Transition function of non deterministic pushdown automata

Question

I was reading book on Automata Theory by Peter Linz.

He gives transition function of the non deterministic finite automata as follows:

$\delta:Q\times (\Sigma\cup\{\lambda\})\rightarrow 2^Q$

But the transition function of non deterministic pushdown automata is given as:

$\delta:Q\times(\Sigma\cup\{\lambda\})\times\Gamma\rightarrow$ set of finite subsets of $Q\times \Gamma^*$

I understand it talks about "finite subsets", because $Q\times \Gamma^*$ is infinite and can have infinite subsets, should it be $2^{Q\times \Gamma^*}$? That is, should the transition function be:

$\delta:Q\times(\Sigma\cup\{\lambda\})\times\Gamma\rightarrow$ set of finite subsets of $2^{Q\times \Gamma^*}$

PS: $Q$ is set of states in automata, $\Sigma$ is an alphabet, $\lambda$ is empty symbol, $\Gamma$ is stack alphabet

Answer 1

$2^S$ is the set of all subsets of $S$, not only the finite ones but the infinite ones also. If $S$ is finite, there are no infinite subsets, and “$2^S$” means the same as "finite subsets of $S$”.

But if $S$ is infinite, $2^S$ includes some infinite subsets also. Then “Finite subsets of $S$” means something different.

Linz wants the output of the transition function to be a finite set in both cases. For the NDPDA he has to say so explicitly. For the NFA he doesn't need to say it because the NFA is already finite.