Blog

2024

An Abstract Nonsense for Gödel Incompleteness

Although Gödel’s proof of his incompleteness is renowned for the formal construction of the sentence “I cannot be proven,” there are ways to illuminate this within the framework of abstract nonsense. From the nLab point of view, this is a consequence of Lawvere’s fixed point theorem, as shown in Lawvere 1969 or Yanofsky 2003. However, since I am not entirely sure if I understood the work correctly, I will present my own re-invention of the wheel below.

18 minute read

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2023

Playing with the Ax–Grothendieck Theorem

We discuss a construction question that naturally arises from the Ax–Grothendieck theorem. In particular, this comes from a field automorphism between the complex numbers and the ultraproduct of prime algebraically closed fields of positive characteristics.

12 minute read

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Elliptic curves of bidegree (2,2)

One way to see an elliptic curve is to view it as a smooth bidegree (2,2) curve in $\mathbb{P}^1\times\mathbb{P}^1$. This fact itself comes from the adjunction formula, but we suggest a way to derive a bidegree (2,2) formula from the Weierstrass equation. Based on that, we see how this connects with the tropicalized actions of Vieta involutions on elliptic curves.

11 minute read

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2022

What to say for IVT or EVT classes

This note is to record some tales that may be adequate while introducing Intermediate Value Theorem (IVT) and Extreme Value Theorem (EVT) for continuous functions. Some ideas behind the proofs, a relation with long division algorithm, etc. are introduced.

15 minute read

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개평법

전통적인 산술로 2차방정식의 수치해를 구하는 방법에 대한 소개 및 여기서 도출되는 몇 가지 ‘숫자 놀이’에 대한 글.

11 minute read

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Berkovich Line, part 3

This series of posts is aimed to be a rudiment of the Berkovich line, a locally compact space that “fills up” a non-archimedean field $K$. This post is the last of the series and continues from part 2.

26 minute read

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Berkovich Line, part 2

This series of posts is aimed to be a rudiment of the Berkovich line, a locally compact space that “fills up” a non-archimedean field $K$. This post continues from part 1.

12 minute read

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A Formula of Hyperbolic Distance

The idea of this post is to (1) introduce a formula to measure a distance on the upper half plane with the ‘standard’ hyperbolic metric, and (2) record some ideas of deriving this.

4 minute read

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Berkovich Line, part 1

This series of posts is aimed to be a rudiment of the Berkovich line, a locally compact space that “fills up” a non-archimedean field $K$. The materials here are more or less my summary of the lecture note by (M. Baker 2007), with some extra materials that I found meaningful to append.

12 minute read

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「호몰로지」 용어에 대한 일고(一考)

대한수학회 수학용어집에 따르면, homology, cohomology는 각각 음값을 그대로 쓴 ‘호몰로지’ ‘코호몰로지’라 번역하는 것이 표준이다. 다만, 보다 원어의 의미를 잘 생각해 보고자, 이 용어가 어떤 맥락에서 도입됐는지 등에 대해 알아 본 내용을 담아 보았다.

4 minute read

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중국인의 나머지 정리 2

앞선 포스트에서는 부분분수 분해와 중국인의 나머지 정리 사이의 관계를 짚어보았으나, 막상 부분분수 분해를 중국인의 나머지 정리의 결과인 것으로 묘사한 바 있었다. 그러나 계산의 흐름을 보면 둘이 동치라 생각하는 편이 자연스러운데, 이 후속 포스트에서는 그 점을 명확히 하고자 한다.

2 minute read

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2021

Some Code Fragments

This is to record some Python macro that I have used to draw some TikZ figures:

less than 1 minute read

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How this blog works

This ‘web log’ (etymology of ‘blog,’ as far as I know of) is intended to be a batch of notes that I make upon academic studies. Among the notes that I feel worth sharing, to be precise.

less than 1 minute read

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