Given two transcendental numbers, is their difference algebraic?

Question

It is straightforward that a transcendental plus an algebraic is transcendental. I wonder if, given two transcendentals x and y we may find such algebraic a that x+a=y?

Answer 1

You answered yourself: if $x+a=y$, then $x,y$ are a pair of transcendentals that differ by an algebraic.

This is of course not the case for any pair $x,y$, because there are many more transcendentals (uncountable) than there are algebraics (countable).

Answer 2

You should be able to see the following:

Each transcendental belongs to a countable set of transcendentals which differ from itself and from each other by an algebraic number.

Since these sets of transcendentals are countable, and a countable collection of countable sets is countable, there must be an uncountable number of such sets to include all the transcendentals.