Inaccessible cardinal

An inaccessible cardinal is an uncountable regular limit cardinal.T. Jech, Set Theory (p.51). The smallest inaccessible cardinal is sometimes called the inaccessible cardinal \(I\). The definition may differ depending on whether GCH holds. This number cannot be proven to exist within ZFC set theory.
Breaking down the definition, a (strongly) inaccessible cardinal \(\alpha\) must be:
* Uncountable: \(\alpha \geq \omega_1\).
* Regular: \(\alpha\) cannot be expressed as the limit of a set \(S\) of smaller ordinals, where the of \(S\) is less than \(\alpha\). This can alternately be phrased as \(\alpha=\)\(\textrm{cof}\)\((\alpha)\). From a cardinal perspective, we can informally say that it cannot be divided into smaller set of smaller sets.
* Strong limit: \(\alpha = \beth_\gamma\) for a limit ordinal \(\gamma\), using the following hierarchy of s:
** \(\beth_0 = \aleph_0\)
** \(\beth_{\alpha + 1} = 2^{\beth_\alpha}\) (cardinal exponentiation)
** \(\beth_\alpha = \sup\{\beta < \alpha : \beth_\beta\}\)
For a cardinal \(\kappa\), "\(\kappa\) is a (strong) limit cardinal" is also equivalent to "\(\kappa\) is closed under the map \(\lambda \tau.2^\tau\) with domain and codomain \(\textrm{Card}\)". 
If we replace "strong limit cardinal" with "(weak) limit cardinal" by replacing "beth numbers" with "s"), we get weakly inaccessible cardinals. The distinction between strongly and weakly inaccessible cardinals only matters if we don't assume the (GCH). Under GCH, all limit cardinals are strong limit cardinals and may just be called limit cardinals.
== Degrees of Inaccessibility ==
One definition of 1-inaccessible is that a cardinal \(\kappa\) is 1-inaccessible if \(\kappa\) is inaccessible and a limit of inaccessiblesE. Carmody, Force to change large cardinal strength (p.10, 13) (2015), a cardinal \(\kappa\) that is \(\kappa\)-inaccessible is called hyper-inaccessible, and a cardinal \(\kappa\) that is \(‟\textrm{hyper}”^\kappa\)-inaccessible is called richly inaccessible. Under this definition the process continues, and higher level inaccessible cardinals can be called utterly inaccessible, deeply inaccessible, truly inaccessible, eternally inaccessible, vastly inaccessible, and so on.
A cardinal \(\kappa\) is \(\alpha\)-hyper-inaccessible if and only if \(\kappa\) is hyper-inaccessible, and for every \(\beta < \alpha\), the cardinal \(\kappa\) is a limit of \(\beta\)-hyper-inaccessible cardinals.
== Representation ==
The \(\alpha\)-th weakly inaccessible is usually denoted by either \(I_\alpha\)M. Rathjen, Ordinal notations based on a weakly Mahlo cardinal (p.2) Arch. Math. Logic (1990) 29:249 263 or \(I(\alpha)\), however Rathjen uses "0-inaccessible" to mean "regular". Now, we can use the extensions above: \(I_\alpha(\beta)\) or \(I(\alpha,\beta)\) is the \(1+\beta\)-th \(\alpha\)-inaccessible cardinal. Then, the first hyper-inaccessible is denoted \(I(1,0,0)\), and the extension works similar to the Veblen function. Such a system would mark the first richly inaccessible at \(I(1,0,0,0)\), the first utterly inaccessible at \(I(1,0,0,0,0)\), and so on.
== Properties ==
An important property of inaccessible cardinals is that if \(I \colon \Lambda \rightarrow\textrm{Ord}\) is the enumeration of inaccessible cardinals, where \(\Lambda\) is either \(\textrm{Ord}\) or an ordinal, \(I\) might not be continuous (i.e. \(I(\alpha)=\textrm{sup}(\{I(\beta):\beta<\alpha\})\) isn't necessarily true for all non-zero limit ordinals \(\alpha\)). Indeed, if \(\Lambda > \omega\), then \(\textrm{sup}(\{I(\beta):\beta<\omega\}\) is of cofinality \(\omega\), and hence does not coincide with \(I(\omega)\). On the other hand, if \(\Lambda \leq \omega\), then it is continuous because there is no non-zero limit ordinal below \(\omega\).
GCH aside, if ZFC is consistent, neither weakly nor strongly inaccessible cardinals can be proven to exist within it. A stronger theory, such as , can prove their existence. ZFC + "there exists a weakly inaccessible cardinal" is believed to be consistent.
Although there is no standard notion of whether a given cardinal is "large", some consider the first inaccessible cardinal (if it exists), \(I\), to be the threshold for large cardinals. That is, all cardinals less than \(I\) are small, and all cardinals at least \(I\) are large in this context.
For an \(\alpha\)-inaccessible cardinal \(\kappa\), \(\forall\beta<\alpha:V_\kappa\vDash (\text{ZFC+}\exists\text{A }\beta\text{-inaccessible cardinal})\)Cantor's Attic, [cantorsattic.info/Inaccessible Inaccessible]
==Collapsing functions using inaccessible cardinals==
The inaccessible cardinals are most relevant to googology through ordinal collapsing functions such as Jäger's function, and ordinal notations associated to them.
With respect to the system of fundamental sequences associated to Jäger's function, the function \(\alpha \mapsto \psi_{I(1,0)}(\alpha)\) enumerates fixed points of \(\beta \mapsto \Omega_\beta\) for small inputs. Therefore \(\psi_{I(1,0)}(0)\) is the least omega fixed point, and \(\psi_{I(1,0)}(1)\) is the second least omega fixed point.
== External links ==
*Inaccessible cardinal on Cantor's Attic (link doesn't work)
*Inaccessible cardinal on Wikipedia
== See also ==
== Sources ==
ja:到達不能基数ru:Недоступный кардинал
Category:Transfinite ordinals