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Part of the book series: Synthese Library ((SYLI,volume 476))

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We conjecture that the existence of some fast-growing functions implies in a simple way the existence of some inaccessible cardinals. This note expands some previous work by the second author.

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  1. 1.

    For notation and basic facts, see da Costa and Doria (2017).

  2. 2.

    This is a much quoted result which has never appeared in print despite its obvious importance. There are proofs in particular cases. We decided to offer this pedestrian but general proof which has been checked by M. Guillaume. The proof we exhibit is quite general and intuitive; it has originally appeared in Carnielli and Doria (2008). To avoid misunderstandings we give it verbatim—we are the authors, anyway.

  3. 3.

    Kleene made F into the centralpiece of his incompleteness phenomenon, see S. C. Kleene, “General recursive functions of the natural numbers” (Kleene, 1936).


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It is a pleasure to dedicate these sketchy doodles to our good friend and colleague Décio Krause. We hope it contains the hidden pearl we think we have perceived in those ideas.

The construction we present here originated in an e-mail exchange between the authors and G. Kreisel on the fast-growth properties of the so-called counterexample function to P = NP, however the quasi-trivial machines are the authors’ own responsibility. We also discussed it with M. Guillaume who helped us with his usual acumen.

And of course the title refers to the 1953 movie featuring Burt Lancaster, Montgomery Clift and Deborah Kerr—and to Ian Stewart’s book From Here to Infinity (Stewart, 2009).

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daCosta, N.C.A., Doria, F.A. (2023). From Here to Eternity. In: Arenhart, J.R.B., Arroyo, R.W. (eds) Non-Reflexive Logics, Non-Individuals, and the Philosophy of Quantum Mechanics. Synthese Library, vol 476. Springer, Cham.

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