[MrMoenty]

You are right, this reasoning depends on only allowing formal systems with finitely many symbols. Turing himself actually gave justification for this in a beautiful footnote in his 1936 paper [1]:

page 249:

> I shall also suppose that the number of symbols which may be printed is finite. If we were to allow an infinity of symbols, then there would be symbols differing to an arbitrarily small extent.

And then in the footnote:

> If we regard a symbol as literally printed on a square we may suppose that the square is 0 < x < 1, 0 < y < 1. The symbol is defined as a set of points in this square, viz. the set occupied by printer's ink. If these sets are restricted to be measurable, we can define the "distance" between two symbols as the cost of transforming one symbol into the other if the cost of moving unit area of printer's ink unit distance is unity, and there is an infinite supply of ink at x = 2. y = 0. With this topology the symbols form a conditionally compact space.

[1] https://www.cs.virginia.edu/~robins/Turing_Paper_1936.pdf