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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / b. Axiom of Extensionality I

[axiom concerning what makes a set]

9 ideas
Members define a unique class, whereas defining characteristics are numerous [Russell]
     Full Idea: There is only one class having a given set of members, whereas there are always many different characteristics by which a given class may be defined.
     From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], II)
In modal set theory, sets only exist in a possible world if that world contains all of its members [Stalnaker]
     Full Idea: One principle of modal set theory should be uncontroversial: a set exists in a given possible world if and only if all of its members exist at that world.
     From: Robert C. Stalnaker (Mere Possibilities [2012], 2.4)
     A reaction: Does this mean there can be no set containing all of my ancestors and future descendants? In no world can we coexist.
Extensionality: ∀x ∀y (∀z (z ∈ x ↔ z ∈ y) → x = y) [Kunen]
     Full Idea: Axiom of Extensionality: ∀x ∀y (∀z (z ∈ x ↔ z ∈ y) → x = y). That is, a set is determined by its members. If every z in one set is also in the other set, then the two sets are the same.
     From: Kenneth Kunen (Set Theory [1980], §1.5)
Extensional sets are clearer, simpler, unique and expressive [Maddy]
     Full Idea: The extensional view of sets is preferable because it is simpler, clearer, and more convenient, because it individuates uniquely, and because it can simulate intensional notions when the need arises.
     From: Penelope Maddy (Believing the Axioms I [1988], §1.1)
     A reaction: [She cites Fraenkel, Bar-Hillet and Levy for this] The difficulty seems to be whether the extensional notion captures our ordinary intuitive notion of what constitutes a group of things, since that needs flexible size and some sort of unity.
The Axiom of Extensionality seems to be analytic [Maddy]
     Full Idea: Most writers agree that if any sense can be made of the distinction between analytic and synthetic, then the Axiom of Extensionality should be counted as analytic.
     From: Penelope Maddy (Believing the Axioms I [1988], §1.1)
     A reaction: [Boolos is the source of the idea] In other words Extensionality is not worth discussing, because it simply tells you what the world 'set' means, and there is no room for discussion about that. The set/class called 'humans' varies in size.
Axiom of Extensionality: for all sets x and y, if x and y have the same elements then x = y [George/Velleman]
     Full Idea: The Axiom of Extensionality says that for all sets x and y, if x and y have the same elements then x = y.
     From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3)
     A reaction: This seems fine in pure set theory, but hits the problem of renates and cordates in the real world. The elements coincide, but the axiom can't tell you why they coincide.
Extensionality: Two sets are equal if and only if they have the same elements [Clegg]
     Full Idea: Axiom of Extension: Two sets are equal if and only if they have the same elements.
     From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15)
A set may well not consist of its members; the empty set, for example, is a problem [Rumfitt]
     Full Idea: There seem strong grounds for rejecting the thesis that a set consists of its members. For one thing, the empty set is a perpetual embarrassment for the thesis.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 8.4)
     A reaction: Rumfitt also says that if 'red' has an extension, then membership of that set must be vague. Extensional sets are precise because their objects are decided in advance, but intensional (or logical) sets, decided by a predicate, can be vague.
A set can be determinate, because of its concept, and still have vague membership [Rumfitt]
     Full Idea: Vagueness in respect of membership is consistent with determinacy of the set's identity, so long as a set's identity is taken to consist, not in its having such-and-such members, but in its being the extension of the concept A.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 8.4)
     A reaction: To be determinate, it must be presumed that there is some test which will decide what falls under the concept. The rule can say 'if it is vague, reject it' or 'if it is vague, accept it'. Without one of those, how could the set have a clear identity?