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Not to be confused with a game named True Infinity.


Absolute infinity is supposedly the limit of all transfinite ordinals. However, 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 \(\color{red}{\Omega}\).[1]

Absolute infinity cannot be also treated as a set of all ordinals, because it leads to Burali-Forti paradox. 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.[2]

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.

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.[3]

Sources

  1. Saibian, Sbiis. Forbidden List of Infinite Numbers. Retrieved 2018-04-19.
  2. Emlightened. Little Bigeddon
  3. Saibian, Sbiis. Infinite Numbers Infinitely. Retrieved 2018-04-19.

See also

Ordinals, ordinal analysis and set theory

Basics: cardinal numbers · normal function · ordinal notation · ordinal numbers · fundamental sequence · ordinal collapsing function · transfinite induction
Theories: Robinson arithmetic · Presburger arithmetic · Peano arithmetic · KP · second-order arithmetic · ZFC
Countable ordinals: \(\omega\) · \(\varepsilon_0\) · \(\zeta_0\) · \(\Gamma_0\) · \(\vartheta(\Omega^3)\) · \(\vartheta(\Omega^\omega)\) · \(\vartheta(\Omega^\Omega)\) · \(\vartheta(\varepsilon_{\Omega + 1}) = \psi(\Omega_2)\) · \(\psi(\Omega_\omega)\) · \(\psi(\varepsilon_{\Omega_\omega + 1})\) · \(\psi(\psi_I(0))\)‎ · \(\omega_1^\mathfrak{Ch}\) · \(\omega_1^\text{CK}\) · \(\omega_\alpha^\text{CK}\) · \(\lambda,\zeta,\Sigma,\gamma\) · List of countable ordinals
Ordinal hierarchies: Fast-growing hierarchy · Slow-growing hierarchy · Hardy hierarchy · Middle-growing hierarchy · N-growing hierarchy
Ordinal collapsing functions Madore · Buchholz · Jäger
Uncountable cardinals: \(\omega_1\) · omega fixed point · inaccessible cardinal \(I\) · Mahlo cardinal \(M\) · weakly compact cardinal \(K\) · indescribable cardinal · rank-into-rank cardinal
Classes: \(V\) · \(L\) · \(\textrm{On}\) · Class (set theory)
Other concepts: Veblen function · absolute infinity

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