Thanks for an introduction of Tarski and Truth into such a forum. Though I have a suspicion something is either trivialized or wrong about the formula as it is stated above, since what you state between apostrophes 'L <=> ¬T(g(L)' is not equivalent to what you are writing in English after 'That is', i.e.,

L is true if and only if the truth predicate applied to the Gödel number of L is false.

To write this statement above symbolically, we would have to first agree to the statements:

'L' is true <=> L

and

'L' is false <=> ¬L.

You have tacitly assumed this. And in so doing, used Tarski's truth convention in trying to explain it. Said otherwise, you have defined truth in the metalanguage in order to show it cannot be defined in the object-language. So in the end, truth is defined after all, just not where we were looking for it.

This much said, enjoyed the read.

Scully