Let $V$ be the class of all sets, $\text{Ord}$ be the class of all ordinals, $\text{InOrd}$ be the class of all initial ordinals, and $f:V \to \text{Ord}$ be the function that sends each set to its corresponding Hartogs number.

As a result, $f(a)=a+1$ for all $a \in \Bbb N$ and thus $f \restriction \Bbb N$ is injective.

I find it difficult to imagine the Hartogs numbers of infinite sets.

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> **My question:** Are $f,f\restriction \text{Ord},f\restriction \text{InOrd}$ injective?
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Thank you in advance for your help!