OP: How can we define Reality = {x: x ∈ Reality} and make it be informative?
Short answer: Actuality = {x: x ∈ Reality && x.exists}
Kant defines something as 'real' if it is possible, in contrast to being 'actual' if it is brought to existence by cognition. He does not deal with broad scientific realism because his antinomies have demonstrated incoherence with infinity and finitude in time and space; so he proceeds from what he is certain of: his experiential cognition.
As regards the existence of things, Aristotle defines 'a thing' as composed of concept and predicates: e.g. apple + red. Kant revolutionises this by declaring that existence is not a predicate: i.e. from Kant's Critique of Pure Reason A598/B626, original here:
Being is evidently not a real predicate, that is, a conception of
something which is added to the conception of some other thing. It is
merely the positing of a thing, or of certain determinations in it.
Logically, it is merely the copula of a judgement.
The copula constitutes the "is" in the experiential observation that "the apple is red": the cognitive judgement involved in joining concept with predicate makes the existence of the red apple.
Prior to encountering the red apple Kant holds open the real possibility of apples of any possible colour (for an apple). He might really find a green apple. The sum of all these available, real possibilities he calls reality. For example, as Kant sits under a tree in reality he is mindful that an apple might fall on his head. He has to navigate reality on his daily walk looking out for real possible cyclists speeding round the next corner looking for apples.
His world of possibilities is noumenal while the things he encounters in actuality are phenomenal. The foregoing hopefully elucidates the following, from the Critique of Pure Reason:-
Ch. III. Section ii. Of the Transcendental Ideal (Prototypon Trancendentale).
If, therefore, a transcendental substratum lies at the foundation of
the complete determination of things—a substratum which is to form the
fund from which all possible predicates of things are to be supplied,
this substratum cannot be anything else than the idea of a sum-total
of reality (omnitudo realitatis). In this view, negations are nothing
but limitations—a term which could not, with propriety, be applied to
them, if the unlimited (the all) did not form the true basis of our
conception.
This conception of a sum-total of reality is the conception of a thing
in itself, regarded as completely determined; and the conception of an
ens realissimum is the conception of an individual being, inasmuch as
it is determined by that predicate of all possible contradictory
predicates, which indicates and belongs to being. It is, therefore, a
transcendental ideal which forms the basis of the complete
determination of everything that exists, and is the highest material
condition of its possibility—a condition on which must rest the
cogitation of all objects with respect to their content. Nay, more,
this ideal is the only proper ideal of which the human mind is
capable; because in this case alone a general conception of a thing is
completely determined by and through itself, and cognized as the
representation of an individuum.
Heidegger also casts some light in The Basic Problems of Phenomenology, Chapter One Kant's Thesis: Being Is Not A Real Predicate, page 34
The concept of reality and the real in Kant does not have the meaning
most often intended nowadays when we speak of the reality of the
external world or of epistemological realism. Reality is not
equivalent to actuality, existence, or extantness. It is not identical
with existence, although Kant indeed uses the concept "objective
reality" identically with existence.
The Kantian meaning of the
term "reality" is the one that is appropriate to the literal sense of
the word. In one place Kant translates "reality" very fittingly by
"thingness," "thing-determinateness." The real is what pertains to
the res. When Kant talks about the omnitudo realitatis, the totality
of all realities, he means not the whole of all beings actually extant
but, just the reverse, the whole of all possible thing-determinations,
the whole of all thing-contents or real-contents, essences, possible
things. Accordingly, realitas is synonymous with Leibniz' term
possibilitas, possibility. Realities are the what-contents of possible
things in general without regard to whether or not they are actual, or
"real" in our modern sense. The concept of reality is equivalent to
the concept of the Platonic idea as that pertaining to a being which
is understood when I ask: Ti esti, what is the being?
Hopefully that gives a definition of 'real' without circularity. Heidegger carries forward Kant's understanding of the existence of things, but mainly turns his attention to the existential nature of the observer/cogito: Dasein, and then the foundation of Dasein. The subjective nature of 'things' is maintained though, including the real as possibility.
OP: So, in some sense, there is a larger collection, call it real-unreal-surreal, that has as a subset the real objects.
Instead: There is a larger collection: real possibilities (reality), that has a subset of actual objects (actuality).
