Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-03T14:53:52.308Z Has data issue: false hasContentIssue false
  • English
  • Français

Singular Cardinals and the PCF Theory

Published online by Cambridge University Press:  15 January 2014

Thomas Jech
Thomas Jech*
Affiliation:
Department of Mathematics, The Pennsylvania State University University Park, PA 16802.E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Extract

§1. Introduction. Among the most remarkable discoveries in set theory in the last quarter century is the rich structure of the arithmetic of singular cardinals, and its deep relationship to large cardinals. The problem of finding a complete set of rules describing the behavior of the continuum function 2ℵα for singular ℵα's, known as the Singular Cardinals Problem, has been attacked by many different techniques, involving forcing, large cardinals, inner models, and various combinatorial methods. The work on the singular cardinals problem has led to many often surprising results, culminating in a beautiful theory of Saharon Shelah called the pcf theory (“pcf” stands for “possible cofinalities”). The most striking result to date states that if 2ℵn < ℵω for every n = 0, 1, 2, …, then 2ℵω < ℵω4.

In this paper we present a brief history of the singular cardinals problem, the present knowledge, and an introduction into Shelah's pcf theory. In Sections 2, 3 and 4 we introduce the reader to cardinal arithmetic and to the singular cardinals problems. Sections 5, 6, 7 and 8 describe the main results and methods of the last 25 years and explain the role of large cardinals in the singular cardinals problem. In Section 9 we present an outline of the pcf theory.

§2. The arithmetic of cardinal numbers. Cardinal numbers were introduced by Cantor in the late 19th century and problems arising from investigations of rules of arithmetic of cardinal numbers led to the birth of set theory. The operations of addition, multiplication and exponentiation of infinite cardinal numbers are a natural generalization of such operations on integers. Addition and multiplication of infinite cardinals turns out to be simple: when at least one of the numbers κ, λ is infinite then both κ + λ and κ·λ are equal to max {κ, λ}. In contrast with + and ·, exponentiation presents fundamental problems. In the simplest nontrivial case, 2κ represents the cardinal number of the power set P(κ), the set of all subsets of κ. (Here we adopt the usual convention of set theory that the number κ is identified with a set of cardinality κ, namely the set of all ordinal numbers smaller than κ.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 1 , Issue 4 , December 1995 , pp. 408 - 424
Copyright
Copyright © Association for Symbolic Logic 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Baumgartner, J. E. and Prikry, K., On a theorem of Silver, Discrete Mathematics, vol. 14 (1976), pp. 17–21.Google Scholar
[2] Baumgartner, J. E. and Prikry, K., Singular cardinals and the generalized continuum hypothesis, American Mathematical Monthly, vol. 84 (1977), pp. 108–113.Google Scholar
[3] Bukovský, L., The continuum problem and the powers of alephs, Commentationes Math-ematicae Universitatis Carolinae, vol. 6 (1965), pp. 181–197.Google Scholar
[4] Bukovský, L., Changing cofinality of ℵ2 , Set theory and hierarchy theory (Marek, W. et al., editors), Lecture Notes in Mathematics, vol. 537, Springer-Verlag, Berlin, 1976, pp. 37–49.Google Scholar
[5] Burke, M. and Magidor, M., Shelah's pcf theory and its applications, Annals of Pure and Applied Logic, vol. 50 (1990), pp. 207–254.CrossRefGoogle Scholar
[6] Cantor, G., Über unendliche, lineare Punktmannigfaltigkeiten, Mathematische Annalen, vol. 21 (1883), pp. 545–591.Google Scholar
[7] Cohen, P. J., The independence of the continuum hypothesis I, Proceedings of the National Academy of Sciences, U.S.A., vol. 50 (1963), pp. 1143–1148.Google Scholar
[8] Cohen, P. J., The independence of the continuum hypothesis II, Proceedings of the National Academy of Sciences, U.S.A., vol. 51 (1964), pp. 105–110.Google Scholar
[9] Cummings, J., A model in which GCH holds at successors but fails at limits, Transactions of the American Mathematical Society, vol. 329 (1992), pp. 1–39.Google Scholar
[10] Devlin, K. and Jensen, R., Marginalia to a theorem of Silver, Logic Conference, Kiel 1974, Proceedings (Müller, G. H. et al., editors), Lecture Notes in Mathematics, vol. 499, Springer-Verlag, Berlin, 1975, pp. 115–142.Google Scholar
[11] Dodd, A. J. and Jensen, R. B., The core model, Annals of Mathematical Logic, vol. 20 (1981), pp. 43–75.Google Scholar
[12] Dodd, A. J. and Jensen, R. B., The covering lemma for K, Annals of Mathematical Logic, vol. 22 (1982), pp. 1–30.Google Scholar
[13] Easton, W. B., Powers of regular cardinals, Annals of Mathematical Logic, vol. 1 (1970), pp. 139–178.Google Scholar
[14] Foreman, M. and Woodin, W. H., The generalized continuum hypothesis can fail everywhere, Annals of Mathematics, vol. 133 (1991), pp. 1–35.Google Scholar
[15] Galvin, F. and Hajnal, A., Inequalities for cardinal powers, Annals of Mathematics, vol. 101 (1975), pp. 491–498.CrossRefGoogle Scholar
[16] Gitik, M., Changing cofinalities and the nonstationary ideal, Israel Journal of Mathematics, vol. 56 (1986), pp. 280–314.Google Scholar
[17] Gitik, M., The negation of the singular cardinal hypothesis from o(κ) = κ++ , Annals of Pure and Applied Logic, vol. 43 (1989), pp. 209–234.CrossRefGoogle Scholar
[18] Gitik, M., The strength of the failure of the singular cardinal hypothesis, Annals of Pure and Applied Logic, vol. 51 (1991), pp. 215–240.CrossRefGoogle Scholar
[19] Gitik, M., On measurable cardinals violating the continuum hypothesis, Annals of Pure and Applied Logic, vol. 63 (1993), pp. 227–240.Google Scholar
[20] Gitik, M., Blowing up the power of a singular cardinal, Annals of Pure and Applied Logic, to appear.Google Scholar
[21] Gitik, M. and Magidor, M., The singular cardinals hypothesis revisited, Set theory and the continuum (Judah, H. et al., editors), Springer-Verlag, Berlin, 1992, pp. 243–279.Google Scholar
[22] Gitik, M. and Magidor, M., Extender based forcing, The Journal of Symbolic Logic, vol. 59 (1994), pp. 445–460.Google Scholar
[23] Gitik, M. and Mitchell, W. J., Indiscernible sequences for extenders, and the singular cardinal hypothesis, to appear.Google Scholar
[24] Gitik, M. and Shelah, S., On certain indestructibility of strong cardinals and a question of Hajnal, Archive for Mathematical Logic, vol. 28 (1989), pp. 35–42.CrossRefGoogle Scholar
[25] Gödel, K., Consistency-proof for the generalized continuum hypothesis, Proceedings of the National Academy of Sciences, U.S.A., vol. 25 (1939), pp. 220–224.Google Scholar
[26] Gödel, K., What is Cantor's continuum problem?, American Mathematical Monthly, vol. 54 (1947), pp. 515–525.Google Scholar
[27] Hechler, S. H., Powers of singular cardinals and a strong form of the negation of the generalized continuum hypothesis, Zeitschrift für Mathematische Logik, vol. 19 (1973), pp. 83–84.CrossRefGoogle Scholar
[28] Jech, T., Properties of the gimel function and a classification of singular cardinals, Fundamenta Mathematicae, vol. 81 (1973), pp. 57–64.Google Scholar
[29] Jech, T., On the cofinality of countable products of cardinal numbers, A tribute to Paul Erdős (Baker, A. et al., editors), Cambridge University Press, Cambridge, 1990, pp. 289–305.CrossRefGoogle Scholar
[30] Jech, T., Singular cardinal problem: Shelah's theorem on 2ℵω , Bulletin of the London Mathematical Society, vol. 24 (1992), pp. 127–139.Google Scholar
[31] Jech, T. and Prikry, K., On ideals of sets and the power set operation, Bulletin of the American Mathematical Society, vol. 82 (1976), pp. 593–595.Google Scholar
[32] Jech, T. and Shelah, S., On a conjecture of Tarskion products of cardinals, Proceedings of the American Mathematical Society, vol. 112 (1991), pp. 1117–1124.CrossRefGoogle Scholar
[33] Jech, T. and Shelah, S., Possible pcf algebras, The Journal of Symbolic Logic, to appear.Google Scholar
[34] Jensen, R. B., The fine structure of the constructible hierarchy, Annals of Mathematical Logic, vol. 4 (1972), pp. 229–308.CrossRefGoogle Scholar
[35] Jensen, R. B., Inner models and large cardinals, this Bulletin, vol. 1 (1995), pp. xxx–xxxx.Google Scholar
[36] Kanamori, A., Higher set theory, Springer-Verlag, Berlin, 1994.Google Scholar
[37] König, J., Zum Kontinuumproblem, Mathematische Annalen, vol. 60 (1905), pp. 177–180.Google Scholar
[38] Kunen, K., On the GCH at measurable cardinals, Logic Colloquium '69 (Gandy, R. O. and Yates, C. E. M., editors), North-Holland, Amsterdam, 1971, pp. 107–110.CrossRefGoogle Scholar
[39] Magidor, M., Chang's conjecture and powers of singular cardinals, The Journal of Symbolic Logic, vol. 42 (1977), pp. 272–276.Google Scholar
[40] Magidor, M., On the singular cardinals problem I, Israel Journal of Mathematics, vol. 28 (1977), pp. 1–31.Google Scholar
[41] Magidor, M., On the singular cardinals problem II, Annals of Mathematics, vol. 106 (1977), pp. 517–547.Google Scholar
[42] Magidor, M., Changing cofinality of cardinals, Fundamenta Mathematicae, vol. 99 (1978), pp. 61–71.Google Scholar
[43] Mathias, A. R. D., Surrealist landscape with figures, Periodica Mathematica Hungarica, vol. 10 (1979), pp. 109–175.Google Scholar
[44] Mitchell, W. J., The core model for sequences of measures I, Mathematical Proceedings of the Cambridge Philosophical Society, vol. 95 (1984), pp. 229–260.Google Scholar
[45] Mitchell, W. J., On the singular cardinal hypothesis, Transactions of the American Mathematical Society, vol. 329 (1992), pp. 507–530.Google Scholar
[46] Namba, K., Independence proof of (ω, ωα)-distributive law in complete Boolean algebras, Commentarii Mathematici Universitatis Sancti Pauli, vol. 19 (1970), pp. 1–12.Google Scholar
[47] Prikry, K., Changing measurable into accessible cardinals, Dissertationes Mathematicae, vol. 68 (1970), pp. 5–52.Google Scholar
[48] Radin, L. B., Adding closed cofinal sequences to large cardinals, Annals of Mathematical Logic, vol. 22 (1982), pp. 243–261.Google Scholar
[49] Rowbottom, F., Some strong axioms of infinity incompatible with the axiom of constructibility, Annals of Mathematical Logic, vol. 3 (1971), pp. 1–44.Google Scholar
[50] Scott, D., Measurable cardinals and constructible sets, Bulletin de l'Académie Polonaise des Sciences, vol. 9 (1961), pp. 521–524.Google Scholar
[51] Shelah, S., Jonsson algebras in successor cardinals, Israel Journal of Mathematics, vol. 30 (1978), pp. 57–64, [Sh 68].Google Scholar
[52] Shelah, S., A note on cardinal exponentiation, The Journal of Symbolic Logic, vol. 45 (1980), pp. 56–66, [Sh 71].Google Scholar
[53] Shelah, S., Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin, 1982.CrossRefGoogle Scholar
[54] Shelah, S., The singular cardinals problem: independence results, Surveys in set theory (Mathias, A. R. D., editor), Cambridge University Press, Cambridge, 1983, pp. 116–134, [Sh 137].Google Scholar
[55] Shelah, S., On power of singular cardinals, Notre Dame Journal of Formal Logic, vol. 27 (1986), pp. 263–299, [Sh 111].Google Scholar
[56] Shelah, S., More on powers of singular cardinals, Israel Journal of Mathematics, vol. 59 (1987), pp. 299–326, [Sh 256].Google Scholar
[57] Shelah, S., Successors of singulars, cofinalities of reduced products of cardinals and productivity of chain conditions, Israel Journal of Mathematics, vol. 62 (1988), pp. 213–256, [Sh 282].CrossRefGoogle Scholar
[58] Shelah, S., Products of regular cardinals and cardinal invariants of products of Boolean algebras, Israel Journal of Mathematics, vol. 70 (1990), pp. 129–187, [Sh 345].Google Scholar
[59] Shelah, S., Cardinal arithmetic for skeptics, Bulletin of the American Mathematical Society, vol. 26 (1992), pp. 197–210, [Sh 400a].Google Scholar
[60] Shelah, S., More on cardinal arithmetic, Archive for Mathematical Logic, vol. 32 (1993), pp. 399–428, [Sh 410].Google Scholar
[61] Shelah, S., Cardinal arithmetic, Oxford Logic Guides, Oxford University Press, Oxford, 1994.CrossRefGoogle Scholar
[62] Shelah, S., Advances in cardinal arithmetic, Proceedings of the Banff Conference in Alberta, to appear, [Sh 420].Google Scholar
[63] Shelah, S., Further cardinal arithmetic, Israel Journal of Mathematics, to appear, [Sh 430].Google Scholar
[64] Shelah, S., The generalized continuum hypothesis revisited, to appear, [Sh 460].Google Scholar
[65] Shelah, S., The pcf-theorem revisited, A special volume dedicated to Paul Erdős, edited by R. Graham and J. Nešetřil, to appear, [Sh 506].Google Scholar
[66] Silver, J. H., Some applications of model theory in set theory, Annals of Mathematical Logic, vol. 3 (1971), pp. 45–110.Google Scholar
[67] Silver, J. H., On the singular cardinals problem, Proceedings of the International Congress of Mathematicians, 1974, pp. 265–268.Google Scholar
[68] Solcvay, R. M., Strongly compact cardinals and the GCH, Proceedings of the Tarski symposium, American Mathematical Society, Providence, RI, 1974, pp. 365–372.Google Scholar
[69] Tarski, A., Quelques théorèmes sur les alephs, Fundamenta Mathematicae, vol. 7 (1925), pp. 1–14.Google Scholar