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===6.13. Issue 2. The Status of Sets===

===6.13. Issue 2. The Status of Sets===

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That the word "set" is being used indiscriminately for completely different notions and that this is the source of the apparent paradoxes of this young branch of science, that, moreover, set theory itself can no more dispense with axiomatic assumptions than can any other exact science and that these assumptions, just as in other disciplines, are subject to a certain arbitrariness, even if they lie much deeper here — I do not want to represent any of this as something new. (Julius Konig, 1905).

<p>That the word &ldquo;set&rdquo; is being used indiscriminately for completely different notions and that this is the source of the apparent paradoxes of this young branch of science, that, moreover, set theory itself can no more dispense with axiomatic assumptions than can any other exact science and that these assumptions, just as in other disciplines, are subject to a certain arbitrariness, even if they lie much deeper here &mdash; I do not want to represent any of this as something new.</p>

| align="right" | Julius König (1905), &ldquo;On the Foundations of Set Theory and the Continuum Problem&rdquo;, in Jean van Heijenoort (ed., 1967)

Set theory is not as young as it used to be, and not half as naive as it was when this statement was originally made, but the statement itself is just as apt in its application to the present scene and just as fresh in its lack of novelty as it was then.  In the current setting, though, I am not so concerned with potentially different theoretical notions of a set that are represented by conventionally different axiom systems as I am with the actual diversity of practical notions that are used to deal with sets under each of the three NOSs identified.

Set theory is not as young as it used to be, and not half as naive as it was when this statement was originally made, but the statement itself is just as apt in its application to the present scene and just as fresh in its lack of novelty as it was then.  In the current setting, though, I am not so concerned with potentially different theoretical notions of a set that are represented by conventionally different axiom systems as I am with the actual diversity of practical notions that are used to deal with sets under each of the three norms of significance identified.

Even though all three NOSs 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 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.

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.