So you're arguing that a set can exist even if its elements don't? Because we can surely give a unique description of the set of real numbers (the unique complete ordered field up to iso) and this description forces it to be uncountable
Also the definition that you gave is complete nonsense to a Constructivist. And the reasoning that forces it to be uncountable in classical set theory requires reasoning that also makes implicit assumptions you are probably not aware of.
Also if we want to be precise I'd like to hear your definition of "finite definition", the only definition of "definability" I'm familiar with is relative to a model of a theory and if ZFC has models at all then it has countable pointwise definable models, but I guess that won't satisfy your idea of "finite definition"
The fact that this may not be a definition from my point of view is irrelevant - I was making a statement about what classical mathematics implies, and from the point of view of classical mathematics this is a perfectly reasonable definition.
Since the number of such statements is countable, and only some of them define real numbers, the set of such real numbers is countable. Being countable it is a set of measure 0, and therefore the "almost all" that I stated follows immediately.