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Circumscribing with sets
 Artificial Intelligence
, 1987
"... Abstract: Sets can play an important role in circumscription’s ability to deal in a general way with certain aspects of commonsense reasoning. Aresult of Kueker indicates that sentences that intuitively one would want circumscription to prove, are nonetheless not so provable in a formal setting devo ..."
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Abstract: Sets can play an important role in circumscription’s ability to deal in a general way with certain aspects of commonsense reasoning. Aresult of Kueker indicates that sentences that intuitively one would want circumscription to prove, are nonetheless not so provable in a formal setting devoid of sets. Furthermore, when sets are introduced, firstorder circumscription handles these cases very easily, obviating the need for secondorder circumscription. The Aussonderungs axiom of ZF set theory plays an intuitive role in this shift back to a firstorder language. descriptors: commonsense, circumscription, sets I.
Commonsense set theory
 MetaLevel Architectures and Reflection. North
, 1988
"... Abstract: It is argued that set theory provides a powerful addition to commonsense reasoning, facilitating expression of metaknowledge, names, and selfreference. Difficulties in establishing a suitable language to include sets for such purposes are discussed, as well as what appear to be promising ..."
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Abstract: It is argued that set theory provides a powerful addition to commonsense reasoning, facilitating expression of metaknowledge, names, and selfreference. Difficulties in establishing a suitable language to include sets for such purposes are discussed, as well as what appear to be promising solutions. Ackermann’s set theory as well as a more recent theory involving universal sets are discussed in terms of their relevance to commonsense.
Why sets?
 PILLARS OF COMPUTER SCIENCE: ESSAYS DEDICATED TO BORIS (BOAZ) TRAKHTENBROT ON THE OCCASION OF HIS 85TH BIRTHDAY, VOLUME 4800 OF LECTURE NOTES IN COMPUTER SCIENCE
, 2008
"... Sets play a key role in foundations of mathematics. Why? To what extent is it an accident of history? Imagine that you have a chance to talk to mathematicians from a faraway planet. Would their mathematics be setbased? What are the alternatives to the settheoretic foundation of mathematics? Besi ..."
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Sets play a key role in foundations of mathematics. Why? To what extent is it an accident of history? Imagine that you have a chance to talk to mathematicians from a faraway planet. Would their mathematics be setbased? What are the alternatives to the settheoretic foundation of mathematics? Besides, set theory seems to play a significant role in computer science; is there a good justification for that? We discuss these and some related issues.
A Framework for Formalizing Set Theories Based on the Use of Static Set Terms
"... To Boaz Trakhtenbrot: a scientific father, a friend, and a great man. Abstract. We present a new unified framework for formalizations of axiomatic set theories of different strength, from rudimentary set theory to full ZF. It allows the use of set terms, but provides a static check of their validity ..."
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To Boaz Trakhtenbrot: a scientific father, a friend, and a great man. Abstract. We present a new unified framework for formalizations of axiomatic set theories of different strength, from rudimentary set theory to full ZF. It allows the use of set terms, but provides a static check of their validity. Like the inconsistent “ideal calculus ” for set theory, it is essentially based on just two settheoretical principles: extensionality and comprehension (to which we add ∈induction and optionally the axiom of choice). Comprehension is formulated as: x ∈{x  ϕ} ↔ϕ, where {x  ϕ} is a legal set term of the theory. In order for {x  ϕ} to be legal, ϕ should be safe with respect to {x}, where safety is a relation between formulas and finite sets of variables. The various systems we consider differ from each other mainly with respect to the safety relations they employ. These relations are all defined purely syntactically (using an induction on the logical structure of formulas). The basic one is based on the safety relation which implicitly underlies commercial query languages for relational database systems (like SQL). Our framework makes it possible to reduce all extensions by definitions to abbreviations. Hence it is very convenient for mechanical manipulations and for interactive theorem proving. It also provides a unified treatment of comprehension axioms and of absoluteness properties of formulas. 1
Global Reflection Principles
, 2012
"... Reflection Principles are commonly thought to produce only strong axioms of infinity consistent with V = L. It would be desirable to have some notion of strong reflection to remedy this, and we have proposed Global Reflection Principles based on a somewhat Cantorian view of the universe. Such princi ..."
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Reflection Principles are commonly thought to produce only strong axioms of infinity consistent with V = L. It would be desirable to have some notion of strong reflection to remedy this, and we have proposed Global Reflection Principles based on a somewhat Cantorian view of the universe. Such principles justify the kind of cardinals needed for, inter alia, Woodin’s ΩLogic. 1 To say that the universe of all sets is an unfinished totality does not mean objective undeterminateness, but merely a subjective inability to finish it. Gödel, in Wang, [17] 1 Reflection Principles in Set Theory Historically reflection principles are associated with attempts to say that no one notion, idea, or statement can capture our whole view of the universe of sets V = ⋃ α∈On Vα where On is the class of all ordinals. That no one idea can pin down the universe of all sets has firm historical roots (see the quotation from Cantor later or the following): The Universe of sets cannot be uniquely characterized (i.e. distinguished from all its initial segments) by any internal structural property of the membership relation in it, which is expressible in any logic of finite or transfinite type, including infinitary logics of any cardinal number. Gödel: Wang ibid. Indeed once set theory was formalized by the (first order version of) the axioms and schemata of Zermelo with the additions of Skolem and Fraenkel, it was seen that reflection of first order formulae ϕ(v0, , vn) in the language of set theory L∈ ˙ could actually be proven:
Implicit dynamic function introduction and its connections to the foundations of mathematics
"... Abstract: We discuss a feature of the natural language of mathematics – the implicit dynamic introduction of functions – that has, to our knowledge, not been captured in any formal system so far. If this feature is used without limitations, it yields a paradox analogous to Russell’s paradox. Hence a ..."
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Abstract: We discuss a feature of the natural language of mathematics – the implicit dynamic introduction of functions – that has, to our knowledge, not been captured in any formal system so far. If this feature is used without limitations, it yields a paradox analogous to Russell’s paradox. Hence any formalism capturing it has to impose some limitations on it. We sketch two formalisms, both extensions of Dynamic Predicate Logic, that innovatively do capture this feature, and that differ only in the limitations they impose onto it. One of these systems is based on a novel theory of functions that interprets ZFC, and thus exhibits interesting connections to the foundations of mathematics.
HigherOrder Dynamic Predicate Logic
"... Dynamic Predicate Logic (DPL) [3] is a formalism whose syntax is identical to that of standard firstorder predicate logic (PL), but whose semantics is defined in such a way that the dynamic nature of natural language quantification is captured in the formalism: 1. If a farmer owns a donkey, he beat ..."
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Dynamic Predicate Logic (DPL) [3] is a formalism whose syntax is identical to that of standard firstorder predicate logic (PL), but whose semantics is defined in such a way that the dynamic nature of natural language quantification is captured in the formalism: 1. If a farmer owns a donkey, he beats it. 2. PL: ∀x∀y (farmer(x) ∧ donkey(y) ∧ owns(x, y) → beats(x, y)) 3. DPL: ∃x (farmer(x) ∧ ∃y (donkey(y) ∧ owns(x, y))) → beats(x, y) In PL, 3 is not a sentence, since the final occurences of x and y are free. In DPL, a variable may be bound by a quantifier even if it is outside its scope. The semantics is defined in such a way that 3 is equivalent to 2. So in DPL, 3 captures the meaning of 1 while being more faithful to its syntax than 2. 2 Dynamic introduction of function symbols In natural language mathematical texts, function symbols may be introduced dynamically: Suppose that, for each vertex v of K, there is a vertex g(v) of L such