### Post by reden on Dec 26, 2022 19:48:54 GMT

We have previously observed that there are energetical differences between

"1000 googolplexians",

"Rayo's Number",

"Large Number Garden Number",

"Most Optimal Number",

"Largest Number Ever",

"Infinity",

"Infinity^∞",

"∞",

"∞^∞"

And so forth. There are degrees to Infinity as yinyang discovered. This thread is dedicated to documenting them and other related research such as Cardinal Numbers.

Copypaste from VSBattles wiki, from where I found the concepts:

1-A: Outerverse level

Characters or objects that can affect structures with a number of dimensions equal to the cardinal aleph-2, which in practical terms also equals a level that completely exceeds Low 1-A structures to the same degree that they exceed High 1-B and below. This can be extrapolated to larger cardinal numbers as well, such as aleph-3, aleph-4, and so on, and works in much the same way as 1-C and 1-B in that regard. Characters that stand an infinite number of steps above baseline 1-A are to have a + modifier in their Attack Potency section (Outerverse level+).

High 1-A: High Outerverse level

Characters or objects that can affect structures that are larger than what the logical framework defining 1-A and below can allow, and as such exceed any possible number of levels contained in the previous tiers, including an infinite or uncountably infinite number. Practically speaking, this would be something completely unreachable to any 1-A hierarchies.

A concrete example of such a structure would be an inaccessible cardinal, which in simple terms is a number so large that it cannot be reached ("accessed") by smaller numbers, and as such has to be "assumed" to exist in order to be made sense of or defined in a formal context (Unlike the standard aleph numbers, which can be straightforwardly put together using the building blocks of set theory). Even just the amount of infinite cardinals between the first inaccessible cardinal and aleph-2 (Which defines 1-A) is greater than cardinals such as aleph-0, aleph-1, aleph-2, aleph-3, etc., and even many aleph numbers whose index is an infinite ordinal.. More information on the concept is available on this page.

Tier 0: Boundless

Characters or objects that can affect structures which completely exceed the logical foundations of High 1-A, much like it exceeds the ones defining 1-A and below, meaning that all possible levels of High 1-A are exceeded, even an infinite or uncountably amount of such levels. This tier has no endpoint, and can be extended to any higher level just like the ones above.

Being "omnipotent" or any similar reasoning is not nearly enough to reach this tier on its own; however, such statements can be used as supporting evidence in conjunction with more substantial information.

--------

Tiering System Explanation Page

Contents

1 Introduction

2 Terminology

3 Cardinal Numbers

4 See also

Introduction

The following is an explanation of the upper bounds of this wiki's Tiering System, namely the parts encompassed by Low 1-A and above, although the concepts presented here do have great influence on the functionings of much lower tiers. For information on the primary measuring stick used to categorize those, please see this page instead.

Power Set: The set of all subsets of a given set X, commonly denoted as 2X or P(X). An example is the power set of {1, 3, 4}, which equals {∅, {1}, {3}, {4), {1, 3}, {1, 4}, {3, 4}, {1, 3, 4}}

κ: Placeholder for infinite cardinals.

Cardinal Numbers

Infinity is larger than you think - Numberphile

The video above is a brief explanation of the concept of countable and uncountable infinite sets, a very fundamental element in Set Theory, and which is of great importance to understand most of the concepts introduced in the following explanation. Hence, it is highly advisable that you watch it if the ideas presented here are unfamiliar.

How_To_Count_Past_Infinity

This video also offers a more detailed explanation of the more general idea of cardinal numbers, and is also highly recommended.

Firstly, to properly introduce greater sizes of infinity, it is extremely important to establish a crucial distinction, namely, the one between Cardinal Numbers and Ordinal Numbers.

In essence, cardinals numbers are used to denote the exact amount of objects contained within a given set, formally called the set's cardinality, for example, the cardinality of a set of four apples is 4.

On the other hand, ordinal numbers can be defined as being used to denote numbers as well-ordered sets, or in layman's terms, collections that possess a defined smallest element. Under the standard construction of ordinals, formalized by John Von Neumann, any given natural number is an ordinal, and the well-ordered set of all ordinals smaller than itself. For example:

4 = {0, 1, 2, 3}

7 = {0, 1, 2, 3, 4, 5, 6}

When approaching finite sets of objects, these two notions walk hand in hand, and there is no practical distinction between them. However, when dealing with infinite collections, these two are split, and there turns out to be a very palpable difference between denoting order in sets and denoting the quantity of objects contained in them.

After all finite ordinal numbers are exhausted, there comes the first infinite ordinal number: ω, which can be mathematically defined as equivalent to the set of all natural numbers, or more precisely:

ω = {0, 1, 2, 3, 4, 5...}

Exactly after ω, comes ω+1, which is defined as:

ω+1 = {0, 1, 2, 3, 4, 5... ω}

Then ω+2:

ω+2 = {0, 1, 2, 3, 4, 5... ω, ω+1}

And so on and so forth, until any further operations applied over ω (See these pages for more information). However, it is important to know that, while all of these ordinal numbers come after ω, and are by technicality, "bigger" than it, the sets which they represent each possess the same quantity of objects, and thus the same cardinality. Indeed, they are countable sets, and thus are all represented by the same cardinal number: Aleph-Naught, denoted as ℵ0.

After ℵ0, comes the smallest uncountable cardinal number: ℵ1, which is itself indexed by the ordinal number ω1, the set of all countable ordinal numbers. For the purposes of the Tiering System, it is accepted as an Axiom that ℵ1 is the cardinality of the set of all real numbers, and thus equal to the power set of ℵ0, and the same principle is generalized unto any higher cardinal number.

This hierarchy is then extended unto Aleph Numbers whose subscript can be defined as being correspondent to any higher number, be it finite or infinite: ℵ2, ℵ3, ℵ4... ℵω, ℵω+1, ℵω+2, and so on and so forth, with each succeeding cardinal being equal to the power set of the previous one.

Fixed Points

Mathematically speaking, a fixed point of the aleph hierarchy is essentially a given cardinal κ such that κ = ℵκ. In plain english, this essentially means that an aleph-fixed point is an infinite cardinal number whose size is so large that it is unchangeable through transformations, and equal to the amount of cardinals below it.

To further elaborate on this: It is useful to think about cardinal fixed points by thinking about the exact amount of cardinals preceding any given infinite number and associate it with their subscript: That is, there are no infinite cardinals below ℵ0, one below ℵ1, two below ℵ2... ℵ0 many infinite cardinals below ℵω, and so on, and so forth.

Now, suppose that there is an infinite cardinal whose subscript is essentially an infinite cascade of alephs, endlessly reiterating themselves downwards. As the subscript itself contains infinitely-many alephs in it, removing one of them from it would not alter the number itself in any way, thus making this cardinal into a number whose size is such that the number of infinite cardinals smaller than it is equal to the cardinal itself.

Large Cardinals

Large Cardinals are, speaking very roughly, certain cardinal numbers with the property of being very "large". That is, they are cardinals numbers whose existence is not provable from the standard axioms of most set theories, and thus have to be explicitly defined on them by the addition of another axiom postulating their existence, much like how an infinite set (ω) cannot be obtained from the standard tools and operations of arithmetic, and thus its existence needs to be added separately.

The Hierarchy of Large Cardinals is fairly extensive and contains many exotic-named numbers, but the most conventional and easily-understood entrypoint to it are the Inaccessible Cardinals: Infinite cardinal numbers that are uncountable, regular (cannot be defined as the union of quantities smaller than themselves) and strong limit (cannot be attained through repeated power set operations), which in layman's terms, means that they cannot be reached from lesser cardinals in any way, shape or form, standing beyond their scope entirely. The following is a good illustration of an Inaccessible Cardinal's relation to lesser numbers:

"1000 googolplexians",

"Rayo's Number",

"Large Number Garden Number",

"Most Optimal Number",

"Largest Number Ever",

"Infinity",

"Infinity^∞",

"∞",

"∞^∞"

And so forth. There are degrees to Infinity as yinyang discovered. This thread is dedicated to documenting them and other related research such as Cardinal Numbers.

Copypaste from VSBattles wiki, from where I found the concepts:

1-A: Outerverse level

Characters or objects that can affect structures with a number of dimensions equal to the cardinal aleph-2, which in practical terms also equals a level that completely exceeds Low 1-A structures to the same degree that they exceed High 1-B and below. This can be extrapolated to larger cardinal numbers as well, such as aleph-3, aleph-4, and so on, and works in much the same way as 1-C and 1-B in that regard. Characters that stand an infinite number of steps above baseline 1-A are to have a + modifier in their Attack Potency section (Outerverse level+).

High 1-A: High Outerverse level

Characters or objects that can affect structures that are larger than what the logical framework defining 1-A and below can allow, and as such exceed any possible number of levels contained in the previous tiers, including an infinite or uncountably infinite number. Practically speaking, this would be something completely unreachable to any 1-A hierarchies.

A concrete example of such a structure would be an inaccessible cardinal, which in simple terms is a number so large that it cannot be reached ("accessed") by smaller numbers, and as such has to be "assumed" to exist in order to be made sense of or defined in a formal context (Unlike the standard aleph numbers, which can be straightforwardly put together using the building blocks of set theory). Even just the amount of infinite cardinals between the first inaccessible cardinal and aleph-2 (Which defines 1-A) is greater than cardinals such as aleph-0, aleph-1, aleph-2, aleph-3, etc., and even many aleph numbers whose index is an infinite ordinal.. More information on the concept is available on this page.

Tier 0: Boundless

Characters or objects that can affect structures which completely exceed the logical foundations of High 1-A, much like it exceeds the ones defining 1-A and below, meaning that all possible levels of High 1-A are exceeded, even an infinite or uncountably amount of such levels. This tier has no endpoint, and can be extended to any higher level just like the ones above.

Being "omnipotent" or any similar reasoning is not nearly enough to reach this tier on its own; however, such statements can be used as supporting evidence in conjunction with more substantial information.

--------

Tiering System Explanation Page

Contents

1 Introduction

2 Terminology

3 Cardinal Numbers

4 See also

Introduction

The following is an explanation of the upper bounds of this wiki's Tiering System, namely the parts encompassed by Low 1-A and above, although the concepts presented here do have great influence on the functionings of much lower tiers. For information on the primary measuring stick used to categorize those, please see this page instead.

Power Set: The set of all subsets of a given set X, commonly denoted as 2X or P(X). An example is the power set of {1, 3, 4}, which equals {∅, {1}, {3}, {4), {1, 3}, {1, 4}, {3, 4}, {1, 3, 4}}

κ: Placeholder for infinite cardinals.

Cardinal Numbers

Infinity is larger than you think - Numberphile

The video above is a brief explanation of the concept of countable and uncountable infinite sets, a very fundamental element in Set Theory, and which is of great importance to understand most of the concepts introduced in the following explanation. Hence, it is highly advisable that you watch it if the ideas presented here are unfamiliar.

How_To_Count_Past_Infinity

This video also offers a more detailed explanation of the more general idea of cardinal numbers, and is also highly recommended.

Firstly, to properly introduce greater sizes of infinity, it is extremely important to establish a crucial distinction, namely, the one between Cardinal Numbers and Ordinal Numbers.

In essence, cardinals numbers are used to denote the exact amount of objects contained within a given set, formally called the set's cardinality, for example, the cardinality of a set of four apples is 4.

On the other hand, ordinal numbers can be defined as being used to denote numbers as well-ordered sets, or in layman's terms, collections that possess a defined smallest element. Under the standard construction of ordinals, formalized by John Von Neumann, any given natural number is an ordinal, and the well-ordered set of all ordinals smaller than itself. For example:

4 = {0, 1, 2, 3}

7 = {0, 1, 2, 3, 4, 5, 6}

When approaching finite sets of objects, these two notions walk hand in hand, and there is no practical distinction between them. However, when dealing with infinite collections, these two are split, and there turns out to be a very palpable difference between denoting order in sets and denoting the quantity of objects contained in them.

After all finite ordinal numbers are exhausted, there comes the first infinite ordinal number: ω, which can be mathematically defined as equivalent to the set of all natural numbers, or more precisely:

ω = {0, 1, 2, 3, 4, 5...}

Exactly after ω, comes ω+1, which is defined as:

ω+1 = {0, 1, 2, 3, 4, 5... ω}

Then ω+2:

ω+2 = {0, 1, 2, 3, 4, 5... ω, ω+1}

And so on and so forth, until any further operations applied over ω (See these pages for more information). However, it is important to know that, while all of these ordinal numbers come after ω, and are by technicality, "bigger" than it, the sets which they represent each possess the same quantity of objects, and thus the same cardinality. Indeed, they are countable sets, and thus are all represented by the same cardinal number: Aleph-Naught, denoted as ℵ0.

After ℵ0, comes the smallest uncountable cardinal number: ℵ1, which is itself indexed by the ordinal number ω1, the set of all countable ordinal numbers. For the purposes of the Tiering System, it is accepted as an Axiom that ℵ1 is the cardinality of the set of all real numbers, and thus equal to the power set of ℵ0, and the same principle is generalized unto any higher cardinal number.

This hierarchy is then extended unto Aleph Numbers whose subscript can be defined as being correspondent to any higher number, be it finite or infinite: ℵ2, ℵ3, ℵ4... ℵω, ℵω+1, ℵω+2, and so on and so forth, with each succeeding cardinal being equal to the power set of the previous one.

Fixed Points

Mathematically speaking, a fixed point of the aleph hierarchy is essentially a given cardinal κ such that κ = ℵκ. In plain english, this essentially means that an aleph-fixed point is an infinite cardinal number whose size is so large that it is unchangeable through transformations, and equal to the amount of cardinals below it.

To further elaborate on this: It is useful to think about cardinal fixed points by thinking about the exact amount of cardinals preceding any given infinite number and associate it with their subscript: That is, there are no infinite cardinals below ℵ0, one below ℵ1, two below ℵ2... ℵ0 many infinite cardinals below ℵω, and so on, and so forth.

Now, suppose that there is an infinite cardinal whose subscript is essentially an infinite cascade of alephs, endlessly reiterating themselves downwards. As the subscript itself contains infinitely-many alephs in it, removing one of them from it would not alter the number itself in any way, thus making this cardinal into a number whose size is such that the number of infinite cardinals smaller than it is equal to the cardinal itself.

Large Cardinals

Large Cardinals are, speaking very roughly, certain cardinal numbers with the property of being very "large". That is, they are cardinals numbers whose existence is not provable from the standard axioms of most set theories, and thus have to be explicitly defined on them by the addition of another axiom postulating their existence, much like how an infinite set (ω) cannot be obtained from the standard tools and operations of arithmetic, and thus its existence needs to be added separately.

The Hierarchy of Large Cardinals is fairly extensive and contains many exotic-named numbers, but the most conventional and easily-understood entrypoint to it are the Inaccessible Cardinals: Infinite cardinal numbers that are uncountable, regular (cannot be defined as the union of quantities smaller than themselves) and strong limit (cannot be attained through repeated power set operations), which in layman's terms, means that they cannot be reached from lesser cardinals in any way, shape or form, standing beyond their scope entirely. The following is a good illustration of an Inaccessible Cardinal's relation to lesser numbers: