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The Mathematical Import Of Zermelo's WellOrdering Theorem
 Bull. Symbolic Logic
, 1997
"... this paper, the seminal results of set theory are woven together in terms of a unifying mathematical motif, one whose transmutations serve to illuminate the historical development of the subject. The motif is foreshadowed in Cantor's diagonal proof, and emerges in the interstices of the inclusi ..."
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this paper, the seminal results of set theory are woven together in terms of a unifying mathematical motif, one whose transmutations serve to illuminate the historical development of the subject. The motif is foreshadowed in Cantor's diagonal proof, and emerges in the interstices of the inclusion vs. membership distinction, a distinction only clarified at the turn of this century, remarkable though this may seem. Russell runs with this distinction, but is quickly caught on the horns of his wellknown paradox, an early expression of our motif. The motif becomes fully manifest through the study of functions f :
SETS AND CLASSES AS MANY by
"... Set theory is sometimes formulated by starting with two sorts of entities called individuals and classes, and then defining a set to be a class as one, that is, a class which is at the same time an individual, as indicated ..."
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Set theory is sometimes formulated by starting with two sorts of entities called individuals and classes, and then defining a set to be a class as one, that is, a class which is at the same time an individual, as indicated
1 Frege's Theorem in a Constructive Setting 1
"... any set E, if there exists a map ν from the power set of E to E satisfying the condition ∀XY [ ν(X) = ν(Y) ⇔ X ≈ Y] 2, then E has a subset which is the domain of a model of Peano's axioms for the natural numbers. (This result is proved explicitly, using classical reasoning, in section 3 of [1 ..."
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any set E, if there exists a map ν from the power set of E to E satisfying the condition ∀XY [ ν(X) = ν(Y) ⇔ X ≈ Y] 2, then E has a subset which is the domain of a model of Peano's axioms for the natural numbers. (This result is proved explicitly, using classical reasoning, in section 3 of [1].) My purpose in this note is to strengthen this result in two directions: first, the premise will be weakened so as to require only that the map ν be defined on the family of (Kuratowski) finite subsets of the set E, and secondly, the argument will be constructive, i.e., will involve no use of the law of excluded middle. To be precise, we will prove, in constructive (or intuitionistic) set theory3, the following Theorem. Let ν be a map with domain a family of subsets of a set E to E satisfying the following conditions: (i) ∅ ∈ dom(ν) (ii) ∀U∈dom(ν) ∀x ∈ E–U U ∪{x} ∈ dom(ν) (iii) ∀UV ∈ dom(ν) ν(U) =ν(V) ⇔ U ≈ V. Then we can define a subset N of E which is the domain of a model of Peano's axioms.