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From 1931 until late in his life (at least 1970) Godel called for the pursuit of new axioms for mathematics to settle both undecided number-theoretical propositions (of the form obtained in his incompleteness results) and undecided set-theoretical propositions (in particular CH). As to the nature of these, Godel 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 non-decidable) rationale for the choice of the latter. Despite the intense exploration of the \higher in nite" in the last 30-odd 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 set-theoretical consequences.
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Citations of this work BETA
Solomon Feferman & Thomas Strahm (2000). The Unfolding of Non-Finitist Arithmetic. Annals of Pure and Applied Logic 104 (1-3):75-96.
Sebastian Eberhard (2014). A Feasible Theory of Truth Over Combinatory Algebra. Annals of Pure and Applied Logic 165 (5):1009-1033.
Pavel Pudlák (1999). A Note on Applicability of the Incompleteness Theorem to Human Mind. Annals of Pure and Applied Logic 96 (1-3):335-342.
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