inaccessible cardinal

Informally, an inaccessible cardinal number is an uncountable cardinal number which cannot be expressed in terms of smaller cardinal numbers. Consider ℵ₀. The set of smaller cardinal numbers is precisely the set of finite numbers. We cannot reach ℵ₀ (or higher) by using cardinals like 2^k (where k is a finite number) or ∑_i=1^ka_i (where k and a₁,...,a_k are all finite). This makes ℵ₀ fundamentally different from what comes below it; it introduces a new concept of size (which we call "infinite").

Formally, a cardinal number a is called inaccessible iff:

• a is uncountable.
• ∀b<a. 2^b<a. (a is a strong limit cardinal)
• if S is a set of cardinal numbers and |S|<a and ∀s∈S.s<a then ∑S<a. (a is a regular cardinal)

ℵ₀ is explicitly excluded from being inaccessible: we're looking for a new concept of size, and merely being "infinite" is old hat. ℵ₁ is not inaccessible, since 2^ℵ₀≥ℵ₁ (it's not a strong limit cardinal; neither is any successor cardinal). ℵ_ω is not inaccessible, since ℵ_ω=ℵ₀ + ℵ₁ + ... (a sum of ℵ₀ smaller cardinal numbers).

So where are the inaccessible cardinals? We don't know if there are any. It is consistent with ZFC that inaccessible cardinals exist, and it is also consistent with ZFC that no inaccessible cardinals exist. Just like ℵ₀ requires a special axiom (the axiom of infinity) to exist, so too do other "new concepts of infinity". This may be aesthetically pleasing, but it is hardly mathematically satisfying. Certain ("stronger") other "large cardinal axioms" would imply the existence of inaccessible cardinals, and also give pleasing results (on very small cardinals!); this might be a reason to accept such a large cardinal axiom.

Weakly inaccessible cardinals are (or rather, may be) a slightly weaker concept.