While the paper does not explicitly state it, he shows that π and e are computable. Those cannot be expressed on a fixed-length tape.

There is also a demonstration why an infinite number of symbols does not give more computability over a finite number of symbols. I believe this only makes sense if the tape length is infinite.

I'm surprised to see that, but all subsequent authors have required it to be infinite (as a condition to distinguish it from other models of computation), and Turing himself later referred to it as infinite:

https://en.wikipedia.org/wiki/Turing_machine#Physical_descri...