Inaccessible Cardinal

What is an inaccessible cardinal?

1) A number that no matter how much effort is put, it cannot be reached from below. The only way to reach it is by declaring it's existance.

2) An uncountable number

3) A number bigger than any finite or infinite number.

a) Is it larger than  ℵ₀ ? Yes, it is.

b) Is it larger than any aleph? Yes, it is.

'''Even though an Inaccessible is larger than any other finite or infinite number, a shadow of this number could be found below it. And that is  ℵ₀. You can't reach  ℵ₀ from below either. '''

Some people say that  '''ℵ₀ is also an inaccessible number, but some say that for a number to be inaccessible it also has to be uncountable.  ℵ₀ is very countable.'''

0,1,2,3...........

'''While an inaccessible cannot be counted. Where do you even start? '''

An inaccessible number is written like this:  Ѳ

Precise definition:

In set theory, an uncountable cardinal is inaccessible if it cannot be obtained from smaller cardinals by the usual operations of cardinal arithmetic. More precisely, a cardinal κ is strongly inaccessible if it is uncountable, it is not a sum of fewer than κ cardinals that are less than κ, and implies.

The term "inaccessible cardinal" is ambiguous. Until about 1950 it meant "weakly inaccessible cardinal", but since then it usually means "strongly inaccessible cardinal". An uncountable cardinal is weakly inaccessible if it is a regular weak limit cardinal. It is strongly inaccessible, or just inaccessible, if it is a regular strong limit cardinal (this is equivalent to the definition given above). Some authors do not require weakly and strongly inaccessible cardinals to be uncountable (in which case is strongly inaccessible). Weakly inaccessible cardinals were introduced by Hausdorff (1908), and strongly inaccessible ones by Sierpiński & Tarski (1930) and Zermelo (1930).

Every strongly inaccessible cardinal is also weakly inaccessible, as every strong limit cardinal is also a weak limit cardinal. If the generalized continuum hypothesis holds, then a cardinal is strongly inaccessible if and only if it is weakly inaccessible.

(aleph-null) is a regular strong limit cardinal. Assuming the axiom of choice, every other infinite cardinal number is regular or a (weak) limit. However, only a rather large cardinal number can be both and thus weakly inaccessible.

An ordinal is a weakly inaccessible cardinal if and only if it is a regular ordinal and it is a limit of regular ordinals. (Zero, one, and are regular ordinals, but not limits of regular ordinals.) A cardinal which is weakly inaccessible and also a strong limit cardinal is strongly 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.