According to what I heard of, an ordinal is constructed by taking an union of {$\alpha$} $\cup$ $\alpha$ where $\alpha$ is a predecessor ordinal.
If so, how can there be a set that is not an ordinal?
According to what I heard of, an ordinal is constructed by taking an union of {$\alpha$} $\cup$ $\alpha$ where $\alpha$ is a predecessor ordinal.
If so, how can there be a set that is not an ordinal?
$\varnothing$ is an ordinal; we call it $0$.
$0\cup\{0\}=\varnothing\cup\{\varnothing\}=\{\varnothing\}=\{0\}$ is an ordinal; we call it $1$.
$1\cup\{1\}=\{\varnothing\}\cup\{\{\varnothing\}\}=\{\varnothing,\{\varnothing\}\}=\{0,1\}$ is an ordinal; we call it $2$.
$2\cup\{2\}=\{0,1\}\cup\{2\}=\{0,1,2\}$ is an ordinal; we call it $3$.
$\{0,2\}$ is not an ordinal: it is not $x\cup\{x\}$ for any ordinal $x$.
An even simpler example of a set that is not an ordinal would be $\{\{\emptyset \}\}$. We have $\emptyset \in \{\emptyset\} \in \{\{\emptyset \}\}$ but $\emptyset \notin \{\{\emptyset \}\}$.
The power set of an ordinal is usually not an ordinal, except for the cases of $0$ and $1$.
For example, $\mathcal P(\omega)$ is not an ordinal.