User blog:Rgetar/Definitions update

Due to changes (1, 2, 3) of my array notation (1, 2, 3) it is needed to update definitions.

Functions of array
eo means "element of"

eo(X; Y) is element of array X at coordinates Y

est means "element of set to"

est(X; Y; α) is array X with element at coordinates Y set to α

beo means "base element of"

If array ≠ 0 then its base elements are non-zero elements. If array = 0 then its base element is element at coordinates 0, that is 0. Base elements are enumerated from right to left beginning from 1.

beo(X; n) is n-th base element of array X

best means "base element set to"

best(X; n; α) is array X with n-th base element set to α

cobeo means "coordinates of base element of"

cobeo(X; n) is coordinates of n-th base element of array X

cobest means "coordinates of base element set to"

cobest(X; n; Y) is array X with coordinates of n-th base element set to Y

nobe means "number of base elements"

nobe(X) is number of base elements of array X

sobe means "set of base elements"

sobe(X) is set of base elements of array X

(old designation: sonze(X; Y))

isobe means "iterated set of base elements"

isobe(X) is sobe(X) ∪ {sobe(cobeo(X; n))}, 1 ≤ n ≤ nobe(X)

(old designation: isonze(X; Y))

leo means "last element of"

leo(X) is eo(X; 0)

lest means "last element set to"

lest(X; α) is est(X; 0; α)

X0

X0 = lest(X; 0)

X-1

If leo(X) = α + 1 then X-1 = lest(X; α)

fbeo means "first base element of"

fbeo(X) is beo(X; nobe(X))

lbeo means "last base element of"

lbeo(X) is beo(X; 1)

cofbeo means "coordinates of first base element of"

cofbeo(X) is cobeo(X; nobe(X))

(old designation: cofrewnzeloi(X))

colbeo means "coordinates of last base element of"

colbeo(X) = X' is cobeo(X; 1)

(old designation: X', still used)

lrt means "left rest"

lrt(X) is array X without its last base element

X = lrt(X), lbeo(X)

rrt means "right rest"

rrt(X) is array X without its first base element

X = fbeo(X), rrt(X)

ileo means "iterated last element of"

ileo(X) is leo(X) ∪ ileo(X')

fbest means "first base element set to"

fbest(X; α) is best(X; nobe(X); α)

lbest means "last base element set to"

fbest(X; α) is best(X; 1; α)

cofbest means "coordinates of first base element set to"

cofbest(X; Y) is cobest(X; nobe(X); Y)

colbest means "coordinates of last base element set to"

colbest(X; Y) is cobest(X; 1; Y)

Note: X* is now invalid and is not used.

Additional designations
(X; a; b) = a, if X' ≠ 0

(X; a; b) = b, if X' = 0

or

(X; a; b) = a, if X{·}a depends on a

(X; a; b) = b, if X{·}a does not depend on a

Elements of array with negative coordinates (<-α>) should be ignored.

[-1]a = a

X{·}a
X{·}a = {lbest(X; β}, &lt;Y&gt;(X'; 1; a)}, β < lbeo(X), Y ∈ X'{·}a

Expanded view (without additional designations (X; a; b) and <-α>):

X{·}a = {lbest(X; β}}, if X' = 0, β < lbeo(X)

X{·}a = {lbest(X; β}, &lt;Y&gt;a}, if X' ≠ 0, X" = 0, β < lbeo(X), Y ∈ X'{·}a

X{·}a = {lbest(X; β}, &lt;Y&gt;1}, if X' ≠ 0, X" ≠ 0, β < lbeo(X), Y ∈ X'{·}a

Ordinal array function [X]a
[0]a = a + 1

[X]a = sup([(X; -1; X0)][Y]a), Y ∈ X{·}a

Expanded view (without additional designations (X; a; b) and [-1]a):

[0]a = a + 1

[X]a = sup([X0][Y]a), if X' = 0, Y ∈ X{·}a

[X]a = sup([Y]a), if X' ≠ 0, Y ∈ X{·}a

Generalized Veblen function
φ(X) = α is (1 + leo(X))-th common fixed point of all functions α = φ(Y), Y ∈ X0{·}α

δ
δ = 0, if leo(X) = 0

δ = φ(X-1) + 1, if leo(X) ≠ 0

X[n]α
X[n]α = lbest(X; lbeo(X)[n]), if lbeo(X) - limit ordinal

X[n]α = colbest(X; X'[n]α), if lbeo(X) - successor ordinal, leo(X') - not successor ordinal

X[n]α = lbest(X; lbeo(X)-1), α, if lbeo(X) - successor ordinal, leo(X') - successor ordinal

Fundamental sequences
1. To get fundamental sequence of Cantor normal form, replace its last term with fundamental sequence of the last term.

2. φ(α+1)[n] = φ(α)·n

Rest of rules are for X not a successor ordinal:

3. φ(X)[n] = φ(X[n]0), if leo(X) - limit ordinal

or, in a more understandable form,

3. φ(X)[n] = φ(lest(X; leo(X)[n])), if leo(X) - limit ordinal

4. φ(X)[n] = φ(lest(X0[n]0; δ)), if leo(X) is successor ordinal or zero, ileo(X0) ∋ limit ordinal

5. φ(X)[n] = φ(X0[n]φ(X)[n-1]) for n ≥ 0 and φ(X)[-1] = δ, if leo(X) is successor ordinal or zero, ileo(X0) ∌ limit ordinal