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'''Infra notation''' is a notation for large numbers made by Wikia user [[User:Nirvana Supermind|Nirvana Supermind]].<ref>{{cite web|last=Nirvana Supermind|url=https://integralview.wordpress.com/2020/12/08/infra-notation/|title=Infra Notation - Integral View}}</ref>. |
'''Infra notation''' is a notation for large numbers made by Wikia user [[User:Nirvana Supermind|Nirvana Supermind]].<ref>{{cite web|last=Nirvana Supermind|url=https://integralview.wordpress.com/2020/12/08/infra-notation/|title=Infra Notation - Integral View}}</ref>. |
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The notation is currently composed of 1 parts, which is as follows: |
The notation is currently composed of 1 parts, which is as follows: |
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− | * [https://integralview.wordpress.com/2020/12/08/basic-infra-notation/ Basic Infra Notation] (BIN)<ref name="first">{{cite web||last=Nirvana Supermind|url=https://integralview.wordpress.com/2020/12/08/basic-infra-notation/|title=Basic Infra Notation - Integral View |
+ | * [https://integralview.wordpress.com/2020/12/08/basic-infra-notation/ Basic Infra Notation] (BIN)<ref name="first">{{cite web||last=Nirvana Supermind|url=https://integralview.wordpress.com/2020/12/08/basic-infra-notation/|title=Basic Infra Notation - Integral View|accessdate=Wed, 09 Dec 2020 10:34:00 GMT}}</ref><ref name="latest">{{cite web||last=Nirvana Supermind|url=https://integralview.wordpress.com/2020/12/08/basic-infra-notation/|title=Basic Infra Notation - Integral View|accessdate=Thu, 10 Dec 2020 02:40:00 GMT}}</ref>. |

* Linear Infra Notation (LIN) |
* Linear Infra Notation (LIN) |
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* Basic Cascading Infra Notation (BCIN) |
* Basic Cascading Infra Notation (BCIN) |
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− | |||

− | It shares many common but irregular features with the creator's preceding works [[Rampant Array Notation]], [[Extensible Illion System]], and [[Quick array notation]], and hence we explains it first. |
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− | |||

− | == Feature == |
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− | '''Unfortunately, this section is a target of the creator's removal.''' |
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− | <ref name="removement 5">[https://googology.wikia.org/wiki/Infra_Notation?type=revision&diff=307249&oldid=307248 A difference page] of this page. (The creator removed the most parts of this article, including the description of the ill-definedness and the alternative definition, by saying "Stop! At least give a reasoning for your revertment. This version is more up-to-date.")</ref> |
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− | <ref name="removement 6">[https://googology.wikia.org/wiki/Infra_Notation?type=revision&diff=307267&oldid=307263 A difference page] of this page. (The creator removed the most parts of this article, including the description of the ill-definedness and the alternative definition, by saying "Your description is not only longer and still contains some outdated information (like fixing issues in the Alternate defintion that I already fixed), but it includes fake informations about how the Original version and Alternate defintion sections are a target of my removement, even though I added their equivalents when I had the time.")</ref> |
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− | <ref name="removement 7">[https://googology.wikia.org/wiki/Infra_Notation?type=revision&diff=307271&oldid=307268 A difference page] of this page. (The creator removed the most parts of this article, including the description of the ill-definedness and the alternative definition, by saying "The precise text where you wrongly say I remove the alternate definition (when in reality I added it in "Pbot's definition") is " Although the creator somewhy repeated to remove the alternative definition,". Also, you give subjective guesses about why I reverted your edits including that I don't understand the ill-definedness, when in reality I was planning to make this all along.")</ref> |
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− | <ref name="removement 8">[https://googology.wikia.org/wiki/Infra_Notation?type=revision&diff=307278&oldid=307277 A difference page] of this article. (The creator removed the most part of this article by saying "Your four sources do not include it, but my revision of the article has a CLEARLY VISIBLE section called "P進大好きbot's definition", which is the same thing. Don't you have anytthing better to do?").</ref> |
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− | <ref name="removement 9">[https://googology.wikia.org/wiki/Infra_Notation?type=revision&diff=307281&oldid=307280 A difference page] of this article. (The creator removed the most part of this article by saying "Will you accept the reality? I have a section which is IDENTICAL to your Alternate definition section. And I initially removed the Alternate definition because I need to remove your version informations, but of course, I would add it back to my version later.").</ref> |
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− | <ref name="removement 10">[https://googology.wikia.org/wiki/Infra_Notation?type=revision&diff=307299&oldid=307296 A difference page] of this page. (The creator silently removed the most part of this page.)</ref> |
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− | <ref name="removement 11">[https://googology.wikia.org/wiki/Infra_Notation?type=revision&diff=307306&oldid=307302 A difference page] of this page. (The creator removed the most part of this page by saying "I have given the clear explanation of what is wrong with your version. Now stay with mine.".)</ref> |
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− | |||

− | We explain the specific features of notations by the creator and the corresponding articles. |
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− | # Many of the articles have similar histroies: |
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− | ## First, the creator creates an article in this wiki on his or her own notations by himself or herself, and insists statements such as well-definedness and analyses. |
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− | ## Second, another user points out errors, ill-definedness, incorrectness of the creator's original statements. |
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− | ## Third, the creator drastically removes descriptions mainly related to ill-definedness, including mathematical proofs of incorrectness of the creator's fake statement and alternative definitions which solve issues in the original definition. |
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− | ## Finally, the unconstructive removements are reverted, and sources of the creator's manipulations are added to the articles. |
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− | # Whenever the creator makes a new notation, they say "it is composed of <number> parts", even if they do not intend to expand it. |
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− | # The creator insists statements on well-definedness, analyses, and intended values by just asserting them or writing something like "this '''most likely''' reaches <ordinal>", but usually the original statements are wrong. Indeed, none of them was well-defined when the creator created the corresponding article in this wiki. When another user points out the incorrectness, then the creator dishonestly removes the description including mathematical proofs of the incorrectness, as if they were correct. Therefore we should not believe the creator's '''"most likely"''' unless he or she gave a proof or a peer-reviewed source. |
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− | See also [[Rampant Array Notation]], [[Extensible Illion System]], and [[Quick array notation]] for the sources of the feature. |
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− | |||

− | Unfortunately, this notation is not a counterexample. Since these features are quite special in googology, we deeply explain them together with full sources. |
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== Basic notation == |
== Basic notation == |
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⚫ | |||

− | Basic Infra Notation (BIN) is the 1st part of Infra Notation. It’s the simplest and weakest one. Unfortunately, it was ill-defined, as we will explain later. |
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+ | All information here only applies to the latest version of the notation. |
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+ | Basic Infra Notation (BIN) is the 1st part of Infra Notation.<ref name="latest" /> An array is of the form I<>, I<a>, or I<a,b>. The numbers inside the angle brackets are called entries. All numbers in the array must be positive integers. It’s the simplest and weakest one. The rules for it are |
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⚫ | |||

− | '''Unfortunately, the most part of this section is a target of the creator's removal.''' |
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− | <ref name="removement 1">[https://googology.wikia.org/wiki/Infra_Notation?type=revision&diff=307183&oldid=307182 A difference page] of this article. (The creator removed the description of the issue and the alternative definition by saying ""Your explanation is incorrect. The empty string is valid expression, it is stated that the array can be I<>. And that expression is also not ill-defined, since I had the foresight to put "I<> = 0"."".)</ref> |
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− | <ref name="removement 2">[https://googology.wikia.org/wiki/Infra_Notation?type=revision&diff=307225&oldid=307208 A difference page] of this page. (The creator silently removed the most parts of this article, including the description of the ill-definedness and the alternative definition.)</ref> |
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− | <ref name="removement 3">[https://googology.wikia.org/wiki/Infra_Notation?type=revision&diff=307238&oldid=307237 A difference page] of this page. (The creator silently removed the most parts of this article, including the description of the ill-definedness and the alternative definition.)</ref> |
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− | <ref name="removement 4">[https://googology.wikia.org/wiki/Infra_Notation?type=revision&diff=307246&oldid=307245 A difference page] of this page. (The creator removed the most parts of this article, including the description of the ill-definedness and the alternative definition, by saying "Why should I keep your version? It is outdated now that I updated the definition to be more likely to be well-defined.")</ref> |
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− | <ref name="removement 5" /> |
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− | <ref name="removement 6" /> |
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− | <ref name="removement 7" /> |
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− | <ref name="removement 8" /> |
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− | <ref name="removement 9" /> |
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− | <ref name="removement 10" /> |
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− | <ref name="removement 11" /> |
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− | Moreover, the creator insisted that this section was not a target of the removal during the removal of this section.<ref name="removement 6" /> To be more precise, the creator is trying to rename this section, and simultaneously removing several contents of this section. |
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− | |||

⚫ | |||

− | A valid expression in this notation is of the form I<>, I<a>, or I<a,b> for '''non-negative integers''' a and b. The rules for a valid expression are:<ref name="first" /> |
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# I<> = 0 |
# I<> = 0 |
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# I<a> = 10<sup>a</sup> |
# I<a> = 10<sup>a</sup> |
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− | # I<#, |
+ | # I<#,1> = I<#> |

− | # I<a,b> = I<I<...<I<a,b-1>,b-1>,...>,b-1> (b>0) |
+ | # I<a,b> = I<I<...<I<a,b-1>,b-1>,...>,b-1> (b>0) with "a" nestings |

− | Where # is a part of said array (a string of entries and commas, it can also be empty). If there are two or more distinct rules to apply to a single expression, the uppermost-numbered rule which is applicable and whose result is a valid expression will be applied. |
+ | Where # is a part of said array (a string of entries and commas, it can also be empty). If there are two or more distinct rules to apply to a single expression, the uppermost-numbered rule which is applicable and whose result is a valid expression will be applied. The creator said that this notation most likely reaches <math>f_\omega(n)</math> in the [[Fast-growing hierarchy]]. |

− | ==== Issue ==== |
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− | Although the phrase "for a times" is ambiguous or is a typo of some precise description, it seems to mean the depth of the nest at least when a is positive. Since the creator tried to remove the sentence above explaining the ambiguity of the phrase "for a times", |
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− | <ref name="removement 1" /> |
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− | <ref name="removement 2" /> |
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− | <ref name="removement 3" /> |
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− | <ref name="removement 4" /> |
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− | <ref name="removement 5" /> |
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− | he or she strongly believes that there is no ambiguity. The creator insisted that this notation most likely reaches \(f_{\omega}(n)\) in the [[Fast-growing hierarchy]]. |
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− | |||

− | As we have explained, the creator always insists something like '''"this notation is well-defined"''' or '''"this notation most likely reaches <ordinal>"''', but those statements are not necessarily true. For example, I<0,1> is ill-defined because there is no rule applied to it. The first three rules are not applicable to it. The fourth rule is applicable to it, but the result is not defined, as the description "for 0 times" itself is ambiguous. According to the creator, the result of the application of rule 4 to I<0,1> is I<>,<ref name="removement 1" /> but they skip over how it is derived. After applying rule 4, it becomes I<I<...<I<0,0>,0>,...>,0> for 0 times. This does not make sense. The creator tends to insists that '''any error pointed by others can be regarded as a typo, and hence does not contribute to the ill-definedness'''. |
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− | <ref name="typo 1">[https://googology.wikia.org/wiki/Talk:Quick_array_notation?type=revision&diff=303924&oldid=303906 A difference page] of the talk page of another article. (The creator said "What is the point of saying a notation is ill-defined because there was a typo or two from my XS-based draft?".)</ref> |
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− | <ref name="typo 2">[https://googology.wikia.org/wiki/Talk:Quick_array_notation?type=revision&diff=303943&oldid=303930 A difference page] of the talk page of another article. (The creator said ""The reason you say "Since there are common errors in the original version and other versions, we start with the explanation of the original version first." is a false because the current notation '''is well-defined'''"" and "'''all mistakes you mentioned are typos, errors and mistakes I made, that do not contribute to the well-definedness or ill-definedness of a notation.'''")</ref> |
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− | If one stands on the creator's viewpoint that '''any error is a typo which does not contribute to the ill-definedness''', then it might be regarded as a '''well-defined''' notation whose computation rule is uniquely applicable to any valid expression. Anyway, there is no reasonable explanation why we should expand I<0,1> as the intended expression I<>. |
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==== Example ==== |
==== Example ==== |
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− | I<3, |
+ | I<3,2> |

− | = I<I<... |
+ | = I<I<...<I<3,1>,1>,...>,1> for 3 times |

− | = I<I<I<3, |
+ | = I<I<I<3,1>,1,1> |

− | = I<I<I<3 |
+ | = I<I<I<3>>> |

− | = 10<sup>10<sup>1000</sup></sup> |
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+ | = 10^10^1000 |
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− | I<0,1> = ill-defined (According to the creator, it expands as I<>.<ref name="removement 1"/>) |
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− | === |
+ | === Old version === |

+ | All information here only applies to the oldest version of the notation. |
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− | '''Unfortunately, this section is a target of the creator's removal.''' |
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+ | |||

− | <ref name="removement 1" /> |
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+ | An old version of the notation had an input of a nonnegative integer array<ref name="first" />. It also had the rule "I<#,0> = I<#>" instead of the current third rule, and a less precise description of "with a nestings". I<0,1> is ill-defined because there is no rule applied to it. The first three rules are not applicable to it. The fourth rule is applicable to it, but the result is not defined, as the description "for 0 times" itself is ambiguous. |
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− | <ref name="removement 2" /> |
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+ | |||

− | <ref name="removement 3" /> |
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+ | However, the creator fixed those issues by changing the input to a positive integer array in the current version. |
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− | <ref name="removement 4" /> |
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⚫ | |||

− | However, the creator insists that this section was not a target of the removal,<ref name="removement 6" /> even though we have the sources. According to the creator, he or she kept this section,<ref name="removement 7" /> but we can easily show that he or she is cheating simply by checking the four sources. After another user pointed out this fact, then the creator changed the statement; the creator '''actually removed''' this section so many times, but created a condensed section instead.<ref name="removement 8" /> Namely, the creator accepted the removement. |
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+ | I<0,1> = ill-defined |
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+ | |||

⚫ | |||

+ | When the issues with the notation had not been fixed, user [[User:P進大好きbot|P進大好きbot]] created an alternate definition of this notation that is well-defined. We define a non-negative integer I<@> for any array @ of non-negative integers in the following recursive way: |
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− | ==== Definition ==== |
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− | It is quite elementary to fix the issue. For example, we define a non-negative integer I<@> for any array @ of '''non-negative integers''' in the following recursive way: |
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# If @ is the empty string, then I<@> = 0. |
# If @ is the empty string, then I<@> = 0. |
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# If @ = "a" for a non-negative integer a, then I<@> = 10<sup>a</sup>. |
# If @ = "a" for a non-negative integer a, then I<@> = 10<sup>a</sup>. |
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## If n = 0, then I<sup>n</sup><#,a,b-1> = a. |
## If n = 0, then I<sup>n</sup><#,a,b-1> = a. |
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## If n ≠ 0, then I<sup>n</sup><#,a,b-1> = I<#,I<sup>n-1</sup><#,a,b-1>,b-1>. |
## If n ≠ 0, then I<sup>n</sup><#,a,b-1> = I<#,I<sup>n-1</sup><#,a,b-1>,b-1>. |
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+ | |||

For example, we have I<0,1> = I<sup>0</sup><0,0> = 0. Fortunately, this result coincides with the creator's intension I<0,1> = I<> = 0. Therefore the restriction of this alternative definition to the subset of the domain consisting of arrays of length at most 2 is perhaps equivalent to the original definition fixed in some intended way. |
For example, we have I<0,1> = I<sup>0</sup><0,0> = 0. Fortunately, this result coincides with the creator's intension I<0,1> = I<> = 0. Therefore the restriction of this alternative definition to the subset of the domain consisting of arrays of length at most 2 is perhaps equivalent to the original definition fixed in some intended way. |
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+ | P進大好きbot's definition is identical to the Original version, except it defines "for b times" using an iterative method that will work for 0. |
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− | ==== Example ==== |
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+ | <br /> |
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− | I<0,1> = I<sup>0</sup><0,0> = 0. |
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− | |||

⚫ | |||

− | '''Unfortunately, this section is a target of the creator's removal.''' |
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− | <ref name="removement 6" /> |
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− | <ref name="removement 7" /> |
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− | <ref name="removement 8" /> |
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− | <ref name="removement 9" /> |
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− | <ref name="removement 10" /> |
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− | <ref name="removement 11" /> |
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− | To be more precise, the creator is trying to rename this section, and simultaneously removing several contents of this section. |
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− | |||

− | After the ill-definedness was pointed out, the creator first could not understand the issue, as he or she dishonestly removed the description and the alternative definition above twice, by insisting that the description was wrong. |
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− | <ref name="removement 1" /> |
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− | <ref name="removement 2" /> |
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− | Later, the creator understood the ill-definedness. Although the creator somewhy repeated to remove the alternative definition |
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− | <ref name="removement 3" /> |
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− | <ref name="removement 4" /> |
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− | <ref name="removement 5" /> |
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− | or rename the section for the alternative definition |
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− | <ref name="removement 6" /> |
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− | <ref name="removement 7" /> |
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− | <ref name="removement 8" /> |
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− | <ref name="removement 9" /> |
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− | <ref name="removement 10" /> |
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− | <ref name="removement 11" /> |
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− | he or she added a second definition. The main change from the original definition is that the domain is restricted to the subset of arrays of length at most 2 whose entries are positive integers instead of non-negative integers. |
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− | |||

− | ==== Definition ==== |
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− | A valid expression in this notation is of the form I<>, I<a>, or I<a,b> for '''positive integers''' a and b. The rules for a valid expression are:<ref name="second" /> |
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− | |||

− | # I<> = 0 |
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− | # I<a> = 10<sup>a</sup> |
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− | # I<#,1> = I<#> |
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− | # I<a,b> = I<I<...<I<a,b-1>,b-1>,...>,b-1> (b>0) for a times |
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− | |||

− | Where # is a part of said array (a string of entries and commas, it can also be empty). If there are two or more distinct rules to apply to a single expression, the uppermost-numbered rule which is applicable and whose result is a valid expression will be applied. Similar to the original definition, the phrase "for a times" is ambiguous or is a typo of some precise description, but it seems to mean the depth of the nest. As we explained above, the creator perhaps considers that there cannot be any ambiguity here, although the ill-definedness of the original definition is due to the ambiguity here. |
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− | |||

− | Since there is no different rule other than the third rule, it is identical to a subsystem of the notation given by the alternative definition except for shifting the second imput (and other higher imputs) by 1, if the ambiguous phrase "for a times" actually means what we have explained above. |
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− | |||

− | ==== Example ==== |
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− | One significant point is that the result of I<3,1> has changed, as the creator replaced the third rule. However, according to the creator, the expansion of I<3,1> has not changed.<ref name="intended value 1">[https://googology.wikia.org/wiki/Infra_Notation?oldid=307243 An old version] of this page. (The creator wrote a wrong computation of I<3,1>.)</ref> Therefore we explain both intended values and actual values. |
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− | |||

− | ===== Intended Value ===== |
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− | I<3,1> |
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− | |||

− | = I<I<...<I<3,0>,0>,...>,0> for 3 times |
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− | |||

− | = I<I<I<3,0>,0>,0><ref name="intended value 1" /> (This is ill-defined because it is an invalid expression in the second definition, but the creator continued the computation.) |
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− | |||

− | I<3,2> |
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− | |||

− | = I<I<…<I<3,1>,1>,…>,1> for 3 times |
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− | |||

− | = I<I<I<3,1>,1,1><ref name="second" /> (This is ill-defined because it is an invalid expression, but the creator continued the computation.) |
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− | |||

− | ===== Actual Value ===== |
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− | * I<3,1> = I<3> = 10<sup>3</sup> = 1000 |
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− | * I<3,2> = I<I<I<3,1>,1>,1> = I<I<1000,1>,1> = I<I<1000>,1> = I<10<sup>1000</sup>,1> = I<10<sup>1000</sup>> = 10<sup>10<sup>1000</sup></sup> |
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− | Therefore the explanation by the creator is still inconsistent. |
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− | |||

− | == See also == |
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− | * [[Rampant Array Notation]] |
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− | * [[Extensible Illion System]] |
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− | * [[Quick array notation]] |
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== Sources == |
== Sources == |
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<references /> |
<references /> |
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− | [[Category:Numbers by Nirvana Supermind]] |
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[[Category:Notations]] |
[[Category:Notations]] |
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[[Category:Functions]] |
[[Category:Functions]] |
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− | [[Category:Incomplete]] |

## Revision as of 10:51, 10 December 2020

**Infra notation** is a notation for large numbers made by Wikia user Nirvana Supermind.^{[1]}.

The notation is currently composed of 1 parts, which is as follows:

- Basic Infra Notation (BIN)
^{[2]}^{[3]}. - Linear Infra Notation (LIN)
- Basic Cascading Infra Notation (BCIN)

## Contents

## Basic notation

### Current version

All information here only applies to the latest version of the notation.

Basic Infra Notation (BIN) is the 1st part of Infra Notation.^{[3]} An array is of the form I<>, I<a>, or I<a,b>. The numbers inside the angle brackets are called entries. All numbers in the array must be positive integers. It’s the simplest and weakest one. The rules for it are

- I<> = 0
- I<a> = 10
^{a} - I<#,1> = I<#>
- I<a,b> = I<I<...<I<a,b-1>,b-1>,...>,b-1> (b>0) with "a" nestings

Where # is a part of said array (a string of entries and commas, it can also be empty). If there are two or more distinct rules to apply to a single expression, the uppermost-numbered rule which is applicable and whose result is a valid expression will be applied. The creator said that this notation most likely reaches in the Fast-growing hierarchy.

#### Example

I<3,2>

= I<I<...<I<3,1>,1>,...>,1> for 3 times

= I<I<I<3,1>,1,1>

= I<I<I<3>>>

= 10^10^1000

### Old version

All information here only applies to the oldest version of the notation.

An old version of the notation had an input of a nonnegative integer array^{[2]}. It also had the rule "I<#,0> = I<#>" instead of the current third rule, and a less precise description of "with a nestings". I<0,1> is ill-defined because there is no rule applied to it. The first three rules are not applicable to it. The fourth rule is applicable to it, but the result is not defined, as the description "for 0 times" itself is ambiguous.

However, the creator fixed those issues by changing the input to a positive integer array in the current version.

#### Example

I<0,1> = ill-defined

### P進大好きbot's definition

When the issues with the notation had not been fixed, user P進大好きbot created an alternate definition of this notation that is well-defined. We define a non-negative integer I<@> for any array @ of non-negative integers in the following recursive way:

- If @ is the empty string, then I<@> = 0.
- If @ = "a" for a non-negative integer a, then I<@> = 10
^{a}. - If @ = "#,0" for a non-empty array # of non-negative integers, then I<@> = I<#>.
- If @ = "#,a,b" for an array # of non-negative integers, non-negative integer a, and a positive integer b, then I<@> = I
^{a}<#,a,b-1>, where I^{n}<#,a,b-1> is the non-negative integer defined for any non-negative integer n in the following recursive way:- If n = 0, then I
^{n}<#,a,b-1> = a. - If n ≠ 0, then I
^{n}<#,a,b-1> = I<#,I^{n-1}<#,a,b-1>,b-1>.

- If n = 0, then I

For example, we have I<0,1> = I^{0}<0,0> = 0. Fortunately, this result coincides with the creator's intension I<0,1> = I<> = 0. Therefore the restriction of this alternative definition to the subset of the domain consisting of arrays of length at most 2 is perhaps equivalent to the original definition fixed in some intended way.

P進大好きbot's definition is identical to the Original version, except it defines "for b times" using an iterative method that will work for 0.

## Sources

- ↑ Nirvana Supermind. Infra Notation - Integral View.
- ↑
^{2.0}^{2.1}Nirvana Supermind. Basic Infra Notation - Integral View. Retrieved Wed, 09 Dec 2020 10:34:00 GMT. - ↑
^{3.0}^{3.1}Nirvana Supermind. Basic Infra Notation - Integral View. Retrieved Thu, 10 Dec 2020 02:40:00 GMT.