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Even though all three norms of significance use set-theoretic constructions, the implicit theories of sets that are involved in their different uses are so varied in their assumptions and intentions that it amounts to a major source of friction between the casual and formal styles to try to pretend that the same subject is being invoked in every case.  In particular, it makes a huge difference whether these sets are treated objectively, as belonging to the OF, or treated syntactically, as belonging to the IF.

Even though all three norms of significance use set-theoretic constructions, the implicit theories of sets that are involved in their different uses are so varied in their assumptions and intentions that it amounts to a major source of friction between the casual and formal styles to try to pretend that the same subject is being invoked in every case.  In particular, it makes a huge difference whether these sets are treated objectively, as belonging to the OF, or treated syntactically, as belonging to the IF.

In practical terms it makes all the difference in the world whether a set is viewed as a set of objects or whether it is viewed as a set of signs.  The same set can be contemplated in each type of placement, but it does not always fit as well into both types of role.  A set of objects is properly a part of the objective framework, and this is intended in its typical parts to model those realities whose laws and vagaries can extend outside the means of an agent's control.  A set of signs is properly part of the interpretive framework, and this is constructed in its typical parts so that its variations and selections are subject to control for the ends of interpretive indication.  The relevant variable is one of control, and the measure of it tells how well matched are the proper placements and the typical assignments that a given set is given.

In practical terms it makes all the difference in the world whether a set is viewed as a set of objects or whether it is viewed as a set of signs.  The same set can be contemplated in each type of placement, but it does not always fit as well into both types of role.  A set of objects is properly a part of the OF, and this is intended in its typical parts to model those realities whose laws and vagaries can extend outside the means of an agent's control.  A set of signs is properly part of the IF, and this is constructed in its typical parts so that its variations and selections are subject to control for the ends of interpretive indication.  The relevant variable is one of control, and the measure of it tells how well matched are the proper placements and the typical assignments that a given set is given.

Things referred to the objective world are not things that one expects to have much control over, at least, not at first, even though a reason for developing a language is to gain more control over events in time.  Things referred to the realm of signs are things that one thinks oneself to have under control, at least, at first, even though their complexity can evolve in time beyond one's powers of oversight.

Things referred to the objective world are not things that one expects to have much control over, at least, not at first, even though a reason for developing a language is to gain more control over events in time.  Things referred to the realm of signs are things that one thinks oneself to have under control, at least, at first, even though their complexity can evolve in time beyond one's powers of oversight.

In an ordinary mathematical context, when one writes out the expression for a finite set in the form "{x1, ..., xn}", then one expects to see the names of objects appearing between the braces.  Furthermore, even if these additional expectations are hardly ever formalized, these objects are typically expected to be the terminal objects of denotative value in the appropriate context of discussion and to inhabit a single order of objective existence.  In other words, it is common to assume that all of the objects named have the same type, with no relations of consequence, functional, semantic, or otherwise, obtaining among them.  As soon as these assumptions are made explicit, of course, it is obvious that they do not have to be so.

In an ordinary mathematical context, when one writes out the expression for a finite set in the form "{x1, ..., xn}", then one expects to see the names of objects appearing between the braces.  Furthermore, even if these additional expectations are hardly ever formalized, these objects are typically expected to be the terminal objects of denotative value in the appropriate context of discussion and to inhabit a single order of objective existence.  In other words, it is common to assume that all of the objects named have the same type, with no relations of consequence, functional, semantic, or otherwise, obtaining among them.  As soon as these assumptions are made explicit, of course, it is obvious that they do not have to be so.