===== First infinite von Neumann ordinal ===== ==== Set ==== | @#FFBB00: definiendum | @#FFBB00: $ \omega_{\mathcal N}$ | ... ----- As is common, I'll also use the symbol $\mathbb N$ to denote the set theoretic object $\omega_{\mathcal N}$. {{Infinite_Jest.jpg?X700}} /* Nikolaj Kuntner 2015 */ === Idea === This is probably the most straightforward way to set up a countably infinite set. === Elaboration === The first requirement says that all elements of $\omega_{\mathcal N}$ are either $\emptyset$ or a successor of another set. The second guaranties that there are no superfluous sets in $\omega_{\mathcal N}$, apart from the ones which are e.g. required to making $\omega_{\mathcal N}$ an ordinal. Put together the axiom says that $\omega_{\mathcal N}$ contains $\emptyset$ and for each established element $m$, it also inductively contains all successors ${\mathrm{succ}}\ m\equiv m\cup\{m\}$. === Notation === To model the [[natural numbers]], one can make the following identifications: * $0\equiv \emptyset$ * $1\equiv {\mathrm{succ}}\ (0)=\emptyset\cup\{\emptyset\}=\{\}\cup\{0\}=\{0\}$ * $2\equiv {\mathrm{succ}}\ (1)=1\cup\{1\}=\{0\}\cup\{1\}=\{0,1\}$ * $3\equiv {\mathrm{succ}}\ (2)=2\cup\{2\}=\{0,1\}\cup\{2\}=\{0,1,2\}$ * $4\equiv {\mathrm{succ}}\ (3)=\dots$ Using our common language conception of natural numbers we can say: Each number contains the numbers which are less than itself, i.e. $n$ is $\{0,1,2,3,4,\dots,n-1\}$. Being an [[Ordinal number|ordinal]], we can model the order relation of the natural numbers via set inclusion $k