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Not to be confused with Extensible-E System.
 Extensible Illion System Based On -illions

Extensible Illion System is a notation by Nirvana Supermind for generating large -illions[1]. According to the creator's description "It was made to coin new -illion numbers past the ones I covered in my last post.",[1] where the phrase "last post" is linked to the page[2] of the creator's extension of Bower's illion numbers, and the description "If this post gets more popular, I may extend them to 7 or 8 tiers later (thus reaching the tetrational level) or even make an extensible -illion notation that goes beyond linear omega level.",[2] it is a continuation of Bower's naming system.

After some time and several technical issues, the creator clarified that this system has been scrapped.[3][4]

It is divided into extensions, and there is currently just one:

1. Primitive Illion System [5][6][7][8]

Although the creator insisted that this was well-defined, there are several issues with the definition, as we will explain later. Moreover, the creator removed correct and sourced descriptions related to the ill-definedness, including a mathematical proof of the incorrectness of the creator's statement, as if it were correct. [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]

## Primitive Illion System

### Original Definition

The primitive notation takes a base and any amount of arguments, which are non-negative integers. The original rules for it are:[5]

1. 0[b] = 103b+3
2. a[b] = a-1[b,b,b,b…] with b “b”s for a>0
3. a[b,c] = a[a[b,c-1]/1000]
4. a[#,0] = a[#]
5. 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 $$\omega^2$$ in the fast-growing hierarchy, but is ill-defined because of many errors:

1. There is no rule applicable to 0[] and 1[].
2. There are two distinct ways to solve 1[0].
1. If you apply rule 2, then the result will be 0[], which is ill-defined by the reason above.
2. If you apply rule 4, then the result will be 1[], which is ill-defined by the reason above.
3. There are two distinct ways to solve 1[0,0].
1. If you apply rule 3, then the result will be 1[1[0,-1]/1000], which is ill-defined because -1 is negative.
2. If you apply rule 4, then the result will be 1[0], which is ill-defined by the reason above.

### Second Definition

Later, the creator updated the definition by adding the case for the empty array after the errors were pointed out:[6]

1. 0[b] = 103b+3
2. a[] = 1000
3. a[b] = a-1[b,b,b,b…] with b “b”s for a>0
4. a[b,c] = a[a[b,c-1]/1000]
5. a[#,0] = a[#]
6. 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.

Indeed, it is still ill-defined because of the same issues for 1[0,0] (the invalidity of 1[0,-1]). However, if rule 4 were taken to only apply if c>0, like rule 6, then it would be calculated through rule 5, and this problem would not happen as we will explain in "Alternative Definition" section. On the other hand, this is not the case.

In this second definition,[6] 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.

### Third Definition

The creator updated the definition again after this issue was pointed out. The third 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 that "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."[7]

However, referring to the well-definedness itself in order to make a notation well-defined just causes a circular logic. Therefore if we consider the circular logic, it is ill-defined. On the other hand, if we ignore the circular logic as the creator did by replacing the description by "If there are two or more distinct rules to apply to a single expression, the uppermost rule which is applicable and whose result is a valid expression will be applied", then it is well-defined.

### Alternative Definition

Although the creator failed to make the notation well-defined and was just insisting that it was well-defined, there are several solutions by setting complete case classification without overlapping. For example, one natural unofficial 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:

1. If a = 0 and @ = "b" for a non-negative integer b, then a[@] = 103b+3.
2. If @ is empty, then a[@] = 1000
3. If a > 0 and @ = "b" for a non-negative integer b, then a[@] = a-1[b,b,b,b…] with b b's.
4. If @ = "b,c" for a non-negative integer b and a positive integer c, then a[@] = a[a[b,c-1] 10-3].
5. If @ = "#,0" for a non-empty array # of non-negative integers, then a[@] = a[#].
6. 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 fourth rule is not applicable to it since c is not positive, therefore it is solved by the fifth rule.

### Current Definition

The current definition, created after the alternative definition was given, refers to the new priority order of rules: "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." However, the creator stated that the "latest" definition includes the rule "If there are two or more distinct rules to apply to a single expression, the uppermost rule which is applicable and whose result is a valid expression will be applied".[19] The quoted description differs from the one in the third definition, which was the latest among the cited definitions at the time, and also from the one in the current definition. Anyway, as we will clearly explain later in "Example" section, the explanation by the creator is still inconsistent with the written rules.

### Example

Here are examples of intended values and actual values. The reason why we distinguish them is simply because the intended values are known to be incorrect.

#### Intended Values

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[1015/1000] = 0[1012].

0[1012]=103*1012+3

0[4,103*1012+3]=0[0[4,103*1012+3-1]/1000], etc.

0[4,1,1] = 0[4,10^12] = 1010^(3*10^12+3)3*10^12+3 ?

The intended value of 0[4,1,1] has mystery. First, it was written "0[4,1,1] = 1010^(3*10^12+3)3*10^12+3".[20] Since the result should be a power of 10, another user marked this result as a doubtful one. However, the creator removed the doubt several times as if it were the correct value by saying that the creator ''''remove NO INFORMATION except for the ill defined part and "intended to be....", not a SINGLE BIT OF IT AT ALL, I am just adding new information and removing the ill-defined part. That's all. NOTHING. ELSE."".[14]

Later, the creator agreed that it was actually a typo. Then what was the actual intended value? Since the creator expanded the equality as "0[4,1,1] = 0[4,10^12] = 1010^(3*10^12+3)3*10^12+3"[13], it would be an intended value. However, there is another issue. Why does 0[4,1,1] = 0[4,1012] hold?

Although the notation was ill-defined, we can expand it using written rules together with the written other examples. For example, we have the first line 0[4,1,1] = 0[4,0[4,1]], but it seems to be incompatible with the expanded equality 0[4,1,1] = 0[4,1012]. Does 0[4,0[4,1]] = 0[4,0[1012]] = 0[4,103*1012+3] coincide with 0[4,1012]?

Expanding 0[4,1,1] gives 0[4,0[1012]], which is NOT 0[4,1012], as shown above.

Since those two equalities are written by the creator, who strongly insists the wrongness of the doubt and removed the doubt so many times, the creator must intend 0[4,1012] = 0[4,103*1012+3]. As a consequence, either one of the following holds:

1. The creator's explanation is still incorrect, although the creator removed the doubt so many times.
2. The written definition is still not compatible with the creator's intention.

So let us check the intended value again. \begin{eqnarray*} & & 0[4,0,1] = 0[4,0[4,0,0]] = 0[4,0[4,0]] = 0[4,0[4]] \\ & = & 0[4,10^{3 \times 4 + 3}] = 0[4,10^{15}] > 0[4,10^{12}] \end{eqnarray*} As the computation shows, even 0[4,0,1] does not coincide with 0[4,1012]. How can we obtain the value 0[4,1012] or 1010^(3*10^12+3)3*10^12+3? We will explain further on actual values instead of those incorrect intended values.

#### Actual Values

When we show estimation of large numbers, it is good to use induction on arguments instead of writing a table of intended values, because they do not necessarily coincide with the actual values, as we have observed the creator's failures above.

Let $$\mathbb{N}$$ denote the set of non-negative integers. We define a map \begin{eqnarray*} f \colon \mathbb{N} & \to & \mathbb{N} \\ n & \mapsto & f(n) \end{eqnarray*} in the following recursive way:

1. If $$n = 0$$, then $$f(n) = 4$$.
2. If $$n > 0$$, then $$f(n) = 1000^{f(n-1)}$$.

By $$1 < 4 < 1000$$, we immediately obtain $$1000 \uparrow \uparrow n < f(n) < 1000 \uparrow \uparrow (n+1)$$ and $$1000 \uparrow \uparrow (n+1) < 1000^{f(n) + 1} < (1000 \uparrow \uparrow (n+2)) \times 1000$$ for any $$n \in \mathbb{N}$$.

Proposition
For any $$n \in \mathbb{N}$$, $$0[4,n]$$ coincides with $$1000^{f(n)+1}$$.
Proof
It is elementary to show the assertion by the induction on $$n$$.
If $$n = 0$$, then we have

\begin{eqnarray*} 0[4,n] = 0[4,0] = 0[4] = 10^{3 \times 4 + 3} = 10^{15} \end{eqnarray*}

and

\begin{eqnarray*} 1000^{f(n) + 1} = 1000^{f(0)+1} = 1000^{4+1} = 1000^5 = (10^3)^5 = 10^{3 \times 5} = 10^{15}. \end{eqnarray*}

Therefore $$0[4,n]$$ coincides with $$1000^{f(n)+1}$$.
If $$n > 0$$, then we have

\begin{eqnarray*} & & 0[4,n] = 0[0[4,n-1] \times 10^{-3}] = 0[1000^{f(n-1)+1} \times 10^{-3}] \\ & = & 0[1000^{f(n-1)}] = 10^{3 \times 1000^{f(n-1)}+3} = (10^3)^{f(n)+1} = 1000^{f(n)+1} \end{eqnarray*}

by the induction hypothesis. Thus the assertion holds. □

Combining Proposition and the inequalities above, we immediately obtain the following:

Corollary
For any $$n \in \mathbb{N}$$, the inequality $$1000 \uparrow \uparrow (n+1) < 0[4,n] < (1000 \uparrow \uparrow (n+2)) \times 1000$$ holds.

Now let us go back to the creator's mysterious values 0[4,1012] and 1010^(3*10^12+3)3*10^12+3. First, we should guess the meaning of the ambiguous expression 1010^(3*10^12+3)3*10^12+3. It must be either one of the following:

1. 10(3 ↑↑ 103×1012 + 3)×1012+3 ?
2. 10((3×1012) ↑↑ 103×1012 + 3)+3 ?
3. 10(3×1012+3) ↑↑ 103×1012 + 3 ?

In any case, the value is greater than $$10^{3 \uparrow \uparrow 10^{3 \times 10^{12} + 3}}$$, and hence is greater than $$10^{1000 \uparrow \uparrow (1000^{10^{12}} \times 333)}$$ as $$3 \uparrow \uparrow 3 = 3^{3^3} = 3^{27}$$, which coincides with 7,625,597,484,987, is greater than $$1000$$.

By the corollary above, we have \begin{eqnarray*} 0[4,10^{12}] < (1000 \uparrow \uparrow (10^{12}+2)) \times 1000. \end{eqnarray*} However, we also have \begin{eqnarray*} & & 10^{1000 \uparrow \uparrow (1000^{10^{12}} \times 333)} > 1000 \uparrow \uparrow (1000^{10^{12}} \times 333) \\ & > & 1000 \uparrow \uparrow (10^{12}+3) > (1000 \uparrow \uparrow (10^{12}+2)) \times 1000. \end{eqnarray*} Combining those inquealities, we obtain \begin{eqnarray*} & & 10^{1000 \uparrow \uparrow (1000^{10^{12}} \times 333)} > 0[4,10^{12}], \end{eqnarray*} which contradicts the creator's explanation "0[4,1012] = 1010^(3*10^12+3)3*10^12+3". Therefore the creator's explanation is still inconsistent.

## Sources

1. https://integralview.wordpress.com/2020/10/12/extensible-illion-system-index/
2. https://integralview.wordpress.com/2020/10/01/extended-tier-4-to-6-illions/
3. A difference page of. (The creator wrote "(scrapped)".)
4. A difference page of the talk page of. (The creator said "I have scrapped those notations and have stopped working on them in the favor of quick array notation which was well-defined.")
5. The original definition (Retrieved at UTC 7:00 on 15/10/2020)
6. The second definition (Retrieved at UTC 23:00 on 15/10/2020)
7. The third definition (Retrieved at UTC 3:00 on 27/10/2020)
8. The fourth definition (Retrieved at UTC 4:30 on 15/11/2020)
9. A difference page of this article. (The creator removed the original definition and the retrieval dates.)
10. A difference page of this article. (The creator removed the original definition.)
11. A difference page of this article. (The creator said "I add your correct sourced information back" as if the creator removed nothing, while the creator actually removed the original definition and added a description "The issues got fixed" even though the issues were not fixed.)
12. A difference page of this article. (The creator removed the original definition and the retrieval date, and added a description "these issues got fixed" again saying "Just realize the notation is NOT ill-defined anymore and stop", while the issue was not fixed.)
13. A difference page of this article. (The creator said "I kept EVERYTHING, just it is not ill defined anymore", while the creator actually removed the original definition and the description of the ill-definedness, although it was still ill-defined.)
14. A difference page of this article. (The creator said ""I remove NO INFORMATION except for the ill defined part and "intended to be....", not a SINGLE BIT OF IT AT ALL, I am just adding new information and removing the ill-defined part. That's all. NOTHING. ELSE."", while the creator actually removed the issue which made the notation ill-defined.)
15. A difference page of this article. (The creator removed the issue and added "This makes makes the notation well-defined.", although the notation was still ill-defined. Moreover, the creator replaced the description of the priority order of application of rules, although it was not written in the source with the written retrieval date.)
16. A difference page of this article. (The creator said ""I am NOT "hiding the fact". Anyone can go peek in the revision history, and seehow the first revision of the notation is ill-defined, or see the next revision by you which explains it. I am just updating the definition. Do you understand that YOU are the one who "doesn't understand what a source is"? You're staying away from the main issue, NO ONE SAID retrieval dates are needed in the citation policy, so it is better to remove the confusing info."" by ignoring all discussions in the talk page and the fact that the creator repeated to remove many descriptions, and removed the original definition, the issues, and so on. Also, the creator changed the retrieval date of the original definition. Needless to say, removing the retrieval date of a personal website makes it an unreliable source, while the creator repeats to state that we do not have to put retrieval dates. See also the site policy.)
17. A difference page of this article. (The creator removed several links above by saying "Reflecting the changes in the talk page" and ignoring requirement to stop removing sourced and correct informations.)
18. A differential page of this article. (The creator silently removed many constructive descriptions such as mathematical arguments, sources of the informations, and so on.)
19. A difference page of a talk page. (The creator said ""Also, it should be stated that “If there are two or more distinct rules to apply to a single expression, the uppermost rule which is applicable and whose result is a valid expression will be applied” rule is the latest one so the current notation is unanimously well-defined, so the “insisting” is false."".)