Real Number Construction From Dedekind Cuts

A geometrically intuitive approach

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. [Read More]

Some Infinite Cardinality Identities

Working with infinite sets

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 $A$. [Read More]

CSB Theorem

A visual argument for 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. [Read More]