set theory on Solarized Sublime Sekai
https://vincenttam.gitlab.io/tags/set-theory/
Recent content in set theory on Solarized Sublime SekaiHugo -- gohugo.ioThu, 27 Sep 2018 20:48:40 +0200Real Number Construction From Dedekind Cuts
https://vincenttam.gitlab.io/post/2018-09-27-real-number-construction-from-dedekind-cuts/
Thu, 27 Sep 2018 20:48:40 +0200https://vincenttam.gitlab.io/post/2018-09-27-real-number-construction-from-dedekind-cuts/Goal To gain a real understanding on real numbers.
Analytical construction I “swallowed” the Compleness Axiom, then I worked on exercises on $\sup$ and $\inf$, and then the $\epsilon$-$\delta$ criterion for limits, before completing $\Q$ with Cauchy sequences.
I’ve also heard about the completion of a metric space in a more general setting. My professor once said that it suffices to view this proof once throughout lifetime: the proof itself wasn’t very useful.Some Infinite Cardinality Identities
https://vincenttam.gitlab.io/post/2018-09-25-some-infinite-cardinality-identities/
Tue, 25 Sep 2018 23:02:24 +0200https://vincenttam.gitlab.io/post/2018-09-25-some-infinite-cardinality-identities/Purpose This post aims at recapturing the main ideas of the formal proofs that I’ve read. It never tries to replace them. You may consult the references if you need any of them.
Some notations Unless otherwise specified, all cardinalities here are infinite. Denote $\mathfrak{a} = \card{A}$, $\mathfrak{b} = \card{B}$ and $\mathfrak{i} = \card{I}$.
Sum $\mathfrak{a} + \mathfrak{b} = \card{A \cup B}$ provided that $A \cap B =\varnothing$. Product $\mathfrak{a} \, \mathfrak{b} = \card{A \times B}$ Power $\mathfrak{a}^\mathfrak{i} = \card{A^I}$, where $A^I = \lbrace f \mid f: I \to A \rbrace$ denotes the set of functions from $I$ to $A$.CSB Theorem
https://vincenttam.gitlab.io/post/2018-08-29-csb-theorem/
Wed, 29 Aug 2018 11:25:24 +0200https://vincenttam.gitlab.io/post/2018-08-29-csb-theorem/This isn’t a substitute for books .
Reminder $A \preceq B$: $A$ can be “injected” into $B$. $A \sim B$: $A$ and $B$ share the same cardinality. $A \prec B$: $A$ can be “injected” into $B$, but it’s “smaller” than $B$. A finite set can be “counted” from one to some nonnegative integer. Infinite is the “antonym” of finite. Wolf’s proof When I first saw this proof in Robert S.