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

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how I learned about cofinality from Denis Maxudov.

Denis Maxudov wrote in rule 4 that if α = Ων + 1 then cof(α) = Ων + 1 and α[η] = η.

Since I did not understand cofinality, I wrote: "By the way, α[η] is fundamental sequence of α, and η is natural number?"..., "α[η] = η is fundamental sequence of ω, and all Ων + 1 are uncountable ordinals. Am I right?"

Then Denis Maxudov explained me that there are fundamental sequences of length larger than ω (so, I realized that η in α[η] are not always natural numbers).

I think it can be started from fundamental sequences and that they can be of different "length". (I use the following definition of fundamental sequence: it is strictly increasing ordinal sequence of minimal possible "length" such as given ordinal is minimal ordinal larger than any element of this sequence). And cofinality is this "length".

I think, first explain about fundamental sequences of length ω for countable ordinals and how they can be used to make larger countable ordinals from smaller (for example, using Veblen function), and that for any limit countable ordinal there are such fundamental sequences. Then explain that there are so large limit ordinals (beginning from least uncountable ordinal Ω), that they cannot be obtained this way. So it is impossible to construct fundamental sequence of length ω for Ω, but for such cases we may introduce "larger" fundamental sequences.

There are 3 kinds of ordinals:
 * 0, which cofinality is 0 (its only fundamental sequence is empty);
 * successor ordinals (x + 1), which cofinality is 1 (and with only fundamental sequence, which consists of one element x + 1[0] = x);
 * limit ordinals, which cofinality is infinite.

Limit ordinals also may differ from each other by length of their fundamental sequence. For example, ω, Ω and Ω2 have fundamental sequences, which can be expressed using the same formula but they differ by cofinality: (We cannot construct fundamental sequence of Ω with ω elements, and fundamental sequence of Ω2 with ω or Ω elements).
 * α[n] = n,
 * for ω cofinality is ω, so n < ω;
 * for Ω cofinality is Ω, so n < Ω;
 * for Ω2 cofinality is Ω2, so n < Ω2.

I think it can be useful to beginners to explore cofinalities of ordinals from smaller to larger, for example So, some Ω numbers are reqular, and some are singular: Ωω is least singular Ω number: its cardinality is Ωω, but cofinality is ω.
 * cof(0) = 0
 * cof(1) = 1
 * cof(2) = 1
 * cof(3) = 1
 * cof(4) = 1
 * cof(5) = 1
 * cof(ω) = ω - from this point till Ω cardinality is ω, but cofinality alternates: 1 for successors, and ω for limits
 * cof(ω + 1) = 1
 * cof(ω + 2) = 1
 * cof(ω + 3) = 1
 * cof(ω + 4) = 1
 * cof(ω + 5) = 1
 * cof(ω2) = ω
 * cof(ω2 + 1) = 1
 * cof(ω2 + 2) = 1
 * cof(ω3) = ω
 * cof(ω4) = ω
 * cof(ω5) = ω
 * cof(ω2) = ω
 * cof(ω2 + 1) = 1
 * cof(ω2 + 2) = 1
 * cof(ω2 + ω) = ω
 * cof(ω2 + ω2) = ω
 * cof(ω22) = ω
 * cof(ω23) = ω
 * cof(ω3) = ω
 * cof(ω4) = ω
 * cof(ωω) = ω
 * cof(ωω + 1) = 1
 * cof(ε0) = ω
 * cof(ε0 + 1) = 1
 * cof(Ω) = Ω - from this point till Ω2 cardinality is Ω, but cofinality alternates: 1 for successors, ω or Ω for limits
 * cof(Ω + 1) = 1
 * cof(Ω + 2) = 1
 * cof(Ω + ω) = ω
 * cof(Ω + ω + 1) = 1
 * cof(Ω + ε0) = ω
 * cof(Ω + ε0 + 1) = 1
 * cof(Ω2) = Ω
 * cof(Ω2 + 1) = 1
 * cof(Ω2 + ω) = ω
 * cof(Ω3) = Ω
 * cof(Ω4) = Ω
 * cof(Ω5) = Ω
 * cof(Ωω) = ω
 * cof(Ω(ω + 1)) = Ω
 * cof(Ωε0) = ω
 * cof(Ω(ε0 + 1)) = Ω
 * cof(Ω2) = Ω
 * cof(Ωω) = ω
 * cof(ΩΩ) = Ω
 * cof(Ω2) = Ω2 - from this point till Ω3 cardinality is Ω2, but cofinality alternates: 1 for successors, ω, Ω or Ω2 for limits
 * cof(Ω2 + 1) = 1
 * cof(Ω2 + ω) = ω
 * cof(Ω2 + Ω) = Ω
 * cof(Ω22) = Ω2
 * cof(Ω3) = Ω3
 * cof(Ω4) = Ω4
 * cof(Ω5) = Ω5
 * cof(Ωω) = ω - not Ωω, since Ωω has fundamental sequence of length ω: ω, Ω, Ω2, Ω3, ...
 * cof(ΩΩ) = Ω
 * cof(Ω2) = Ω2 - from this point till Ω3 cardinality is Ω2, but cofinality alternates: 1 for successors, ω, Ω or Ω2 for limits
 * cof(Ω2 + 1) = 1
 * cof(Ω2 + ω) = ω
 * cof(Ω2 + Ω) = Ω
 * cof(Ω22) = Ω2
 * cof(Ω3) = Ω3
 * cof(Ω4) = Ω4
 * cof(Ω5) = Ω5
 * cof(Ωω) = ω - not Ωω, since Ωω has fundamental sequence of length ω: ω, Ω, Ω2, Ω3, ...
 * cof(Ω5) = Ω5
 * cof(Ωω) = ω - not Ωω, since Ωω has fundamental sequence of length ω: ω, Ω, Ω2, Ω3, ...
 * cof(Ωω) = ω - not Ωω, since Ωω has fundamental sequence of length ω: ω, Ω, Ω2, Ω3, ...
 * cof(Ωx) = Ωx - regular
 * cof(Ωx) < Ωx - singular

If x is successor ordinal, then Ωx is regular.

If x is limit ordinal, then Ωx is usually singular, but not always: there are limit x such as Ωx is regular. Such Ωx are called weakly inaccessible cardinals.