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The Mark2 Theorem Prover
"... ABSTRACT1 @ ?x: ?x = 68 Id @ ?x ["ID"] *)  s "?x";  rri "ID";ex();  p "ABSTRACT1@?x"; (* ABSTRACT2 @ ?x: ?f @ ?a = COMP != ?f @ (ABSTRACT @ ?x) =? ?a ["COMP"] *)  s "?f@?a";  rri "COMP";  right(); right(); ri "ABSTRACT@?x";  prove "ABSTRACT2@?x"; (* ABSTRACT3 @ ?x: ?a & ? ..."
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ABSTRACT1 @ ?x: ?x = 68 Id @ ?x ["ID"] *)  s "?x";  rri "ID";ex();  p "ABSTRACT1@?x"; (* ABSTRACT2 @ ?x: ?f @ ?a = COMP != ?f @ (ABSTRACT @ ?x) =? ?a ["COMP"] *)  s "?f@?a";  rri "COMP";  right(); right(); ri "ABSTRACT@?x";  prove "ABSTRACT2@?x"; (* ABSTRACT3 @ ?x: ?a & ?b = RAISE0 =? ((ABSTRACT @ ?x) =? ?a) & (ABSTRACT @ ?x) =? ?b [] *)  s "?a&?b";  right();  ri "ABSTRACT@?x";  up();left();  ri "ABSTRACT@?x";  top();  ri "RAISE0";  prove "ABSTRACT3@?x"; (* ABSTRACT4 @ ?x: ?a = [?a] @ ?x [] *)  s "?a";  ri "BIND@?x"; ex(); 69  p "ABSTRACT4@?x"; (* ABSTRACT@term will (attempt to) express a target term as a function of its parameter "term" *) (* ABSTRACT @ ?x: ?a = (ABSTRACT4 @ ?x) =?? (ABSTRACT3 @ ?x) =?? (ABSTRACT2 @ ?x) =?? (ABSTRACT1 @ ?x) =? ?a ["COMP","ID"] *)  s "?a";  ri "ABSTRACT1@?x";  ari "ABSTRACT2@?x";  ari "ABSTRACT3@?x";  ari "ABSTRACT4@?x";  p "ABSTRACT@?x"; (* REDUCE will reverse the effect of ABSTRACT; it will "evaluate" functions built by ABSTRACT *) (* REDUCE: ?f @ ?x = (ABSTRACT4 @ ?x) !!= ((RL @ REDUCE) *? RAISE0) !!= ((RIGHT @ REDUCE) *? COMP) =?? ID =? ?f @ ?x ["COMP","ID"] *)  dpt "REDUCE";  s "?f@?x";  ri "ID";  ari "(RIGHT@REDUCE)*?COMP";  arri "(RL@REDUCE)*?RAISE0";  arri "ABSTRACT4@?x";  prove "REDUCE"; (* old approach to hypotheses *) (* equational forms of tactics given without proof; the proofs of the tactics involve no actual rewriting *) PIVOT: (?a = ?b)  ?T , ?U = (RIGHT @ LEFT @ EVAL) =? HYP =? (?a = ?b) 70  ((BIND @ ?a) =? ?T) , ?U ["HYP"] REVPIVOT: (?a = ?b)  ?T , ?U = (RIGHT @ LEFT @ EVAL) =? HYP != (?a = ?b)  ((BIND @ ?b) =? ?T) , ?U ["HYP"] We now present examples of the use of thes...
Object Oriented Mathematics
"... This paper shows that OO principles can be used to enhance the rigour of mathematical notation without loss of brevity and clarity. It is well known that traditional mathematical notation is not completely formal. This is so because mathematicians and other users of mathematical notation tend to ..."
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This paper shows that OO principles can be used to enhance the rigour of mathematical notation without loss of brevity and clarity. It is well known that traditional mathematical notation is not completely formal. This is so because mathematicians and other users of mathematical notation tend to sacrifice exactness to obtain brevity and clarity. The mathematician thereby leaves to the reader to guess the meaning of each formula presented based on the written and unwritten rules of the particular field of research. This works perfectly well for communication between researchers in the same field, but may be an obstacle for communication between researchers form different fields or for newcomers such as students. The lack of rigour in mathematical notation may also be an obstacle when mathematical phenomena are to be simulated on computers, where the programmer has to fill out the gaps in the notation. It is generally believed that complete formal rigour leads to an explosio...
Identity of indiscernibles in Quine's "New Foundations" and related theories
"... this paper refers to some witness to the appropriate instance of the comprehension axiom, and does not implicitly assume the existence of a unique or even a canonical witness) ..."
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this paper refers to some witness to the appropriate instance of the comprehension axiom, and does not implicitly assume the existence of a unique or even a canonical witness)
The Theory of the Foundations of Mathematics
, 2002
"... form of Godel's first theorem: Let P be a set of Godel numbers of all the provable sentences. If the set # is expressible in correct, then there is a true sentence of not provable in L. ..."
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form of Godel's first theorem: Let P be a set of Godel numbers of all the provable sentences. If the set # is expressible in correct, then there is a true sentence of not provable in L.
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
4, SOME FORMAL SYSTEMS FOR THE UNLIMITED THEORY OF STRUCTURES AND CATEGORIES
"... Abstract. In the informal unlimited theory of structures and (particularly) categories, one considers unrestricted statements concerning structures such as that the substructure relation on all structures of a given kind forms a partially ordered structure. or that the collection of all categories f ..."
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Abstract. In the informal unlimited theory of structures and (particularly) categories, one considers unrestricted statements concerning structures such as that the substructure relation on all structures of a given kind forms a partially ordered structure. or that the collection of all categories forms a category with arbitrary These sorts of propositio~s are not accounted for difunctors as its morphisms. The aim of the present work is to give a founrectly by currently accepted means. dation for the theory of structures including such unlimited statementsmore or less as they are presented to usby means of certain formal systems. The theories studied here are based on an extension of Quinels idea of stratification. Their use is justified bya consistency proof. adapting methods of Jensen. These systems are successful for the basic aim to a considerable extent. but they suffer a specific defect which prevents them from being fully successful. Some possible alternatives are also suggested.
M. H. Newman’s Typability Algorithm for LambdaCalculus
, 2006
"... This article is essentially an extended review with historical comments. It looks at an algorithm published in 1943 by M. H. A. Newman, which decides whether a lambdacalculus term is typable without actually computing its principal type. Newman’s algorithm seems to have been completely neglected by ..."
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This article is essentially an extended review with historical comments. It looks at an algorithm published in 1943 by M. H. A. Newman, which decides whether a lambdacalculus term is typable without actually computing its principal type. Newman’s algorithm seems to have been completely neglected by the typetheorists who invented their own rather different typability algorithms over 15 years later. 1
THE AXIOM SCHEME OF ACYCLIC COMPREHENSION
"... Abstract. A “new ” criterion for set existence is presented, namely, that a set {x  φ} should exist if the multigraph whose nodes are variables in φ and whose edges are occurrences of atomic formulas in φ is acyclic. Formulas with acyclic graphs are stratified in the sense of New Foundations, so co ..."
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Abstract. A “new ” criterion for set existence is presented, namely, that a set {x  φ} should exist if the multigraph whose nodes are variables in φ and whose edges are occurrences of atomic formulas in φ is acyclic. Formulas with acyclic graphs are stratified in the sense of New Foundations, so consistency of the set theory with weak extensionality and acyclic comprehension follows from the consistency of Jensen’s system NFU. It is much less obvious, but turns out to be the case, that this theory is equivalent to NFU: it appears at first blush that it ought to be weaker. This paper verifies that acyclic comprehension and stratified comprehension are equivalent, by verifying that each axiom in a finite axiomatization of stratified comprehension follows from acyclic comprehension.
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