User blog:Rgetar/Definitions update

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

Arrays are written as sequence of &lt;c&gt;e pairs, separated with commas, where c is coordinates of element, and e is element at this coordinates. Coordinates also can be arrays. Sometimes coordinates can be omitted. If an array is not zero, all its zero elements may be omitted.

Arrays are written in order of decreasing of coordinates. And, if coordinates are arrays, first compared elements with larger coordinates, then with smaller.

Example of array:

1, 6, 0, 0, 3, 0, 0, 0, 5, 0 =

<9>1, <8>6, <7>0, <6>0, <5>3, <4>0, <3>0, <2>0, <1>5, <0>0 =

<9>1, <8>6, <5>3, <1>5 =

<9>1, 6, <5>3, <1>5

Functions of array
1. eo means "element of"

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

2. est means "element of set to"

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

3. 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

4. best means "base element set to"

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

5. cobeo means "coordinates of base element of"

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

6. 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

7. nobe means "number of base elements"

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

8. sobe means "set of base elements"

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

(old designation: sonze(X; Y))

9. isobe means "iterated set of base elements"

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

(old designation: isonze(X; Y))

10. leo means "last element of"

leo(X) is eo(X; 0)

11. lest means "last element set to"

lest(X; α) is est(X; 0; α)

12. X0

X0 = lest(X; 0)

13. X-1

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

14. fbeo means "first base element of"

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

15. lbeo means "last base element of"

lbeo(X) is beo(X; 1)

16. cofbeo means "coordinates of first base element of"

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

(old designation: cofrewnzeloi(X))

17. colbeo means "coordinates of last base element of"

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

(old designation: X', still used)

18. lrt means "left rest"

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

X = lrt(X), lbeo(X)

19. rrt means "right rest"

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

X = fbeo(X), rrt(X)

20. ileo means "iterated last element of"

ileo(X) is leo(X) ∪ ileo(X')

21. fbest means "first base element set to"

fbest(X; α) is best(X; nobe(X); α)

22. lbest means "last base element set to"

fbest(X; α) is best(X; 1; α)

23. cofbest means "coordinates of first base element set to"

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

24. colbest means "coordinates of last base element set to"

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

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

(X; a; b)
(X; a; b) = \(\left\{\begin{array}{lcr} a \quad \text{if} \; X' ≠ 0\\ b \quad \text{if} \; X' = 0\\ \end{array}\right. \)

or

(X; a; b) = \(\left\{\begin{array}{lcr} a \quad \text{if} \; X\{·\}a \; \text{depends on} \; a\\ b \quad \text{if} \; X\{·\}a \; \text{does not depend on} \; a\\ \end{array}\right. \)

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

[-1]a
[-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