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Set theory:)

Informally, an *inaccessible* cardinal number is an uncountable cardinal number which cannot be expressed in terms of smaller cardinal numbers. Consider ℵ_{0}. The set of smaller cardinal numbers is precisely the set of finite numbers. We cannot reach ℵ_{0} (or higher) by using cardinals like 2^{k} (where k is a finite number) or ∑_{i=1}^{k}a_{i} (where k and a_{1},...,a_{k} are all finite). This makes ℵ_{0} 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:

This is just a codification of the above.

ℵ_{0} is explicitly excluded from being inaccessible: we're looking for a *new* concept of size, and merely being "infinite" is old hat. ℵ_{1} is *not* inaccessible, since 2^{ℵ0}≥ℵ_{1} (it's not a strong limit cardinal; neither is any successor cardinal). ℵ_{ω} is *not* inaccessible, since ℵ_{ω}=ℵ_{0} + ℵ_{1} + ... (a sum of ℵ_{0} 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 ℵ_{0} 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.