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An inaccessible cardinal (or strongly inaccessible cardinal) is a cardinal number that is an uncountable regular strong limit cardinal. 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 \(\alpha\). 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)
    • \(\beth_\alpha = \sup\{\beta < \alpha : \beth_\beta\}\)

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 generalized continuum hypothesis (GCH). Under GCH, all limit cardinals are strong limit cardinals.

Properties Edit

GCH aside, if ZFC is consistent, neither weakly nor strongly inaccessible cardinals can be proven to exist within it. A stronger theory, TG set theorycan prove their existence. ZFC + "there exists a weakly inaccessible cardinal" is believed to be consistent.

Often, the first inaccessible cardinal (if it exists), denoted \(I\), is considered the threshold for large cardinals. That is, all cardinals less than \(I\) are small, and all cardinals at least \(I\) are large.

Collapsing functions using inaccessible cardinalsEdit

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

The function \(\alpha \mapsto \psi_I(\alpha)\) enumerates the fixed points of \(\beta \mapsto \Omega_\beta\), so \(\psi_I(0)\) is the omega fixed point, and \(\psi_I(1)\) is the second fixed point of \(\beta \mapsto \Omega_\beta\), and so on.

See also Edit

Ordinals, ordinal analysis and set theory

Basics: cardinal numbers · normal function · ordinal notation · ordinal numbers
Theories: Presburger arithmetic · Peano arithmetic · second-order arithmetic · ZFC
Countable ordinals: \(\omega\) · \(\varepsilon_0\) · \(\zeta_0\) · \(\Gamma_0\) · \(\vartheta(\Omega^3)\) · \(\vartheta(\Omega^\omega)\) · \(\vartheta(\Omega^\Omega)\) · \(\vartheta(\varepsilon_{\Omega + 1})\) · \(\psi(\Omega_\omega)\) · \(\psi(\varepsilon_{\Omega_\omega + 1})\) · \(\psi(\psi_I(0))\)‎ · \(\omega_1^\mathfrak{Ch}\) · \(\omega_1^\text{CK}\) · \(\lambda,\zeta,\Sigma,\gamma\) · List of countable ordinals
Ordinal hierarchies: Fast-growing hierarchy · Slow-growing hierarchy · Hardy hierarchy · Middle-growing hierarchy · N-growing hierarchy
Uncountable cardinals: \(\omega_1\) · omega fixed point · inaccessible cardinal \(I\) · Mahlo cardinal \(M\) · weakly compact cardinal \(K\) · indescribable cardinal · rank-into-rank cardinal · more...

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