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Gödel's program for new axioms: Why, where, how and what?
 IN GODEL '96
, 1996
"... From 1931 until late in his life (at least 1970) Gödel called for the pursuit of new axioms for mathematics to settle both undecided numbertheoretical propositions (of the form obtained in his incompleteness results) and undecided settheoretical propositions (in particular CH). As to the nature of ..."
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From 1931 until late in his life (at least 1970) Gödel called for the pursuit of new axioms for mathematics to settle both undecided numbertheoretical propositions (of the form obtained in his incompleteness results) and undecided settheoretical propositions (in particular CH). As to the nature of these, Gödel made a variety of suggestions, but most frequently he emphasized the route of introducing ever higher axioms of in nity. In particular, he speculated (in his 1946 Princeton remarks) that there might be a uniform (though nondecidable) rationale for the choice of the latter. Despite the intense exploration of the "higher infinite" in the last 30odd years, no single rationale of that character has emerged. Moreover, CH still remains undecided by such axioms, though they have been demonstrated to have many other interesting settheoretical consequences. In this paper, I present a new very general notion of the "unfolding" closure of schematically axiomatized formal systems S which provides a uniform systematic means of expanding in an essential way both the language and axioms (and hence theorems) of such systems S. Reporting joint work with T. Strahm, a characterization is given in more familiar terms in the case that S is a basic
A New Approach to Predicative Set Theory
"... We suggest a new basic framework for the WeylFeferman predicativist program by constructing a formal predicative set theory PZF which resembles ZF. The basic idea is that the predicatively acceptable instances of the comprehension schema are those which determine the collections they define in an a ..."
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We suggest a new basic framework for the WeylFeferman predicativist program by constructing a formal predicative set theory PZF which resembles ZF. The basic idea is that the predicatively acceptable instances of the comprehension schema are those which determine the collections they define in an absolute way, independent of the extension of the “surrounding universe”. This idea is implemented using syntactic safety relations between formulas and sets of variables. These safety relations generalize both the notion of domainindependence from database theory, and Godel notion of absoluteness from set theory. The language of PZF is typefree, and it reflects real mathematical practice in making an extensive use of statically defined abstract set terms. Another important feature of PZF is that its underlying logic is ancestral logic (i.e. the extension of FOL with a transitive closure operation). 1
Ramified HigherOrder Unification
, 1996
"... While unification in the simple theory of types (a.k.a. higherorder logic) is undecidable, we show that unification in the pure ramified theory of types with integer levels is decidable. Since pure ramified type theory is not very expressive, we examine the impure case, which has an undecidable uni ..."
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While unification in the simple theory of types (a.k.a. higherorder logic) is undecidable, we show that unification in the pure ramified theory of types with integer levels is decidable. Since pure ramified type theory is not very expressive, we examine the impure case, which has an undecidable unification problem even at order 2. However, the decidability result for the pure subsystem indicates that unification terminates more often than general higherorder unification. We present an application to ACA 0 and other expressive subsystems of secondorder Peano arithmetic.
AXIOMATIZING MATHEMATICAL CONCEPTUALISM IN THIRD ORDER ARITHMETIC
"... Abstract. We review the philosophical framework of mathematical conceptualism as an alternative to settheoretic foundations and show how mainstream mathematics can be developed on this basis. The paper includes an explicit axiomatization of the basic principles of conceptualism in a formal system C ..."
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Abstract. We review the philosophical framework of mathematical conceptualism as an alternative to settheoretic foundations and show how mainstream mathematics can be developed on this basis. The paper includes an explicit axiomatization of the basic principles of conceptualism in a formal system CM set in the language of third order arithmetic. This paper is part of a project whose goal is to make a case that mathematics should be disassociated from set theory. The reasons for wanting to do this, which I discuss in greater detail elsewhere ([22]; see also [19] and [23]), involve both the philosophical unsoundness of set theory and its practical irrelevance to mainstream mathematics. Set theory is based on the reification of a collection as a separate object, an elementary philosophical error. Not only is this error obvious, it also has the spectacular consequence of immediately giving rise to the classical set theoretic paradoxes. Of course, these paradoxes are not derivable in the standard axiomatizations of set theory, but that is only because these systems were specifically designed to avoid them. In these systems the paradoxes are blocked by means of ad hoc restrictions on the set concept that have no obvious intuitive justification, which has led to the development of a large literature of attempted rationalizations
MATHEMATICAL CONCEPTUALISM
"... We must remember that in Hilbert’s time all mathematicians were excited about the foundations of mathematics. Intense controversy centered around the problem of the legitimacy of abstract objects... The contrast with today’s foggy atmosphere of intellectual exhaustion and compartmentalization could ..."
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We must remember that in Hilbert’s time all mathematicians were excited about the foundations of mathematics. Intense controversy centered around the problem of the legitimacy of abstract objects... The contrast with today’s foggy atmosphere of intellectual exhaustion and compartmentalization could not be more striking. — Stephen Simpson ([15], p. 350) From the standpoint of mainstream mathematics, the great foundational debates of the early twentieth century were decisively settled in favor of Cantorian set theory, as formalized in the system ZFC (ZermeloFraenkel set theory including the axiom of choice). Although basic foundational questions have never entirely disappeared, it seems fair to say that they have retreated to the periphery of mathematical practice. Sporadic alternative proposals like topos theory or Errett Bishop’s constructivism have never attracted a substantial mainstream following, and Cantor’s universe is generally acknowledged as the arena in which modern mathematics takes place. Despite this history, I believe a strong case can be made for abandoning Cantorian
MATHEMATICAL CONCEPTUALISM
, 2005
"... We must remember that in Hilbert’s time all mathematicians were excited about the foundations of mathematics. Intense controversy centered around the problem of the legitimacy of abstract objects... The contrast with today’s foggy atmosphere of intellectual exhaustion and compartmentalization could ..."
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We must remember that in Hilbert’s time all mathematicians were excited about the foundations of mathematics. Intense controversy centered around the problem of the legitimacy of abstract objects... The contrast with today’s foggy atmosphere of intellectual exhaustion and compartmentalization could not be more striking. — Stephen Simpson ([14], p. 350) From the standpoint of mainstream mathematics, the great foundational debates of the early twentieth century were decisively settled in favor of Cantorian set theory, as formalized in the system ZFC (ZermeloFraenkel set theory including the axiom of choice). Although basic foundational questions have never entirely disappeared, it seems fair to say that they have retreated to the periphery of mathematical practice. Sporadic alternative proposals like topos theory or Errett Bishop’s constructivism have never attracted a substantial mainstream following, and Cantor’s universe is generally acknowledged as the arena in which modern mathematics takes place. Despite this history, I believe a strong case can be made for abandoning Cantorian