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Large Cardinals and the Continuum Hypothesis

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The Hyperuniverse Project and Maximality

Abstract

This is a survey paper which discusses the impact of large cardinals on provability of the Continuum Hypothesis (CH). It was Gödel who first suggested that perhaps “strong axioms of infinity” (large cardinals) could decide interesting set-theoretical statements independent over ZFC, such as CH. This hope proved largely unfounded for CH—one can show that virtually all large cardinals defined so far do not affect the status of CH. It seems to be an inherent feature of large cardinals that they do not determine properties of sets low in the cumulative hierarchy if such properties can be forced to hold or fail by small forcings.

The paper can also be used as an introductory text on large cardinals as it defines all relevant concepts.

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Notes

  1. 1.

    Originally published in R. Honzik, Large cardinals and CH, AUC Philosophica et Historica. Miscellanea Logica IX 2, 35–52 (2013).

  2. 2.

    The cofinality of the size of the continuum must be uncountable.

  3. 3.

    See footnote 27.

  4. 4.

    As regards the intuitive “truth” of such axioms, or why they should be preferable to other types of axioms, see a discussion for instance in [2].

  5. 5.

    Such extensions will always decide new statement, such as Con(ZFC), but these are considered too “logical” and not properly set-theoretical.

  6. 6.

    This axioms states that sets are “well-behaved”; for instance sets x such as x ∈ x are prohibited by this axiom.

  7. 7.

    \(({\mathcal {P}} (\mu ))^{V_\kappa }\) is the powerset of μ in the sense of 〈V κ , ∈〉. Note that for every limit ordinal α, if β < α, then \(({\mathcal {P}} (\beta ))^{V_\alpha } = {\mathcal {P}} ({\beta })\) because \({\mathcal {P}} ({\beta }) \subseteq V_{\beta +1}\), and so \({\mathcal {P}} ({\beta }) \in V_{\beta +2} \subseteq V_\alpha \).

  8. 8.

    The Generalized Continuum Hypothesis which states that for every cardinal μ, 2μ = μ +.

  9. 9.

    But it may be a singular strong limit cardinal.

  10. 10.

    We will not define the notion of a stationary set here; any standard set-theoretical textbook contains this definition. Roughly speaking, a set is stationary in κ if it intersects every continuous enumeration of unboundedly many elements below κ. In particular, every stationary subset of a regular cardinal κ has size κ.

  11. 11.

    We assume AC, the Axiom of Choice, in formulating these generalizations.

  12. 12.

    Note that a priori there is no guarantee that we get anything like a large cardinal in this fashion; the generalization may turn out to be mathematically trivial and uninteresting. The fact that we do get large cardinals seems to indicate that these generalizations are mathematically relevant.

  13. 13.

    For instance “∃ β<α x β φ”, α < κ, quantifies over α-many variables in φ.

  14. 14.

    Notice that κ-completeness can be equivalently expressed as follows: whenever μ < κ and {X α | α < μ} are sets not in U, the the union ⋃ α<μ X α is not in U, either.

  15. 15.

    For every n < ω, \(|{\mathrm {rng}}({f \upharpoonright [A]^n})| = 1\).

  16. 16.

    GCH, the Generalized Continuum Hypothesis, states that for all cardinals κ, 2κ = κ +. SCH, the Singular Cardinal Hypothesis, states that for all singular cardinals κ, 2κ = max(2cf(κ), κ +).

  17. 17.

    This is true for larger cardinals than just inaccessibles.

  18. 18.

    For instance if ZFC + (sC) is consistent, so is ZFC + ¬SCH.

  19. 19.

    Notice that for every X ⊆ ω in V , it is dense in \(\mathbb {P}_{\mathrm {CH}}\) that there exists some α < ω 1 and p such that p restricted to [α, α + ω) is a characteristic function of X. The function defined in a generic extension which takes every α < ω 1 to a subset of ω given by the restriction of the generic filter to [α, α + ω) is therefore onto (2ω)V. It follows that 2ω of V is collapsed to ω 1.

  20. 20.

    Note also that all cardinals ≥|P|+, and hence also μ, remain cardinals in V [G].

  21. 21.

    There are some logical issues here because ZFC does not formalize satisfication for proper classes, and hence one should be careful in saying that some φ holds in M, or that j is elementary. The relativation φ M solves the issue to a certain extent, but it is not entirely optimal (for instance the property “j is elementary” is a schema of infinitely many sentences). Luckily, as always with issues like these, there are ways to make these concepts completely correct from the formal point of view. See for instance [4] for a nice discussion of approaches to formalizing large cardinal concepts which refer to elementary embeddings.

  22. 22.

    This means that G meets every dense open set which is an element of M.

  23. 23.

    If |P| < κ, then there is an isomorphic copy of P which is in V κ .

  24. 24.

    γ M ⊆ M is true if for every sequence of length γ of elements in M, the whole sequence is in M. This a non-trivial requirement because the sequence itself is in general only in V , and not in M.

  25. 25.

    Rather surprisingly, it is still open whether this limiting result holds in ZF.

  26. 26.

    This can be shown using Gödel’s class of constructible sets L, or by forcing.

  27. 27.

    Notice that by (4.5), \([\mathrm {GCH}]_{\equiv _c}\) is equal to \([\nu ]_{\equiv _c}\) for any ν such that ZFC ⊢ ν.

  28. 28.

    PFA, a strengthening of MA—the Martin’s Axiom-, implies 2ω = ω 2 and thus decides CH. However, PFA itself is not a large cardinal axiom in the strict sense. Also, PFA trivially implies 2ω > ω 1 the way it is set up, so what is surprising is that it also implies 2ω ≤ ω 2, and not that it implies failure of CH. MA, on the other hand, is consistent with any reasonable value of 2ω > ω 1.

  29. 29.

    We do know that PFA implies consistency of many Woodin cardinals, and so PFA is sandwiched between “many Woodins” and “supercompact”. But this gap is quite substantial.

  30. 30.

    We consider κ to be large when it is at least inaccessible. If we drop this requirement, the situation is more complex.

References

  1. S. Feferman, J. Dawson, S. Kleene, G. Moore, J. Van Heijenoort, (eds.), Kurt Gödel. Collected Works. Volume II (Oxford University Press, Oxford, 1990)

    Google Scholar 

  2. Sy-D. Friedman, T. Arrigoni, Foundational implications of the inner model hypothesis. Ann. Pure Appl. Logic 163, 1360–1366 (2012)

    Google Scholar 

  3. M. Gitik, The negation of singular cardinal hypothesis from o(κ) = κ ++. Ann. Pure Appl. Logic 43, 209–234 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  4. J.D. Hamkins, G. Kirmayer, N.L. Perlmutter, Generalizations of the Kunen inconsistency. Ann. Pure Appl. Logic 163, 1872–1890 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. D. Hilbert, Mathematical problems. Bull. Am. Math. Soc. 8, 437–479 (1902)

    Article  MathSciNet  MATH  Google Scholar 

  6. R. Honzik, Quick guide to independence results in set theory. Miscellanea Logica 7, 89–131 (2009)

    Google Scholar 

  7. T. Jech, Set Theory (Springer, New York, 2003)

    MATH  Google Scholar 

  8. A. Kanamori, The Higher Infinite (Springer, New York, 2003)

    MATH  Google Scholar 

  9. K. Kunen, Elementary embeddings and infinitary combinatorics. J. Symb. Log. 407–413, 21–46 (1971)

    MathSciNet  MATH  Google Scholar 

  10. K. Kunen, Set Theory: An Introduction to Independence Proofs. (North-Holland, Amsterdam, 1980)

    Google Scholar 

  11. A. Levy, R.M. Solovay, Measurable cardinals and the continuum hypothesis. Isr. J. Math. 5, 234–248 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  12. W.J. Mitchell, Ramsey cardinals and constructibility. J. Symb. Log. 44(2), 260–266 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  13. W.J. Mitchell, Aronszajn trees and the independence of the transfer property. Ann. Math. Logic 5(1), 21–46 (1972/1973)

    Article  MathSciNet  MATH  Google Scholar 

  14. W.J. Mitchell, The core model for sequences of measures. I. Math. Proc. Camb. Phil. Soc. 95, 229–260 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  15. D.S. Scott, Measurable cardinals and constructible sets. Bull. de l’Académie Pol. Sci. 9, 521–524 (1961)

    MathSciNet  MATH  Google Scholar 

  16. S. Shelah, Can you take Solovay’s inaccessible away? Isr. J. Math. 48, 1–47 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  17. A. Sochor, Klasická matematická logika (Karolinum, 2001)

    MATH  Google Scholar 

  18. R.M. Solovay, A model of set theory in which every set of reals is Lebesgue measurable. Ann. Math. 92, 1–56 (1970)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author acknowledges the generous support of JTF grant Laboratory of the Infinite ID35216.

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Authors and Affiliations

  1. Department of Logic, Charles University, Celetná 20, Praha 1, 116 42, Czech Republic

    Radek Honzik

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  1. Radek Honzik
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Correspondence to Radek Honzik .

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Editors and Affiliations

  1. Zukunftskolleg, University of Konstanz, Konstanz, Germany

    Carolin Antos

  2. Kurt Gödel Research Center, University of Vienna, Vienna, Austria

    Sy-David Friedman

  3. Department of Logic, Charles University, Prague, Czech Republic

    Radek Honzik

  4. Kurt Gödel Research Center, University Wien, Wien, Austria

    Claudio Ternullo

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Honzik, R. (2018). Large Cardinals and the Continuum Hypothesis. In: Antos, C., Friedman, SD., Honzik, R., Ternullo, C. (eds) The Hyperuniverse Project and Maximality. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-62935-3_10

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