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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
Constructing Cardinals from below
"... this paper are all formulated in terms of a formula #(X), with only X free. For now, the formula is one in the language of basic set theory and X is a secondorder variable. The corresponding condition is that #(A) is true in R(#) for some A and, for no # # is #(A R(#)) true in R(#). The fo ..."
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Cited by 3 (0 self)
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this paper are all formulated in terms of a formula #(X), with only X free. For now, the formula is one in the language of basic set theory and X is a secondorder variable. The corresponding condition is that #(A) is true in R(#) for some A and, for no # # is #(A R(#)) true in R(#). The formal expression that this condition is an existence condition is the axiom #X[#(X) # ### R(#))] (1) (X) is the result of restricting the first and secondorder bound variables in #(X) to R(#) and R(# + 1), respectively. Axioms of this form have been called reflection principles, because they express the fact that R(#)'s possession of a certain property is reflected by R(#)'s possession of it for some # #