...

And I contend that ANY number which is Real can be expressed by the decimal expansion above. Which clearly qualifies as a formal system.

To be a formal system of the type required, you would also have to specify deterministic rules that could generate the "Ai" values. The key distinction is between "expressed by" and "generated by".

But we do have a formal system for generating those terms. Dedekind Cuts used to enumerate the value of an arbitrary length compared to the unit length.