I haven't heard the latest about periods, but I want to point out a potential fallacy here.  It's said very often that the easy proof of the existence of transcendental numbers (on cardinality grounds) is non-constructive.  But, that's false!  It _is_ constructive.  Given pen, paper and lots of time, I could extract from that argument the decimal expansion $0.a_1 a_2 \ldots$ of the transcendental number that the proof constructs.

See, for example, [these comments](https://mathoverflow.net/questions/10334/what-practical-applications-does-set-theory-have/10394#10394) of Joel David Hamkins.

I suspect that the same is true for periods: that there's an effective enumeration of them, so there's an algorithm for generating the decimal digits of a number that isn't a period.