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Absolute infinity

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The actual infinite was distinguished by three relations: first, as it is realized in the supreme perfection, in the completely independent, extra worldly existence, in God, where I call it the absolute infinite or simply the absolute; second to the extent that it is represented in the dependent, creatural world; third as it can be conceived by thought in the abstract as a mathematical magnitude, number or order type. In the latter two relations, where it obviously reveals itself as limited, capable of further increase, and hence related to the finite, I call it the Transfinite and strictly oppose it with the absolute.[1][2]
  —Georg Cantor
Not to be confused with a game named True Infinity.

Absolute infinity or absolute infinite is intended to serve as the supremum of all transfinite cardinals and ordinals. This idea has been proposed by Georg Cantor, and is denoted as ת or \(\Omega\).[1] Cantor has linked this concept to the Abrahamic God.

The collection of all cardinals is denoted ת (tav), also by Cantor. ת cannot have a cardinality, because it would lead to the Burali-Forti paradox. [3]

A portrait of Georg Cantor

A portrait of Georg Cantor

Absolute infinity cannot be also treated as a set of all ordinals, \(\Omega\) for the same reason. Instead, it can be treated as the proper class of all ordinals, which is usually denoted by \(\textrm{Ord}\) or \(\textrm{On}\). Unlike absolute infinity itself, the class frequently appears in googology. For example, it is used in the transfinite induction and in the definition of Little Bigeddon.[4]

Note that absolute infinity is a well-ordered class by itself if we identify it with \(\textrm{Ord}\). Every initial segment of it is an ordinal.

Sbiis Saibian stated himself that it is "not considered an official transfinite number" and "there is no such thing as a largest number". He denotes this by a red \(\color{red}{\Omega}\).[5] However, the initial idea of absolute infinity by Sbiis Saibian was thinking about it as "the largest infinite number", which is paradoxical and thereby the term "absolute infinity" is not commonly used in the meaning of \(\textrm{Ord}\). In that case, it might be better to informally think about absolute infinity as some indefinitely large uncountable ordinal so that it is larger than any ordinal for which we can pick a reasonably large system of axioms in order to define it. Sbiis Saibian himself made a page showing that there are always larger "absolute infinities" in that sense.[6]

Burali-Forti paradox[]

Assume \(\textrm{Ord}\) to be a set. The following logic follows:

  1. Every set of ordinals has an ordinal greater than any ordinal in the segment.
  2. \(\textrm{Ord}\) is a set, meaning it must have a greater ordinal. Denote this \(\textrm{Ord}^+\).
  3. Because \(\textrm{Ord}\) is the set of every ordinal, and \(\textrm{Ord}^+\) is an ordinal, \(\textrm{Ord}\) contains \(\textrm{Ord}^+\).
  4. This means the set \(\textrm{Ord}\) is greater than \(\textrm{Ord}^+\), but by definition, \(\textrm{Ord}^+\) is greater than \(\textrm{Ord}\). This creates the inequality \(\textrm{Ord}^+>\textrm{Ord}^+\).

This would create a paradox, therefore, \(\textrm{Ord}\) has to be defined as a proper class to work properly.[7]

Sources[]

  1. ↑ 1.0 1.1 1.2 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, 2013. ISBN 3-540-09849-6.
  2. ↑ p. 378 in the reprinted version (1962) Translated from German text shown here by Gregor Nickel: Es wurde das Aktual-Unendliche (A-U.) nach drei Beziehungen unterschieden: erstens, sofern es in der höchsten Vollkommenheit, im völlig unabhängigen außerweltlichen Sein, in Deo realisiert ist, wo ich es Absolut Unendliches oder kurzweg Absolutes nenne; zweitens, sofern es in der abhängigen, kreatürlichen Welt vertreten ist; drittens, sofern es als mathematische Größe, Zahl oder Ordnungstypus vom Denken in abstracto aufgefaßt werden kann. In den beiden letzten Beziehungen, wo es offenbar als beschränktes, noch weiterer Vermehrung fähiges und insofern dem Endlichen verwandtes A.-U. sich darstellt, nenne ich es Transfinitum und setze es dem Absoluten strengstens entgegen.
  3. ↑ Georg Cantor (1932)[1] 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, 2002, ISBN 978-0674324497 These references both purport to be a letter from Cantor to Richard Dedekind
  4. ↑ Emlightened. Little Bigeddon. 
  5. ↑ Saibian, Sbiis. Forbidden List of Infinite Numbers. Retrieved 2018-04-19.
  6. ↑ Saibian, Sbiis. Infinite Numbers Infinitely. Retrieved 2018-04-19.
  7. ↑ https://mathworld.wolfram.com/Burali-FortiParadox.html

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)
Classes: \(\textrm{Card}\) · \(\textrm{On}\) · \(V\) · \(L\) · \(\textrm{Lim}\) · \(\textrm{AP}\) · Class (set theory)

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