Changes: Extensible Illion System

Revision as of 01:55, 16 October 2020

Not to be confused with Extensible-E System.

Extensible Illion System
Growth	$f_{\omega ^{2}}(x)$ ^[1]
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

Primitive Illion System

The primitive notation takes a base and any amount of arguments, which are non-negative integers. The rules for it are:

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 is well-defined, or else the lower rule. Note that an old version of the notation^[2] excluded the second rule and did not include the precedence, making some expressions such as ill-defined:

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.

However, these issues got fixed.

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] = 0[10¹⁵]

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

Sources

[placeholder-1] 1.0 ^1.1 https://integralview.wordpress.com/2020/10/12/extensible-illion-system-index/

[current-2] The original definition (Retrieved at UTC 7:00 on 15/10/2020)

[1]

[2]

@@ Line 2: / Line 2: @@
 {{Infobox
 |Box title = Extensible Illion System
-|Row 1 title = state
+|Row 1 title = Growth
-|Row 1 info = ill-defined
+|Row 1 info = <math>f_{\omega^2}(x)</math><ref name="placeholder"/>
 |Row 2 title = Based On
 |Row 2 info = [[-illion]]s
 }}
-'''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:
+'''Extensible Illion System''' is a notation by [[User:Nirvana Supermind|Nirvana Supermind]] (for generating large -illions)<ref name="placeholder">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
-# Primitive Illion System <ref name="orginal">[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 23:00 on 15/10/2020)]</ref>
-This is ill-defined, as is explained later.
 == Primitive Illion System ==
-The primitive notation takes a base and any amount of arguments, which are non-negative integers. The original rules for it<ref name="original"> are:
+The primitive notation takes a base and any amount of arguments, which are non-negative integers. The rules for it are:
 # 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,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.
+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 is well-defined, or else the lower rule. Note that an old version of the notation<ref name="current">[https://integralview.wordpress.com/2020/10/13/primitive-illion-system/ The original definition (Retrieved at UTC 7:00 on 15/10/2020)]</ref> excluded the second rule and did not include the precedence, making some expressions such as ill-defined:
-This notation qas 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 is no rule applicable to 0[] and 1[].
-# There are two distinct ways to solve 1[0].
+*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 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.
+**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].
+*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 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.
+**If you apply the rule 4, then the result will be 1[0], which is ill-defined by the reason above.
-# The computation of 0[4,1] = 0[0[4,0]/1000] is intended to be 0[1.015], which is ill-defined because 1.015 is not an integer.
-Later, the creator updated the definition by adding the case for the empty array:
+However, these issues got fixed.
-# 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,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.
+<br />
-It is still ill-defined because of the same issues for 1[0,0] (the invalidity of 1[0,-1]) and 0[4,1] (the invalidity of 0[1.015]).
 === Example ===
@@ Line 50: / Line 43: @@
 [4,1] = 0[0[4,0]/1000]
-[4,0] = 0[4] = 1015
+[4,0] = 0[4] = 10<sup>15</sup>
+[4,1] = 0[10<sup>15</sup>]
-[4,1] = 0[1.015], which is ill-defined because 1.015 is not an integer.
-[4,1,1] is intended to be 1010^(3*10^12+3)3*10^12+3, but is ill-defined because 0[4,0[4,1]] includes the ill-defined value 0[4,1].
+[4,1,1] = 10<sup><sup>10^(3*10^12+3)3</sup>3*10^12+3</sup>.
 == Sources ==