A whole of things exists only if it has one or more proper attributes of its own. On our attributivist construction a set of things which are not attributes never exists (as a thing or whole) because it does not have attributes, let alone proper attributes. And even a set of existing attributes (such as mental properties) which is the predicament of a whole (such as a person) does not exist.

Now, some logicians, set-theorists, mathematicians and other theorists are 'infinitely' more realistic than we are. They believe in every set that can be constructed theoretically, and in an abstract world which is at least partially incomprehensible to us 'finite creatures'. Not only would all cardinal numbers one can think of 'exist' but also every other 'real' number. For some of these 'real' numbers there are even no formulas for generating or defining them, which makes them truly incomprehensible. Since there is merely a denumerably infinite number of names and definitions in human language, it is impossible to comprehend every set, given the 'existence' of nondenumerably infinite sets. If this sounds too abstract, another realist infinity creator may take over and talk about physical space as having infinitely many 'concrete points' (whatever that may mean). And if this 'concrete' infinity is not nondenumerable too, there would be at least a nondenumerably infinite number of subsets of these points. 'Thus' even not in the 'concrete' world could every subset of points be defined and comprehended. (This should consequently make us believe that there is necessarily something incomprehensible, and therefore something maximally incomprehensible. The climax of the argument is then reached, when it is 'proved' that a being named "God", and defined as maximally incomprehensible, necessarily 'exists'.)

By way of stipulative definition any abstract construction or infinite sequence, or any factual or potential result of a theoretical exercise, may be called "existing" or "concrete", so long as the theory within the bounds of which it is done is free from contradictions or from proved contradictions. The result, however, is merely the greatest possible exhaustion of the notions of existence and concreteness. Underlying this 'infinite' eagerness of set-theoretical realists is basically the idea that coherence (or the absence of incoherence) would be a sufficient (not just a necessary) condition for truth, and that the 'truth' thus generated would also somehow correspond with reality; or, that it would have something to do with existing things.

According to the meaning-variance thesis mentioned in section 1.2.1 existence means something else in a system in which incomprehensible, nondenumerably infinite sets 'exist' (and maybe also a maximally incomprehensible, partially abstract, partially concrete devil) than in a system in which only things such as the bodies of living beings and artifacts 'exist'. So the belief that gods or unicorns 'exist' will have a different meaning too in the two systems. The very function of the notion of existence, however, is precisely to provide a theory-independent link between a particular theory and that what the theory is about (nonpropositional reality if the theory is of the first order). Its function is definitely not purely theoretical like that of a criterion of validity or consistence. Furthermore, the meaning of existence is also different from that of concreteness, given the vocabulary of the present language which requires, or at least enables, us to make this distinction. It is for these reasons that we reject both a notion of 'existence' which remains entirely intratheoretical and a notion of 'existence' which is superfluous given the concepts of the 'concrete' and the 'material'.