and meanwhile writing detailed notes and comments (see chapter 3 in my Research Document for an example) including a solution to each exercise in the form of an electronic book that counts to 350 pages. To my fondness of it, I have even written down the sketch of a proof for Godel's Second Incompleteness Theorem, which requires a tremendous amount of formal arithmetics, in the book's own terms, referring to other books like J. R. Shoenfield's Mathematical Logic, G. Tourlakis' Lecture in Logic and Set Theory vol. 1 and W. Rautenberg's A Concise Introduction to Mathematical Logic.

//xination: 我會建議敘述的方式要轉換一下，感覺上把書讀很仔細當賣點是OK，但強調每個字句的讀完，會變成有點太過。

(第十段)

Now I have arrived at the current stage of my math journey, encouraged by the magnificent work of Professor XXX at your University. So touched by Professor XXX's part in the goal to realize a large portion of Hilbert’s Program marked by his excellent masterpiece — Subsystems of Second Order Arithmetic — a book I have been reading through, it is my most sincere wish to apply here, hoping to join this research and devoting my effort to it, with an ambition to explore more truths along this journey.

//xination: 總算出現我覺得比較有對申請有用的資訊了，對於想要研究的內容與希望跟隨的教授有詳細的說明，這邊寫的不錯。

//xination: 後來我才瞭解到，原作是由大學讀電機，碩士讀資工，然後想申請數學博班的。

Final work

“The tenacious truth seekers” is my impression to mathematicians. Similar to them of having inexhaustible enthusiasm to delve in the fundamental mathematical elements and theories, I am determined to devote myself into the field of mathematical logic and the foundations of mathematics. After long and thoughtful consideration when had finished the master thesis and then worked as a RA in Academia Sinica, I sincerely wish to enter your Ph.D. program to continue my mathematics research and achieve my full potential.

//xination:加了一個小summary, 提示出master thesis 與 RA.

The math journey of mine started from Euler’s simple but yet profound identity, e^(iπ) + 1 = 0, in my freshman year in XXXU. Not only I was astonished by its pure metaphysical beauty, but also it kindled my ever-lasting interest in studying fundamental principles of math ever since. Later, I found partial answers in E. Landau’s “Foundations of Analysis”, a book of having a huge impact on me. It cultivated me to think mathematical problems in details and developed my passion for doing proofs independently with rigor.

//xination: 把Euler identity 與 Foundation of analysis 的段落結合在一起。並強調在"原作"數學能力上的進步，而不是general fact的敘述。

Besides “Foundations of Analysis”, I learned the insight of the Halting problem and Godel’s incompleteness theorems from an excellent book “Godel: A Life of Logic” by J. L. Casti and W. DePauli. My comprehension of the above topics was also greatly enhanced in the course of “Computing Theory” when studying computer science and information engineering at XXXU for my master degree. And my master thesis was focus on the study of formulating the successor relation, in which I investigated the interrelationship between logic and computation, or more precisely, that between logic and graph theory. I have to thank my supervisor, Professor XXX, who has been built fame in algorithms, for giving me invaluable advices during research, and also I am indebted to Professor XXX for his help from his immense knowledge in logic. Both of them brought me a solid foundation in logic.

//xination:把Logic有關的經驗在這段先說明。

I gained deeper understanding in logic and computation not only from working on my master thesis but also from working in the highest academic institution in Taiwan, Academia Sinica, as a RA for Dr. XXX. During that period of time, I mainly handled logic problems from intuitionistic aspects, invoking automatic theorem provers such as Agda and Coq, and I wrote plenty of programs for proofs by Curry-Howard Correspondence of logic to computation.

//xination:連結上段，並再說Logic更進一步的經驗

To further specialize in logic and to hone my abilities in math, I spent an entire year in meticulously reading through the book “Mathematical Logic” by H.-D. Ebbinghaus, J. Flum and W. Thomas. I scrutinized every single character and wrote down countless notes and comments, and finished every exercise with my own solution. For the sake of fully comprehending Godel’s second incompleteness theorem, I have written a sketch of a proof by doing a tremendous amount of formal arithmetics, referring to other books such as G. Tourlakis’ Lectures in Logic and Set Theory, vol. 1.

//xination: 談論到 "Mathematical Logic" 一書對原作著在Logic上的成長。並改善了行文的語氣，讓別人看起來是很有熱情看讀書，而不是看書是為了炫耀。

Hilbert’s Program, though having been rendered impossible by the incompleteness results, best represents mathematicians’ firm determination to defend mathematics against all possible attackers and skeptics. Therefore when I learned that Professor XXX and his excellent effort to realize a substantial portion of Hilbert’s program by using the technique Reverse Mathematics, I felt so excited, and I wish to join your Ph.D. program and participate in Prof. XXX’s research group for the research of subsystems of second-order arithmetic.

//xination:由Logic轉到未來想學的題材上。

Finally, I always remember during my junior year in XXXU, I suffered from the recurring pain of XXXX, and only when reading the book “Godel: A Life of Logic” can I feel happy and peaceful, since I came into a pristine mathematical world with pure soul without physical pain. I sincerely appreciate the opportunity of entering your Ph.D. where I can continue to discover truth and enjoy the math journey with your excellent faculty.

//xination: 最後再用一個比較個人特質的段落做個氣氛的轉換，並且結尾。

xination