Is an infinite tape alphabet for TM valid?

Question

Can I use an infinite alphabet for the tape in a turing machine?

e.g. with input string as (1, 0)* can I define the symbol 1j as the symbol 1 with j marks on top of it where j in a natural number, to be used on the tape?

Answer 1

Turing machines have a finite tape alphabet.

You can think of a generalization of Turing machines with infinite tape alphabet, but there are two problems:

1. The description of the machine is no longer finite.
2. Such a machine can decide any language over $\{0,1\}$ (exercise).

There are ways around it - for example, we might require the rules to be finitely specifiable in some specific form. The resulting model will then be equivalent to Turing machines (in terms of computability).