$∀\, x \, \{x\in\mathbb N\rightarrow P(x)\}$ can be abbreviated to $∀ \hspace{.1cm} x∈ℕ[P(x)].$ But, I am not sure how I can indicate "concisely" that n and k are natural numbers in ∀n[(∀k < n P(k)) → P(n)], which is strong induction.
To do it verbosely, I could write $∀n∈N[∀k∈N(k<n \rightarrow P(k)) → P(n)]$, but it looks ugly and is difficult to read.
I would write
$$\forall \,x \in\mathbb N\left[x\leq 3\rightarrow P(x)\right] $$
Not much of an improvement, but hey.
How about: $$\forall x\in\mathbb N\cap[0,3][P(x)]$$ where $[0,3]$ means the closed interval from $0$ to $3$.
Or: $$\forall x\in 4[P(x)]$$ since $4=\{0,1,2,3\}$ in set theory.
EDIT: This is in response to the question before the edit
I usually write this as
$$\forall n\in\Bbb N,(\forall k\in\{1\dots n-1\},P(k)\implies P(n))$$
where $\{1\dots n-1\}$ is the set of integers between $1$ and $n-1$ inclusive. Example. If you have a notation for $\Bbb N\cap[1,n)$ instead of $\Bbb N\cap[1,n-1]$ that's even better (but I think would not be as universally recognized as $\{1\dots n-1\}$).
