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4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / a. Sets as existing

[commitment to sets as really existint entities]

8 ideas
Classes are a host of ethereal, platonic, pseudo entities [Goodman]
     Full Idea: I will not willingly use apparatus that peoples the world with a host of ethereal, platonic, pseudo entities.
     From: Nelson Goodman (The Structure of Appearance [1951], II.2), quoted by David Lewis - Parts of Classes 2.1
     A reaction: This represents the big gap that opened up with Goodman's former comrade in arms, Quine. Lewis quotes it in order to ask whether he means ethereal or platonic, as they are very different. I sympathise with Goodman.
The use of plurals doesn't commit us to sets; there do not exist individuals and collections [Boolos]
     Full Idea: We should abandon the idea that the use of plural forms commits us to the existence of sets/classes… Entities are not to be multiplied beyond necessity. There are not two sorts of things in the world, individuals and collections.
     From: George Boolos (To be is to be the value of a variable.. [1984]), quoted by Henry Laycock - Object
     A reaction: The problem of quantifying over sets is notoriously difficult. Try http://plato.stanford.edu/entries/object/index.html.
If singletons are where their members are, then so are all sets [Lewis]
     Full Idea: If every singleton was where its member was, then, in general, classes would be where there members were.
     From: David Lewis (Parts of Classes [1991], 2.1)
     A reaction: There seems to be a big dislocation of understanding of the nature of sets, between 'pure' set theory, and set theory with ur-elements. I take the pure to be just an 'abstraction' from the more located one. The empty set has a puzzling location.
A huge part of Reality is only accepted as existing if you have accepted set theory [Lewis]
     Full Idea: The preponderant part of Reality must consist of unfamiliar, unobserved things, whose existence would have gone unsuspected but for our acceptance of set theory.
     From: David Lewis (Parts of Classes [1991], 2.6)
     A reaction: He is referring to the enormous sets at the far end of set theory, of a size that had never been hitherto conceived. Excellent. Daft to believe in something entirely because you have accepted set theory, with no other basis.
Set theory isn't innocent; it generates infinities from a single thing; but mathematics needs it [Lewis]
     Full Idea: Set theory is not innocent. Its trouble is that when we have one thing, then somehow we have another wholly distinct thing, the singleton. And another, and another....ad infinitum. But that's the price for mathematical power. Pay it.
     From: David Lewis (Parts of Classes [1991], 3.6)
Are sets part of logic, or part of mathematics? [Shapiro]
     Full Idea: Is there a notion of set in the jurisdiction of logic, or does it belong to mathematics proper?
     From: Stewart Shapiro (Foundations without Foundationalism [1991], Pref)
     A reaction: It immediately strikes me that they might be neither. I don't see that relations between well-defined groups of things must involve number, and I don't see that mapping the relations must intrinsically involve logical consequence or inference.
The set theorist cannot tell us what 'membership' is [Chihara]
     Full Idea: The set theorist cannot tell us anything about the true relationship of membership.
     From: Charles Chihara (A Structural Account of Mathematics [2004], 01.5)
     A reaction: If three unrelated objects suddenly became members of a set, it is hard to see how the world would have changed, except in the minds of those thinking about it.
ZFC can prove that there is no set corresponding to the concept 'set' [George/Velleman]
     Full Idea: Sets, unlike extensions, fail to correspond to all concepts. We can prove in ZFC that there is no set corresponding to the concept 'set' - that is, there is no set of all sets.
     From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.4)
     A reaction: This is rather an important point for Frege. However, all concepts have extensions, but they may be proper classes, rather than precisely defined sets.