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Asymptotic Quasi-completeness and ZFC

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Contradictions, from Consistency to Inconsistency

Part of the book series: Trends in Logic ((TREN,volume 47))

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Abstract

The axioms ZFC of first order set theory are one of the best and most widely accepted, if not perfect, foundations used in mathematics. Just as the axioms of first order Peano Arithmetic, ZFC axioms form a recursively enumerable list of axioms, and are, then, subject to Gödel’s Incompleteness Theorems. Hence, if they are assumed to be consistent, they are necessarily incomplete. This can be witnessed by various concrete statements, including the celebrated Continuum Hypothesis CH. The independence results about the infinite cardinals are so abundant that it often appears that ZFC can basically prove very little about such cardinals. However, we put forward a thesis that ZFC is actually very powerful at some infinite cardinals, but not at all of them. We have to move away from the first few and to look at limits of uncountable cardinals, such as \( \aleph _\omega \). Specifically, we work with singular cardinals (which are necessarily limits) and we illustrate that at such cardinals there is a very serious limit to independence and that many statements which are known to be independent on regular cardinals become provable or refutable by ZFC at singulars. In a certain sense, which we explain, the behavior of the set-theoretic universe is asymptotically determined at singular cardinals by the behavior that the universe assumes at the smaller regular cardinals. Foundationally, ZFC provides an asymptotically univocal image of the universe of sets around the singular cardinals. We also give a philosophical view accounting for the relevance of these claims in a platonistic perspective which is different from traditional mathematical platonism.

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Notes

  1. 1.

    According to the common abuse of notation, we call F ‘power set function’, even though it is in fact a class-function.

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Acknowledgements

The first author gratefully acknowledges the help of EPSRC through the grant EP/I00498, Leverhulme Trust through research Fellowship 2014–2015 and l’Institut d’Histoire et de Philosophie des Sciences et des Techniques, Université Paris 1, where she is an Associate Member. The second acknowledges the support of ANR through the project ObMathRe. The authors are grateful to Walter Carnielli for his instructive comments on a preliminary version of the manuscript, and to Marianna Antonutti-Marfori, Drew Moshier and Rachael Schiel for valuable suggestions.

Author information

Authors and Affiliations

  1. School of Mathematics, University of East Anglia, Norwich, NR4 7TJ, UK

    Mirna Džamonja

  2. CNRS, IHPST (CNRS and Université Paris 1, Panthéon-Sorbonne), 13 Rue de Four, 75006, Paris, France

    Marco Panza

  3. Chapman University, Orange, CA, USA

    Marco Panza

Authors
  1. Mirna Džamonja
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Correspondence to Marco Panza .

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Editors and Affiliations

  1. Department of Philosophy, Centre for Logic, Epistemology and the History of Science, University of Campinas-Unicamp, Campinas, São Paulo, Brazil

    Walter Carnielli

  2. Institute of Philosophy and Sociology, Polish Academy of Sciences, Warsaw, Poland

    Jacek Malinowski

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Džamonja, M., Panza, M. (2018). Asymptotic Quasi-completeness and ZFC. In: Carnielli, W., Malinowski, J. (eds) Contradictions, from Consistency to Inconsistency. Trends in Logic, vol 47. Springer, Cham. https://doi.org/10.1007/978-3-319-98797-2_8

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