User blog comment:Scorcher007/About Cofinality, sipmly/@comment-32213734-20200102055227/@comment-28606698-20200102165900

Rgetar wrote correctly. I just want to add the definition of cofinality.

But at first I define what are normal functions:

f:Ord→Ord is a normal function if it is increasing and continuous i.e.

α<β⇒ f(α)<f(β)

f(α)=sup{ f(β)| β < α } for a limit ordinal α.

Then, the cofinality of a limit ordinal α is the least limit ordinal β such that there is a normal function

f: β → α with sup{f(ξ)|ξ< β}=α.

In other words the cofinality of a limit ordinal α is the least length of transfinite strictly increasing sequence such that α is the limit of this sequence.

A sequence with length cof(α) and limit α is a fundamental sequence of the ordinal α.

Also cofinality of an ordinal α can be considered as least ordinal β such that β is the order type of a cofinal subset of α. A subset B of α is cofinal (i.e. it has "the same final - α") if for all ξ∈α there is an ordinal η∈ B such that ξ≤ η.

Each ordinal is identified with the set of all its predecessors.

For example, 42={0,1,2,…,41} and cofinal subset of 42 with smallest order type is {41}. Then cof(42)=1. This is correct for all successor ordinals.

ω*2={0,1,2,…,ω,ω+1,ω+2,…} and cofinal subset of ω*2 with smallest order type is {ω,ω+1,ω+2,…}, that gives cof(ω*2)=ω. This is correct for all countable limit ordinals.

Same way, Ω*2={0,1,2,…, Ω, Ω +1, Ω +2,…} and cofinal subset of Ω *2 with smallest order type is { Ω, Ω +1, Ω +2,…}, that gives cof(Ω *2)= Ω.

And so on.