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Types in logic and mathematics before 1940
 Bulletin of Symbolic Logic
, 2002
"... Abstract. In this article, we study the prehistory of type theory up to 1910 and its development between Russell and Whitehead’s Principia Mathematica ([71], 1910–1912) and Church’s simply typed λcalculus of 1940. We first argue that the concept of types has always been present in mathematics, thou ..."
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Abstract. In this article, we study the prehistory of type theory up to 1910 and its development between Russell and Whitehead’s Principia Mathematica ([71], 1910–1912) and Church’s simply typed λcalculus of 1940. We first argue that the concept of types has always been present in mathematics, though nobody was incorporating them explicitly as such, before the end of the 19th century. Then we proceed by describing how the logical paradoxes entered the formal systems of Frege, Cantor and Peano concentrating on Frege’s Grundgesetze der Arithmetik for which Russell applied his famous paradox 1 and this led him to introduce the first theory of types, the Ramified Type Theory (rtt). We present rtt formally using the modern notation for type theory and we discuss how Ramsey, Hilbert and Ackermann removed the orders from rtt leading to the simple theory of types stt. We present stt and Church’s own simply typed λcalculus (λ→C 2) and we finish by comparing rtt, stt and λ→C. §1. Introduction. Nowadays, type theory has many applications and is used in many different disciplines. Even within logic and mathematics, there are many different type systems. They serve several purposes, and are formulated in various ways. But, before 1903 when Russell first introduced
Strong Axioms of Infinity in ...
"... This paper discusses a sequence of extensions of NFU , Jensen's improvement of Quine's set theory \New Foundations" (NF ) of [16] ..."
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This paper discusses a sequence of extensions of NFU , Jensen's improvement of Quine's set theory \New Foundations" (NF ) of [16]
Quine's NF60 years on
, 1998
"... Sixty years ago in this journal, the distinguished American philosopher W.V. Quine published a novel approach to set theory. The title was New Foundations for Mathematical Logic [6]. The diamond anniversary is being commemorated by a workshop in Cambridge (England) and comes at a time of rapid incre ..."
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Sixty years ago in this journal, the distinguished American philosopher W.V. Quine published a novel approach to set theory. The title was New Foundations for Mathematical Logic [6]. The diamond anniversary is being commemorated by a workshop in Cambridge (England) and comes at a time of rapid increase of interest in the alternatives to the hitherto customary ZermeloFr"ankel set theory, which promises a new lease of life for the axiomatic system now known as `NF'; its creator remains in good health too. Although he is best known to a wider public for his philosophical writings, his most enduring and most concrete legacy for the next fifty years may well turn out to be his most mathematical: he gave us NF. Set theory is the study of sets, which are the simplest of all mathematical entities. Let us illustrate by constrasting sets with groups. Two distinct groups can have the same elements and yet be told apart by the way those elements are related. Sets are distinguished from all other mathematical fauna by the fact that a set is constituted solely by its members: two sets with the same members are the same set. To use a bit of jargon from another age, sets are properties in extension. As a result, all set theories have the axiom of extensionality: (8xy)(x = y! (8z)(z 2 x! z 2 y)): they differ in their views on which properties have extensions. Since set theory first sprang on the scene about a hundred years ago there has been a tendency to attempt to use this simplicity to simplify and illuminate the rest of mathematics by translating (perhaps a better word is implementing) it into set theory. After all, if we can represent all of mathematics as facts about these delightfully simple things, some facts about mathematics might become clear that would otherwise remain obscure. This same simplicity means that set theory is always a good topic on which to try out any new mathematical idea.
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
Alternative Set Theories
, 2006
"... By “alternative set theories ” we mean systems of set theory differing significantly from the dominant ZF (ZermeloFrankel set theory) and its close relatives (though we will review these systems in the article). Among the systems we will review are typed theories of sets, Zermelo set theory and its ..."
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By “alternative set theories ” we mean systems of set theory differing significantly from the dominant ZF (ZermeloFrankel set theory) and its close relatives (though we will review these systems in the article). Among the systems we will review are typed theories of sets, Zermelo set theory and its variations, New Foundations and related systems, positive set theories, and constructive set theories. An interest in the range of alternative set theories does not presuppose an interest in replacing the dominant set theory with one of the alternatives; acquainting ourselves with foundations of mathematics formulated in terms of an alternative system can be instructive as showing us what any set theory (including the usual one) is supposed to do for us. The study of alternative set theories can dispel a facile identification of “set theory ” with “ZermeloFraenkel set theory”; they are not the same thing. Contents 1 Why set theory? 2 1.1 The Dedekind construction of the reals............... 3 1.2 The FregeRussell definition of the natural numbers....... 4
Ambiguous Cardinals ∗
"... A cardinal number κ is said to be “ambiguous ” if it is indiscernible from 2 to the power of κ. In a more specific way, κ is ambiguous if the natural typed structure over a set X of size κ is elementarily equivalent to the natural structure over the power set of X. Some striking results arising from ..."
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A cardinal number κ is said to be “ambiguous ” if it is indiscernible from 2 to the power of κ. In a more specific way, κ is ambiguous if the natural typed structure over a set X of size κ is elementarily equivalent to the natural structure over the power set of X. Some striking results arising from the method Specker used to refute AC in NF will be extracted from the NF context and used to give results in the more usual ZF theory. 1 Strong ambiguity The fact that no set can be equal to the set of its subsets contradicts our primitive intuition that there is nothing more dreamt of in our universe of sets, than things in heaven and earth, i.e. that P(U) is not greater than U. Among the possible different ways out of this predicament, I will consider only the following one: while not insisting that P(U) = U, one should at least expect that U and P(U) could be somehow indiscernible. Taken literally, even this is not true, since one can define a notion of height of a set: h(X) is the cardinal of the set {..., X2, X1, X}, where P(Xi+1) = Xi. Now, since this set is finite 1, h(X) is even iff h(P(X)) is odd. Actually, the way Cantor showed that X ̸ = P(X) was by showing that X  ̸ = P(X). It seems therefore natural to limit the indiscernibility requirement to properties of cardinals. The consistency of ZF plus the existence of a cardinal κ indiscernible from 2 κ, in an unqualified sense, is open. The same question with ZFC is easily settled as we now show.
semantic
, 2012
"... Symmetry motivates a new consistent fragment of NF and an extension of NF with ..."
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Symmetry motivates a new consistent fragment of NF and an extension of NF with