[FOM] Who was the first to accept undefinable individuals in mathematics?

[W. Mueckenheim]

Until the end of the nineteenth century mathematicans dealt with 
definable numbers only. This was the most natural thing in the world. 
An example can be found in a letter from Cantor to Hilbert, dated 
August 6, 1906: "Infinite definitions (that do not happen in finite 
time) are non-things. If Koenigs theorem was correct, according to 
which all finitely definable numbers form a set of cardinality 
aleph_0, this would imply that the whole continuum was countable, and 
that is certainly false." Today we know that Cantor was wrong and 
that an uncountable continuum implies the existence of undefinable numbers.

Who was the first mathematician to deliberately accept undefinable 
individuals like real numbers in mathematics?

Regards, WM