As is well-known, Cantor's continuum problem, namely, what is the cardinality of ? is in... more As is well-known, Cantor's continuum problem, namely, what is the cardinality of ? is independent of the usual ZFC ax- ioms of Set Theory. K. Godel ((12), (13)) suggested that new natural axioms should be found that would settle the problem and hinted at large-cardinal axioms as such. However, shortly after the invention of forcing, it was shown by Levy and Solovay (20) that the prob- lem remains independent even if one adds to ZFC the usual large- cardinal axioms, like the existence of measurable cardinals, or even supercompact cardinals, provided, of course, that these axioms are consistent. While numerous axioms have been proposed that settle the problem-although not always in the same way-from the Axiom of Constructibility to strong combinatorial axioms like the Proper Forcing Axiom or Martin's Maximum, none of them so far has been recognized as a natural axiom and been accepted as an appropriate solution to the continuum problem. In this paper we discuss some heuristic principles, which might be regarded as Meta-Axioms of Set Theory, that provide a criterion for assessing the naturalness of the set-theoretic axioms. Under this criterion we then evaluate several kinds of axioms, with a special emphasis on a class of recently intro- duced set-theoretic principles for which we can reasonably argue that they constitute very natural axioms of Set Theory and which settle Cantor's continuum problem.
As is well-known, Cantor's continuum problem, namely, what is the cardinality of ? is in... more As is well-known, Cantor's continuum problem, namely, what is the cardinality of ? is independent of the usual ZFC ax- ioms of Set Theory. K. Godel ((12), (13)) suggested that new natural axioms should be found that would settle the problem and hinted at large-cardinal axioms as such. However, shortly after the invention of forcing, it was shown by Levy and Solovay (20) that the prob- lem remains independent even if one adds to ZFC the usual large- cardinal axioms, like the existence of measurable cardinals, or even supercompact cardinals, provided, of course, that these axioms are consistent. While numerous axioms have been proposed that settle the problem-although not always in the same way-from the Axiom of Constructibility to strong combinatorial axioms like the Proper Forcing Axiom or Martin's Maximum, none of them so far has been recognized as a natural axiom and been accepted as an appropriate solution to the continuum problem. In this paper we discuss some heuristic principles, which might be regarded as Meta-Axioms of Set Theory, that provide a criterion for assessing the naturalness of the set-theoretic axioms. Under this criterion we then evaluate several kinds of axioms, with a special emphasis on a class of recently intro- duced set-theoretic principles for which we can reasonably argue that they constitute very natural axioms of Set Theory and which settle Cantor's continuum problem.
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