Inaccessible cardinal

From Exampleproblems

In set theory, an uncountable cardinal number κ is called weakly inaccessible if the following two conditions hold.

1. cf(κ) = κ, where cf denotes the cofinality. Such a cardinal is called a regular cardinal.
2. There is no next smaller cardinal number; i.e., for every cardinal γ < κ, there is another cardinal number δ between γ and κ. Such a cardinal number κ is called a limit cardinal.

Every transfinite cardinal number, except for aleph-null which meets those two conditions (but is not weakly inaccessible because it is countable), is either regular or a limit; however, only a rather large cardinal number can be both. If condition 2. above is replaced by

2'. For every cardinal γ < κ, 2^γ < κ (that is, κ is a strong limit cardinal).

then κ is called strongly inaccessible, or just inaccessible. Again, meets this condition, but is not inaccessible because it is countable.

Assuming that ZFC is consistent, the existence of (strongly or weakly) inaccessible cardinals provably cannot be proved in ZFC; inaccessible cardinals are therefore a type of large cardinal. In fact, ZFC cannot even prove that the existence of inaccessible cardinals is consistent with ZFC (because ZFC+"there exists an inaccessible cardinal" proves the consistency of ZFC); however, the assumption that there is no inaccessible cardinal is provably consistent with ZFC (assuming the consistency of ZFC).

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.