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Higher Order Logic
 In Handbook of Logic in Artificial Intelligence and Logic Programming
, 1994
"... Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Definin ..."
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Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Defining data types : : : : : : : : : : : : : : : : : : : : : 6 2.4 Describing processes : : : : : : : : : : : : : : : : : : : : : 8 2.5 Expressing convergence using second order validity : : : : : : : : : : : : : : : : : : : : : : : : : 9 2.6 Truth definitions: the analytical hierarchy : : : : : : : : 10 2.7 Inductive definitions : : : : : : : : : : : : : : : : : : : : : 13 3 Canonical semantics of higher order logic : : : : : : : : : : : : 15 3.1 Tarskian semantics of second order logic : : : : : : : : : 15 3.2 Function and re
System Description: TPS: A Theorem Proving System for Type Theory
, 2000
"... Introduction This is a brief update on the Tps automated theorem proving system for classical type theory, which was described in [3]. Manuals and information about obtaining Tps can be found at http://gtps.math.cmu.edu/tps.html. In Section 2 we discuss some examples of theorems which Tps can now ..."
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Introduction This is a brief update on the Tps automated theorem proving system for classical type theory, which was described in [3]. Manuals and information about obtaining Tps can be found at http://gtps.math.cmu.edu/tps.html. In Section 2 we discuss some examples of theorems which Tps can now prove automatically, and in Section 3 we discuss an example which illustrates one of the many challenges of theorem proving in higherorder logic. We rst provide a brief summary of the key features of Tps . Tps uses Church's type theory [8] (typed calculus) as its logical language. Ws are displayed on the screen and in printed proofs in the notation of this system of symbolic logic. One can use Tps in automatic, semiautomatic, or interactive mode to construct proofs in natural deduction style, and a mixture of these modes of operation is most useful fo
The Mathematical Development Of Set Theory  From Cantor To Cohen
 The Bulletin of Symbolic Logic
, 1996
"... This article is dedicated to Professor Burton Dreben on his coming of age. I owe him particular thanks for his careful reading and numerous suggestions for improvement. My thanks go also to Jose Ruiz and the referee for their helpful comments. Parts of this account were given at the 1995 summer meet ..."
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This article is dedicated to Professor Burton Dreben on his coming of age. I owe him particular thanks for his careful reading and numerous suggestions for improvement. My thanks go also to Jose Ruiz and the referee for their helpful comments. Parts of this account were given at the 1995 summer meeting of the Association for Symbolic Logic at Haifa, in the Massachusetts Institute of Technology logic seminar, and to the Paris Logic Group. The author would like to express his thanks to the various organizers, as well as his gratitude to the Hebrew University of Jerusalem for its hospitality during the preparation of this article in the autumn of 1995.
Choice principles in constructive and classical set theories
 POHLERS (EDS.): PROCEEDINGS OF THE LOGIC COLLOQUIUM 2002
, 2002
"... The objective of this paper is to assay several forms of the axiom of choice that have been deemed constructive. In addition to their deductive relationships, the paper will be concerned with metamathematical properties effected by these choice principles and also with some of their classical models ..."
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The objective of this paper is to assay several forms of the axiom of choice that have been deemed constructive. In addition to their deductive relationships, the paper will be concerned with metamathematical properties effected by these choice principles and also with some of their classical models.
Full FirstOrder Free Variable Sequents and Tableaux in Implicit Induction
 IN IMPLICIT INDUCTION. 8 TH TABLEAU 1999, LNAI 1617
, 1999
"... We show how to integrate implicit inductive theorem proving into free variable sequent and tableau calculi and compare the appropriateness of tableau calculi for this integration with that of sequent calculi. When firstorder validity is introduced to students it comes with some complete calculus ..."
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We show how to integrate implicit inductive theorem proving into free variable sequent and tableau calculi and compare the appropriateness of tableau calculi for this integration with that of sequent calculi. When firstorder validity is introduced to students it comes with some complete calculus. If this calculus happens to be an analytic calculus augmented with a Cut rule like a sequent or tableau calculus the students can compare the formal proofs with the informal ones they are hopefully acquainted with. This is because these calculi can mirror the human proof search process better than others. While knowing a complete calculus does not mean to know much about firstorder theorem proving, the interrelation of a humanoriented calculus and the informal proof search of the students will turn out to be fruitful for their later mathematical work. It is a pity thatwhile nearly all proofs of a working mathematician include inductionnothing comparable for inductive firstorder ...
Arrow’s Theorem, Weglorz ’ Models and the Axiom of Choice [Mathematical Logic Quarterly (2000) 46: 335359]
"... Abstract. Applying Weglorz ’ models of set theory without the axiom of choice, we investigate Arrowtype social welfare functions for infinite societies with restricted coalition algebras. We show that there is a reasonable, nondictatorial social welfare function satisfying “finite discrimination”, ..."
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Abstract. Applying Weglorz ’ models of set theory without the axiom of choice, we investigate Arrowtype social welfare functions for infinite societies with restricted coalition algebras. We show that there is a reasonable, nondictatorial social welfare function satisfying “finite discrimination”, if and only if in Weglorz ’ model there is a free ultrafilter on a set representing the individuals. AMSsubject classification. 03E35 (Consistency and Independence Proofs), 90A08 (Social Choice) Key words. Arrow’s theorem, anonymity, social/ecological welfare functions; axiom of choice, ultrafilters, Weglorz ’ models, permutation models. 1. Introduction. Arrow’s theorem [2], as formulated by [18], is the assertion that the decisive coalitions of a reasonable 1 social welfare function F form an ultrafilter 2 U on the set I of individuals. Here the (two or more) voters in I decide about the “social preference ” (the output of the social welfare function) on three or more
COMPACTNESS IN COUNTABLE TYCHONOFF PRODUCTS AND CHOICE
, 1999
"... Abstract. We study the relationship between the countable axiom of choice and the Tychonoff product theorem for countable families of topological spaces. ..."
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Abstract. We study the relationship between the countable axiom of choice and the Tychonoff product theorem for countable families of topological spaces.