In an interesting article (available here), Timothy Chow proposes that a closed-form number be defined as an element of the smallest subfield of $\mathbb{C}$ that is closed under $\exp$ and a chosen branch of $\log$. It is fun to check that pretty much any number that you might accept as closed-form answer to a calculus problem belongs to this field.
He writes, "My hope is that this definition of closed-form expression for a number will become standard, and that many readers will be lured into working on the many attarctive open problems in this area." The bulk of the article relates his notion of closed-form numbers to standard conjectures in transcendental number theory, most notably < a href="http://en.wikipedia.org/wiki/Schanuel%27s_conjecture">Schanuel's Schanuel's conjecture.
My questions are
To what extent has this notion become accepted as standard?
Are there new results since the time of his writing?
There was a rekindling of interest in Schanuel's conjecture after Boris Zil'ber's categoricity results on algebraically closed exponential fields in characteristic zero. In what way has this changed the status of problems mentioned in Chow's article (if it has)?
