Would it be possible/legal to design a PDA that can use the stack as a way to keep track of the number of inputs seen? (i.e the size of the stack would act as some sort of counter).
What I was thinking:
First transition would require you to push a "1" to the stack
All future transitions would require you to pop the top of the stack (let's call this $x$) and push $x+1$ onto it.
The language of the stack would be the set of all natural numbers excluding 0.
I'm having trouble understanding whether the "push $x+1$" move would work