Every Real number is expressible in terms of differences of two transcendentals Is it true that for every real number $x$ there exist transcendental numbers $\alpha$ and $\beta$ such that $x=\alpha-\beta$?
(it is true if $x$ is an algebraic number). 

Suppose there is some real number x that can't be written as the difference of transcendental numbers.  Then for every transcendental number y there exists a unique algebraic number z = y+x.  But this means there is an injective function f(a) = a+x from the transcendental numbers to the algebraic numbers, which implies the cardinality of the transcendental numbers is less than or equal to that of the algebraic numbers.  But we know this is false because the cardinality of the algebraic numbers is strictly less than that of the transcendentals. 
Q.E.D.
