An inaccessible cardinal (or strongly inaccessible cardinal) is a cardinal number that is an uncountable regular strong limit cardinal.[1] The smallest inaccessible cardinal is sometimes called the inaccessible cardinal \(I\).

Breaking down the definition, an 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 order type 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 beth numbers:
    • \(\beth_0 = \aleph_0\)
    • \(\beth_{\alpha + 1} = 2^{\beth_\alpha}\) (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 "limit cardinal" (replacing "beth numbers" with "aleph numbers"), we get weakly inaccessible cardinals. The distinction between strongly and weakly inaccessible cardinals only matters if we don't assume the generalized continuum hypothesis (GCH). Under GCH, all limit cardinals are strong limit cardinals.

Degrees of Inaccessibility

There are conflicting definitions of 1-inaccessible.[citation needed] One definition is that a cardinal \(\kappa\) is 1-inaccessible if \(\kappa\) is inaccessible and a limit of inaccessibles[2], a cardinal \(\kappa\) that is \(\kappa\)-inaccessible is called hyper-inaccessible[2], and a cardinal \(\kappa\) that is \(``\textrm{hyper}\!"^\kappa\)-inaccessible is called richly inaccessible[2]. Under this definition the process continues, and higher level inaccessible cardinals are called utterly inaccessible, deeply inaccessible, truly inaccessible, eternally inaccessible, vastly inaccessible, and so on.[2]

The term "\(\alpha\)-hyper-inaccessible" is ambiguous, and one definition is that for a cardinal \(\kappa\) and for \(\beta<\alpha\), the set of \(\beta\)-hyper-inaccessible cardinals \(<\kappa\) is unbounded in \(\kappa\).[citation needed]

Representation

The \(\alpha\)-th weakly inaccessible is usually denoted by either \(I_\alpha\)[3] or \(I(\alpha)\), however Rathjen uses "0-inaccessible" to mean "regular". Now, we can use the extensions above: \(I_\alpha(\beta)\)[3] 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 enumerating function 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 limit ordinalscan 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 \(\kappa\), \(\forall\beta<\alpha:V_\kappa\vDash (\text{ZFC+}\exists\text{A }\beta\text{-inaccessible cardinal})\)[4]

Collapsing functions using inaccessible cardinals

The inaccessible cardinals are most relevant to googology through ordinal collapsing functions.

With respect to several ordinal collapsing functions \(\psi\), the function \(\alpha \mapsto \psi_I(\alpha)\) enumerates the fixed points of \(\beta \mapsto \Omega_\beta\). Therefore \(\psi_I(0)\) is the least omega fixed point, \(\psi_I(1)\) is the second omega fixed point.

External links

See also

Basics: cardinal numbers · ordinal numbers · limit ordinals · fundamental sequence · normal form · transfinite induction · ordinal notation
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)‎ · \(\omega_1^\mathfrak{Ch}\) · \(\omega_1^{\text{CK}}\) (Church-Kleene ordinal) · \(\omega_\alpha^\text{CK}\) (admissible ordinal) · recursively inaccessible ordinal · recursively Mahlo ordinal · reflecting ordinal · stable ordinal · \(\lambda,\zeta,\Sigma,\gamma\) (ordinals on infinite time Turing machine) · gap ordinal · List of countable ordinals
Ordinal hierarchies: Fast-growing hierarchy · Slow-growing hierarchy · Hardy hierarchy · Middle-growing hierarchy · N-growing hierarchy
Ordinal functions: enumeration · normal function · derivative · Veblen function · ordinal collapsing function · Bachmann's function · Madore's function · Feferman's theta function · Buchholz's function · Extended Buchholz's function · Jäger's function · Rathjen's psi function · Rathjen's Psi function · Stegert'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: \(V\) · \(L\) · \(\textrm{On}\) · \(\textrm{Lim}\) · \(\textrm{AP}\) · Class (set theory)

Sources

  1. T. Jech, Set Theory (p.51).
  2. 2.0 2.1 2.2 2.3 E. Carmody, Force to change large cardinal strength (p.10, 13) (2015)
  3. 3.0 3.1 M. Rathjen, Ordinal notations based on a weakly Mahlo cardinal (p.2) Arch. Math. Logic (1990) 29:249 263
  4. Cantor's Attic, [cantorsattic.info/Inaccessible Inaccessible][dead link]
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