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\(\mathrm{Card}\) denotes the class of all cardinals.[1] (p.13) It is also commonly denoted as \(\mathrm{CARD}\).[2] (p.250)

Georg Cantor denoted it by ת (tav), the final letter of the Hebrew alphabet [5][6].

Properties[]

  • The class \(\mathrm{Card}\) is a proper class. If it were a set, it would lead to a paradox of the Burali-Forti type.
  • Assuming the axiom of choice, there is a transfinite sequence of cardinal numbers, starting with the natural numbers, and then followed by the aleph numbers.
    • The aleph mapping \(\alpha \rightarrow \aleph_\alpha\) provides a one-to-one correspondance between the ordinals and cardinals. It is the only order-isomorphism between the ordinals and cardinals with respect to membership.
    • The aleph numbers are indexed by the ordinal numbers.
  • If the axiom of choice does not hold, then there exists infinite cardinals \(\kappa\) that are not aleph numbers.

Application[]

  • Each ordinal is associated with a cardinal.
    • Under the von Neumann definition of ordinals, the cardinality of the ordinal \(\alpha\) is the least ordinal with the same cardinality of \(\alpha\).
    • Under this definition, \(\mathrm{Card}\) is a strict subclass of \(\mathrm{On}\).

See Also[]

Basics: cardinal numbers · ordinal numbers · limit ordinals · fundamental sequence · normal form · transfinite induction · ordinal notation · Absolute infinity
Theories: Robinson arithmetic · Presburger arithmetic · Peano arithmetic · KP · second-order arithmetic · ZFC
Model Theoretic Concepts: structure · elementary embedding
Countable ordinals: \(\omega\) · \(\varepsilon_0\) · \(\zeta_0\) · \(\eta_0\) · \(\Gamma_0\) (Feferman–Schütte ordinal) · \(\varphi(1,0,0,0)\) (Ackermann ordinal) · \(\psi_0(\Omega^{\Omega^\omega})\) (small Veblen ordinal) · \(\psi_0(\Omega^{\Omega^\Omega})\) (large Veblen ordinal) · \(\psi_0(\varepsilon_{\Omega + 1}) = \psi_0(\Omega_2)\) (Bachmann-Howard ordinal) · \(\psi_0(\Omega_\omega)\) with respect to Buchholz's ψ · \(\psi_0(\varepsilon_{\Omega_\omega + 1})\) (Takeuti-Feferman-Buchholz ordinal) · \(\psi_0(\Omega_{\Omega_{\cdot_{\cdot_{\cdot}}}})\) (countable limit of Extended Buchholz's function)‎ · \(\omega_1^\mathfrak{Ch}\) (Omega one of chess) · \(\omega_1^{\text{CK}}\) (Church-Kleene ordinal) · \(\omega_\alpha^\text{CK}\) (admissible ordinal) · recursively inaccessible ordinal · recursively Mahlo ordinal · reflecting ordinal · stable ordinal · \(\lambda,\gamma,\zeta,\Sigma\) (Infinite time Turing machine ordinals) · gap ordinal · List of countable ordinals
Ordinal hierarchies: Fast-growing hierarchy · Hardy hierarchy · Slow-growing hierarchy · Middle-growing hierarchy · N-growing hierarchy
Ordinal functions: enumeration · normal function · derivative · Veblen function · ordinal collapsing function · Weak Buchholz's function · Bachmann's function · Madore's function · Feferman's theta function · Buchholz's function · Extended Weak Buchholz's function · Extended Buchholz's function · Jäger-Buchholz function · Jäger's function · Rathjen's psi function · Rathjen's Psi function · Stegert's Psi function · Arai's psi function
Uncountable cardinals: \(\omega_1\) · omega fixed point · inaccessible cardinal \(I\) · Mahlo cardinal \(M\) · weakly compact cardinal \(K\) · indescribable cardinal · rank-into-rank cardinal · Berkeley cardinal · (Club Berkeley cardinal · limit club Berkeley cardinal)
Classes: \(\textrm{Card}\) · \(\textrm{On}\) · \(V\) · \(L\) · \(\textrm{Lim}\) · \(\textrm{AP}\) · Class (set theory)

References[]

  1. Stegert, Jan-Carl. "Ordinal proof theory of Kripke-Platek set theory augmented by strong reflection principles"
  2. Rathjen, Michael. "Ordinal Notations Based on a Weakly Mahlo Cardinal", Archive for Mathematical Logic 29 (1990) 249--263.
  3. Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, Georg Cantor, ed. Ernst Zermelo, with biography by Adolf Fraenkel; orig. pub. Berlin: Verlag von Julius Springer, 1932; reprinted Hildesheim: Georg Olms, 1962, and Berlin: Springer-Verlag, 1980, ISBM 3-540-09849-6.
  4. The Rediscovery of the Cantor-Dedekind Correspondence[dead link], I. Grattan-Guinness, Jahresbericht der Deutschen Mathematiker-Vereinigung 76 (1974/75), pp. 104–139, at p. 126 ff.
  5. Gesammelte Abhandlungen,[3] Georg Cantor, ed. Ernst Zermelo, Hildesheim: Georg Olms Verlagsbuchhandlung, 1962, pp. 443–447; translated into English in From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931, ed. Jean van Heijenoort, Cambridge, Massachusetts: Harvard University Press, 1967, pp. 113–117. These references both purport to be a letter from Cantor to Dedekind, dated July 28, 1899. However, as Ivor Grattan-Guinness has discovered,[4] this is in fact an amalgamation by Cantor's editor, Ernst Zermelo, of two letters from Cantor to Dedekind, the first dated July 28 and the second dated August 3.
  6. The Correspondence between Georg Cantor and Philip Jourdain[dead link], I. Grattan-Guinness, Jahresbericht der Deutschen Mathematiker-Vereinigung 73 (1971/72), pp. 111–130, at pp. 116–117.
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