\(\mathrm{Card}\) 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[]
- ↑ Stegert, Jan-Carl. "Ordinal proof theory of Kripke-Platek set theory augmented by strong reflection principles"
- ↑ Rathjen, Michael. "Ordinal Notations Based on a Weakly Mahlo Cardinal", Archive for Mathematical Logic 29 (1990) 249--263.
- ↑ 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.
- ↑ 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.
- ↑ 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.
- ↑ 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.