Reality as if is doubled in relation to language: The one counterpart of reality is within the language as the representation of the other counterpart of reality being outside the language and existing by itself. Both representation and metaphor are called to support the correspondence between the two twins as an “image and simile”. The mechanism of that correspondence and its formal conditions are investigated by the following construction. Language is reduced to an infinite countable set (A) of its units of meaning, either words or propositions, or whatever others. It includes all possible meanings, which can be ever expressed in the language rather than the existing till now, which would always a finite set.

The external twin of reality is introduced by another set (B) such that its intersection with the above set of language to be empty. The union of them (C=A∪B) exists always so that a one-to-one mapping (f: C↔A) should exist under the condition of the axiom of choice. The mapping (f) produces an image
(B (f)) of the latter set (B) within the former set (A). That image (B (f)) serves as the other twin of reality to model the reality within the language as the exact representation of the reality out of language (modeled as the set B). In the model, the necessity and sufficient condition of that representation between reality both within and out of the language is just the axiom of choice:

If the axiom of choice does not hold, the relation between the sets B(f) and B cannot be defined rigorously as an exact representation but rather as some simile and the vehicle between the two twins of reality can be only metaphor.