수학/집합론 (Set Theory)

[ZFC Set Theory] XVI. 알레프 수 Aleph NumbersBy 초코맛 도비

[lang-en]To see the previous post[/lang-en][lang-ko]이전 글 보러가기[/lang-ko] [lang-en]As mentioned in the previous post, this post will introduce the aleph numbers, a specific sequence of cardinal numbers.[/lang-en] [lang-ko]이전 글에서 언급한 바와 같이, 이 글에서는 특정한 순서의 기수인 알레프 수에 대해 소개하겠습니다.[/lang-ko] [lang-en]To introduce the aleph numbers, we need to define the concept of the "next" cardinal. To achieve this, ..

[ZFC Set Theory] XV. 기수의 산술연산 Cardinal ArithmeticBy 초코맛 도비

[lang-en]To see the previous post[/lang-en][lang-ko]이전 글 보러가기[/lang-ko] [lang-en]As mentioned in the previous post, we will discuss the cardinal arithmetic in this post. The definitions of cardinal arithmetic are as follows:[/lang-en] [lang-ko]이전 글에서 언급했듯이 이번 글에서는 기수의 산술연산에 대해 다룰 것입니다. 기수의 산술연산은 다음과 같이 정의됩니다.[/lang-ko] [def]{1. [lang-en]Cardinal Arithmetic[/lang-en][lang-ko]기수의 산술연산[/lang-ko]}[l..

[ZFC Set Theory] XIV. 기수 Cardinal NumbersBy 초코맛 도비

[lang-en]To see the previous post[/lang-en][lang-ko]이전 글 보러가기[/lang-ko] [lang-en]As noted in the previous post, we will talk about cardinal numbers in this post. Cardinal number is what is commonly called the number of elements of a set. Formally, we can define cardinal numbers as follows:[/lang-en] [lang-ko]저번 글에서 언급했듯이, 이번 글에선 기수에 대해 다룰 것입니다. 기수는 우리가 흔히 집합의 원소의 개수라고 일컫는 그 개념을 말합니다. 보다 엄밀하게는 다음..

[ZFC Set Theory] XIII. 칸토어 정리 Cantor's TheoremBy 초코맛 도비

[lang-en]To see the previous post[/lang-en][lang-ko]이전 글 보러가기[/lang-ko] [lang-en]As noted in the previous post, we will talk about Cantor's theorem. Cantor's Theorem is a fundamental concept in set theory that states that for any given set $X$, the power set of $X$ (denoted as $\mathscr{P}(X)$) is strictly larger than $X$ itself. In other words, there are always more subsets of a set than there ..

[ZFC Set Theory] XII. 칸토어-슈뢰더-베른슈타인 정리 Cantor-Schröder-Bernstein TheoremBy 초코맛 도비

[lang-en]To see the previous post[/lang-en][lang-ko]이전 글 보러가기[/lang-ko] [lang-en]As noted in the previous post, we will address the question, "Is the $\leq$ between the cardinality of sets antisymmetric?" The answer is, YES! The answer is, yes! But what does this mean exactly? Formally, this can be stated as: "If $\lvert X \rvert \leq \lvert Y \rvert$ and $\lvert Y \rvert \leq \lvert X \rvert$ f..

[ZFC Set Theory] XI. 집합의 농도 Cardinality of SetsBy 초코맛 도비

[lang-en]To see the previous post[/lang-en][lang-ko]이전 글 보러가기[/lang-ko] [lang-en]In this post, we will talk about the cardinality of sets. The cardinality of a set is a kind of the "size" of a set, intuitively. We can compare the sizes of two sets using the concept of cardinality. The formal definition of cardinality is as follows:[/lang-en] [lang-ko]이번 글에서는 집합의 농도에 대한 이야기를 해보려 합니다. 집합의 농도는 직관적으..

[ZFC Set Theory] Appendix B. 서수 Ordinal NumbersBy 초코맛 도비

[lang-en]This post deals with random things related with ordinal numbers, which were not covered in the series posts. One day, this post might be quite long. No one knows for sure. :-)[/lang-en] [lang-ko]이 포스트는 공리적 집합론과 관련이 있으나 시리즈 글들에서 다루지 못한 내용들을 다루는 글입니다. 언젠가는 글이 상당히 길어질지도 모르죠. 확실한 건 아무도 모릅니다. :-)[/lang-ko] i. [lang-ko]시리즈 글들[/lang-ko][lang-en]List of Series Posts[/lang-en] VI. 전순서 집합 Well-Or..

[ZFC Set Theory] X. 정합 관계 Well-Founded RelationsBy 초코맛 도비

[lang-en]To see the previous post[/lang-en][lang-ko]이전 글 보러가기[/lang-ko] [lang-en]As commented in the last post, we will talk about the well-founded relations, which generalize the concept of well-ordering. Informally, a well-founded relation is a binary relation that each element can be assigned an ordinal called "rank". The "height" of set can be defined, of course. Or, it can be considered as ..

[ZFC Set Theory] IX. 서수의 산술연산 Arithmetics of OrdinalsBy 초코맛 도비

[lang-en]To see the previous post[/lang-en][lang-ko]이전 글 보러가기[/lang-ko] $\def\Ord{\operatorname{Ord}}$ [lang-en]As commented in the last post, we will talk about the arithmetics of ordinals. Arithmetics of ordinals are defined by the transfinite recursion, which is introduced in the last post. Before explaining about arithmetics of ordinals, for the convenience of description, we shall introduce..

[ZFC Set Theory] VIII. 초한 귀납법 Transfinite InductionBy 초코맛 도비

[lang-en]To see the previous post[/lang-en][lang-ko]이전 글 보러가기[/lang-ko] $\def\Ord{\operatorname{Ord}}\def\ran{\operatorname{ran}}$ [lang-en]In the previous post, we discussed what ordinal numbers are and what their properties are. In this post, we are going to talk about the transfinite induction. The transfinite induction is a kind of extension of the mathematical induction. Now, let's take a m..