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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 :
Inhomogeneity of the urelements in the usual models of NFU
, 2005
"... The simplest typed theory of sets is the multisorted first order system TST with equality and membership as primitive predicates and with sorts (types) indexed by the natural numbers. Atomic formulas are wellformed if they are of one of the forms x n ∈ y n+1; x n = y n. The axioms of TST are exten ..."
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The simplest typed theory of sets is the multisorted first order system TST with equality and membership as primitive predicates and with sorts (types) indexed by the natural numbers. Atomic formulas are wellformed if they are of one of the forms x n ∈ y n+1; x n = y n. The axioms of TST are extensionality (objects of positive type are equal iff they have the same members) and comprehension (“{x n  φ} n+1 exists ” for any formula φ in the language of TST). (this theory has often been incorrectly attributed to Russell, by this author among others: see [17] for a discussion of the actual history of this system). Quine’s New Foundations (NF) ([14]) is obtained from TST by abandoning the types but retaining the same axioms. Note that the comprehension axioms of NF are not all the axioms “{x  φ} exists ” for φ a formula in the language of NF: this would be the inconsistent comprehension axiom of naive set theory. The comprehension axioms of NF are those assertions “{x  φ} exists ” where φ can be obtained from a formula of TST by dropping distinctions of type between variables (without creating any additional identifications between variables).
THE EMPTY SET, THE SINGLETON, AND THE ORDERED PAIR AKIHIRO KANAMORI
, 2002
"... For the modern set theorist the empty set ∅, the singleton {a}, and the ordered pair 〈x, y 〉 are at the beginning of the systematic, axiomatic development of set theory, both as a field of mathematics and as a unifying framework for ongoing mathematics. These notions are the simplest building blocks ..."
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For the modern set theorist the empty set ∅, the singleton {a}, and the ordered pair 〈x, y 〉 are at the beginning of the systematic, axiomatic development of set theory, both as a field of mathematics and as a unifying framework for ongoing mathematics. These notions are the simplest building blocks in the abstract, generative conception of sets advanced by the initial axiomatization of Ernst Zermelo [1908a] and are quickly assimilated long before the complexities of Power Set, Replacement, and Choice are broached in the formal elaboration of the ‘set of ’ {} operation. So it is surprising that, while these notions are unproblematic today, they were once sources of considerable concern and confusion among leading pioneers of mathematical logic like Frege, Russell, Dedekind, and Peano. In the development of modern mathematical logic out of the turbulence of 19th century logic, the emergence of the empty set, the singleton, and the ordered pair as clear and elementary settheoretic concepts serves as a motif that reflects and illuminates larger and more significant developments in mathematical logic:
Contemporary Mathematics Permutations and Wellfoundedness: the True Meaning of the Bizarre Arithmetic of Quine’s NF
"... It is shown that, according to NF, many of the assertions of ordinal arithmetic involving the Tfunction which is peculiar to NF turn out to be equivalent to the truthincertainpermutationmodels of assertions which have perfectly sensible ZFstyle meanings, such as: the existence of wellfounded s ..."
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It is shown that, according to NF, many of the assertions of ordinal arithmetic involving the Tfunction which is peculiar to NF turn out to be equivalent to the truthincertainpermutationmodels of assertions which have perfectly sensible ZFstyle meanings, such as: the existence of wellfounded sets of great size or rank, or the nonexistence of small counterexamples to the wellfoundedness of ∈. Everything here holds also for NFU if the permutations are taken to fix all urelemente. NF is Quine’s system of set theory, axiomatised by extensionality and those instances of the naïve set existence scheme that are stratified. A formula of the language of set theory is stratified if the variables within it can be labelled with integers in such a way that all occurrences of each variable receive the same label, and if x ∈ y appears in it then the label of x must be one less than the label of y, and if x = y occurs then the label of x must be the same as the label of y.
The usual model construction for NFU preserves information
, 2009
"... The “usual ” model construction for NFU (Quine’s New Foundations with urelements, shown to be consistent by Jensen) starts with a model of the usual set theory with an automorphism that moves a rank (this rank is the domain of the model). “Most ” elements of the resulting model of NFU are urelements ..."
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The “usual ” model construction for NFU (Quine’s New Foundations with urelements, shown to be consistent by Jensen) starts with a model of the usual set theory with an automorphism that moves a rank (this rank is the domain of the model). “Most ” elements of the resulting model of NFU are urelements (it appears that information about their extensions is discarded). The surprising result of this paper is that this information is not discarded at all: the membership relation of the original model (restricted to the domain of the model of NFU) is definable in the language of NFU. A corollary of this is that the urelements of a model of NFU obtained by the “usual ” construction are inhomogeneous: this was the question the author was investigating initially. Other aspects of the mutual interpretability of NFU and a fragment of ZFC are discussed in sufficient detail to place