Changes: Extensible Illion System

Revision as of 19:26, 27 October 2020

Not to be confused with Extensible-E System.

Extensible Illion System
No Title	No information
Based On	-illions

Extensible Illion System is a notation by Nirvana Supermind (for generating large -illions)^[1]. It is divided into extensions, and there is currently just one:

Primitive Illion System ^[2]^[3]

This notation is most likely well-defined, but an old version of it was ill-defined earlier^[2].

Primitive Illion System

The primitive notation takes a base and any amount of arguments, which are non-negative integers. Here, # is a portion of the array, which can be empty. The current definition of the notation is^[3]:

0[b] = 10^3b+3
a[] = 1000
a[b] = a-1[b,b,b,b...] with b "b"s for a>0
a[b,c] = a[a[b,c-1]/1000]
a[#,0] = a[#]
a[#,b,c] = a[#,a[#,b,c-1]] for c>0

Here, # is a portion of the array, which can be empty. If there are two or more distinct rules to apply to a single expression, the upper rule will be applied if it can be evaluated, otherwise the lower rule is applied.

Example

0[4,1,1] = 0[4,0[4,1]]

0[4,1] = 0[0[4,0]/1000]

0[4,0] = 0[4] = 10¹⁵

0[4,1,1] = 0[4,10^12] = 10^{^{10^(3*10^12+3)}3*10^12+3}

Sources

[1] ttps://integralview.wordpress.com/2020/10/12/extensible-illion-system-index/

[original-2] 2.0 ^2.1 The original definition (Retrieved at UTC 19:25 on 15/10/2020)

[current-3] 3.0 ^3.1 The latest definition (Retrieved at UTC 7:25 on 27/10/2020)

[1]

[2]

[3]

@@ Line 2: / Line 2: @@
 {{Infobox
 |Box title = Extensible Illion System
-|Row 1 title = state
-|Row 1 info = ill-defined
 |Row 2 title = Based On
 |Row 2 info = [[-illion]]s
@@ Line 10: / Line 8: @@
 '''Extensible Illion System''' is a notation by [[User:Nirvana Supermind|Nirvana Supermind]] (for generating large -illions)<ref>https://integralview.wordpress.com/2020/10/12/extensible-illion-system-index/</ref>. It is divided into extensions, and there is currently just one:
-# Primitive Illion System <ref name="original">[https://integralview.wordpress.com/2020/10/13/primitive-illion-system/ The original definition (Retrieved at UTC 7:00 on 15/10/2020)]</ref><ref name="current">[https://integralview.wordpress.com/2020/10/13/primitive-illion-system/ The current definition (Retrieved at UTC 3:00 on 27/10/2020)]</ref>
+# Primitive Illion System <ref name="original">[https://integralview.wordpress.com/2020/10/13/primitive-illion-system/ The original definition (Retrieved at UTC 19:25 on 15/10/2020)]</ref><ref name="current">[https://integralview.wordpress.com/2020/10/13/primitive-illion-system/ The latest definition (Retrieved at UTC 7:25 on 27/10/2020)]</ref>
-Although the creator insists that this is well-defined, this is ill-defined, as is explained later.
+This notation is most likely well-defined, but an old version of it was ill-defined earlier<ref name="original />.
 == Primitive Illion System ==
-The primitive notation takes a base and any amount of arguments, which are non-negative integers. The original rules for it are:<ref name="original" />
+The primitive notation takes a base and any amount of arguments, which are non-negative integers.  Here, # is a portion of the array, which can be empty. The current definition of the notation is<ref name="current" />:
-# 0[b] = 10<sup>3b+3</sup>
-# a[b] = a-1[b,b,b,b…] with b “b”s for a>0
-# a[b,c] = a[a[b,c-1]/1000]
-# a[#,0] = a[#]
-# a[#,b,c] = a[#,a[#,b,c-1]] for c>0
-Here, # is a portion of the array, which can be empty.
-This notation was intended to reach w^2 in the fast-growing hierarchy, but was ill-defined because of many errors:
-# There is no rule applicable to 0[] and 1[].
-# There are two distinct ways to solve 1[0].
-## If you apply the rule 2, then the result will be 0[], which is ill-defined by the reason above.
-## If you apply the rule 4, then the result will be 1[], which is ill-defined by the reason above.
-# There are two distinct ways to solve 1[0,0].
-## If you apply the rule 3, then the result will be 1[1[0,-1]/1000], which is ill-defined because -1 is negative.
-## If you apply the rule 4, then the result will be 1[0], which is ill-defined by the reason above.
-Later, the creator updated the definition by adding the case for the empty array:<ref name="second">[https://integralview.wordpress.com/2020/10/13/primitive-illion-system/ The current definition (Retrieved at UTC 23:00 on 15/10/2020)]</ref>
 # 0[b] = 10<sup>3b+3</sup>
 # a[] = 1000
-# a[b] = a-1[b,b,b,b…] with b “b”s for a>0
+# a[b] = a-1[b,b,b,b...] with b "b"s for a>0
 # a[b,c] = a[a[b,c-1]/1000]
 # a[#,0] = a[#]
 # a[#,b,c] = a[#,a[#,b,c-1]] for c>0
-Here, # is a portion of the array, which can be empty. If there are two or more distinct rules to apply to a single expression, the upper rule will be applied.
+Here, # is a portion of the array, which can be empty. If there are two or more distinct rules to apply to a single expression, the upper rule will be applied if it can be evaluated, otherwise the lower rule is applied.
-Indeed, it is still ill-defined because of the same issues for 1[0,0] (the invalidity of 1[0,-1]), while the creator still insists that it is well-defined. In this second definition,<ref name="second" /> 1[0,0] should be solved by the forth rule because the rule forces us to apply the upper most rule applicable to it. The priority order of rules is actually what the second definition explains, but the creator is trying to hide this fact as if the priority order were corrected from the beginning.<ref>[https://googology.wikia.org/wiki/Extensible_Illion_System?action=history The history page] of this article.</ref>
-The current definition is the same as the second definition except for the priority order of rules. As the issue on 1[0,0] has been pointed out, the creator tried to avoid the issue by clarifying "If there are two or more distinct rules to apply to a single expression, the upper rule will be applied if it is well-defined, otherwise the lower rule."<ref name="current" /> However, refering to the well-definedness itself in order to make a notation well-defined just causes a circular logic. As a conclusion, the notation is still ill-defined, although the creator insists that it is well-defined.
-Although the creator failed to make the notation well-defined and is just insisting that it is well-defined, there are several simple and elementary solutions by setting complete case classification without overlapping. For example, one natural inofficial alternative definition of a[@] for a non-negative integer a and a (possibly empty) array @ of non-negative integers can be given in the following recursive way:
-# If a = 0 and @ = "b" for a non-negative integer b, then a[@] = 10<sup>3b+3</sup>.
-# If @ is empty, then a[@] = 1000
-# If a > 0 and @ = "b" for a non-negative integer b, then a[@] = a-1[b,b,b,b…] with b b's.
-# If @ = "b,c" for a non-negative integer b and a positive integer c, then a[@] = a[a[b,c-1] 10<sup>-3</sup>].
-# If @ = "#,0" for a non-empty array # of non-negative integers, then a[@] = a[#].
-# If @ = "#,b,c" for a non-empty array # of non-negative integers, a non-negative integer b, and a positive integer c, then a[@] = a[#,a[#,b,c-1]].
-Then the issue on 1[0,0] does not occur, because the forth rule is not applicable to it and the fifth rule is applicable to it.
 === Example ===
 [4,1,1] = 0[4,0[4,1]]
-[4,1] = 0[0[4,0]/1000]
-[4,0] = 0[4] = 10<sup>15</sup>
+[4,1] = 0[0[4,0]/1000]
-[4,1] = 0[10<sup>15</sup>/1000] = 0[10<sup>12</sup>].
+[4,0] = 0[4] = 10<sup>15</sup>
+[4,1,1] = 0[4,10^12] = 10<sup><sup>10^(3*10^12+3)</sup>3*10^12+3</sup>
-[4,1,1] = 0[4,10<sup>12</sup>] is intended to be 1010^(3*10^12+3)3*10^12+3, but the result is doubtful. The result should be a power of 10, but the creator insists that 1010^(3*10^12+3)3*10^12+3, which is 3 module 10, is the correct value.<ref>[https://googology.wikia.org/wiki/Extensible_Illion_System?type=revision&diff=299158&oldid=299081 A difference page] of this article</ref>
 == Sources ==