User blog:Rgetar/Square brackets and I ordinal notation

This notation is like notation from my blog Square brackets OCF.

Purpose of this notation: I'll try to use this notation to make conversion into standard form a bit easier.

Fundamental sequence and cofinality
Definition of fundamental sequence I use: fundamental sequence of ordinal α is strictly increasing sequence of ordinals of minimal length such as α is least ordinal larger than all elements of this sequence.

(Or, we can say that α is least strict upper bound of its fundamental sequence. Least strict upper bound is defined similarly to least upper bound (supremum), but "≥" is replaced with ">").

Definition of cofinality I use: cofinality of ordinal α is length of its fundamental sequence.

Elements of fundamental sequence of ordinal α are enumerated using ordinals, beginning from 0:

α[0], α[1], α[2], α[3], ...

Designation of cofinality of ordinal α: cof(α).

So, cofinality of 0 is 0, and fundamental sequence of 0 is empty set; cofinality of successor ordinal α is 1, and fundamental sequence of successor ordinal α contains only one element α[0] such as α[0] + 1 = α.

incard(α)
incard means "infinite cardinality".

Definition of incard(α):


 * if α < ω then incard(α) = ω;
 * if α ≥ ω then incard(α) = card(α), where card(α) is cardinality of α.

Symbols
This notation uses 3 symbols:
 * ]
 * I
 * I

Strings
This notation uses 3 sorts of strings:
 * empty string
 * I
 * [β]γ, where β and γ are any strings in this notation.

Correspondence "string - ordinal"

 * Empty string is 0.
 * I is least weakly inaccessible cardinal.
 * []γ is successor of γ: []γ = γ + 1.
 * If γ < I then [I]γ is least uncountable cardinal larger than γ.

For other strings this correspondence is defined using fundamental sequences (see next section).

Cofinality and elements of fundamental sequence
 α is empty string (that is α = 0). cof(α) = 0. Fundamental sequence of α does not contain elements. α = [β]γ, cof(β) = 0 (that is β = 0, α = []γ). cof(α) = 1. α[0] = γ. α = [β]γ, cof(β) = 1. cof(α) = ω. α[0] = [β[0]]γ, α[n + 1] = [β[0]]α[n]. α = [β]γ, ω ≤ cof(β) ≤ incard(γ). cof(α) = cof(β). α[n] = [β[n]]α. α = [β]γ, incard(γ) < cof(β) < I. cof(α) = ω. α[n] = [δ[n + 1]]γ, where δ[n + 1] = [β[δ[n]]]δ[n], δ[0] is largest cardinal less than cof(β). α = [β]γ, γ < I, cof(β) = I. cof(α) = [I]γ. α[n] = [β[n]]α. α = I. cof(α) = I. α[n] = n. 

Calculation of incard(α)

 * α is empty string (that is α = 0). incard(α) = (that is incard(α) = ω).
 * α = [β]γ, β < I, γ < I. incard(α) = incard(γ).
 * α = [β]γ, β ≥ I, γ < I. incard(α) = α.
 * α = [β]γ, γ ≥ I. incard(α) = I.

Calculation of δ[0]
Calculation of δ[0] from "5." of section "Cofinality and elements of fundamental sequence": cof(β) > = ω, since ω is least possible incard, but incard(γ) < cof(β); cof(β) ≠ I, so, cof(β) should be calculated using "4." or "6." of section "Cofinality and elements of fundamental sequence", but in "4." cofinality is calculated using cofinality (cof(α) = cof(β)), so, cof(β) should be calculated using "6." of section "Cofinality and elements of fundamental sequence": cof(α) = [I]γ (here β is α); δ[0] is incard of this γ: δ[0] = incard(γ).

Comparison of strings

 * empty string < I
 * γ < [β]γ
 * if β1 < β2 then [β1]γ < [β2]γ
 * if γ < I then [β]γ < I

Note: these rules are not enough for comparison of any [β1]γ1 and [β2]γ2.