Inaccessible cardinal
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In set theory, κ is called weakly inaccessible iff the following two conditions hold.
- 1. κ is an uncountable regular cardinal, i.e. cf(κ) = κ > ω, where cf denotes cofinality.
- 2. κ is a limit cardinal, i.e. for every cardinal μ < κ, μ+ < κ, where μ+ is the successor cardinal of μ.
Aleph-null would meet those two conditions, but it is not weakly inaccessible because it is countable. If the axiom of choice, then every other infinite cardinal number is either regular or a limit. However, only a rather large cardinal number can be both and thus weakly inaccessible.
If condition 2. above is replaced by
- 2'. κ is a strong limit cardinal, i.e. for every cardinal μ < κ, 2μ < κ.
then κ is called strongly inaccessible, or just inaccessible. Again, <math>\aleph_0</math> meets this condition, but is not inaccessible because it is countable.
If the Generalized Continuum Hypothesis holds, then a cardinal is strongly inaccessible if and only if it is weakly inaccessible.
The assumption of the existence of a strongly inaccessible cardinal is sometimes applied in the form of the assumption that one can work inside a Grothendieck universe, the two ideas being intimately connected.
Models and consistency
ZFC implies that Vκ is a model of ZFC whenever κ is strongly inaccessible. And ZF implies that Lκ is a model of ZFC whenever κ is weakly inaccessible. Thus ZF together with "there exists a weakly inaccessible cardinal" implies that ZFC is consistent. Therefore, inaccessible cardinals are a type of large cardinal.
If κ is the smallest strong inaccessible, then Vκ is a standard set model of ZFC which contains no strong inaccessibles. If κ is the smallest ordinal which is either a weak inaccessible in the universe or weakly inaccessible relative to any standard set model, then Lκ is a standard set model of ZFC which contains no weak inaccessibles. Thus if ZFC implies the existence of weakly inaccessible cardinals, then ZFC implies a contradiction.
α-inaccessible cardinals and hyper-inaccessible cardinals
For any ordinal α, a cardinal κ is α-inaccessible iff κ is inaccessible and for every ordinal β < α, the set of β-inaccessibles less than κ is unbounded in κ (and thus of cardinality κ, since κ is regular).
A cardinal κ is hyper-inaccessible iff κ is κ-inaccessible. (It can never be κ+1-inaccessible.)
For any ordinal α, a cardinal κ is α-hyper-inaccessible iff κ is hyper-inaccessible and for every ordinal β < α, the set of β-hyper-inaccessibles less than κ is unbounded in κ.
One could go on this way defining "super-inaccessibles", "ultra-inaccessibles", etc.. But why bother? Let us just skip ahead to Mahlo cardinals.
Using "weakly inaccessible" instead of "inaccessible", similar definitions can be made for "weakly α-inaccessible", "weakly hyper-inaccessible", and "weakly α-hyper-inaccessible".
See also: