Powerscalling : Cosmology of the Raonverse

Cardinality of the Raonverse

Hello everyone, Son Raon here and I'm back with another cosmological chapter for Raonverse.

The reason why I wrote this chapter is mainly due to the fact that I know that some of you - that are either mathematically based scalers or fans of mathematics in general - may have some questions as to what types of Cardinals exist.

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In order to avoid any confusion or contradictory thoughts from both you and I in regards to what I've written up to know let me define what do I mean by Cardinals :

In the science of mathematics, cardinality or cardinal number is a number that defines a set of numbers.

These sets can be finite - where the cardinal number in this is a natural number (a.k.a 0,1,2,3... etc) - or flat out infinite where in the latter case infinite cardinal numbers have been introduced.

The smallest of these infinite cardinals are often denoted with the Hebrew letter Aleph (אָ) and a number that shows their order of magnitude in the infinite Aleph Cardinals.

These cardinals go all the way from Aleph Null (אָ0) which is the smallest replaceable uncountable infinity to Aleph Omega (אָω) which is denoted as the Hebrew letter Aleph and the lowercase version of the Greek letter omega (ω) which is still an uncountable infinity, but far from the largest possible infinity.

Now let me explain which part of the Raonverse Cosmology has Aleph Cardinals in it's scaling.

This part is the spatial components of the verse's "normal" dimensions or as I used to call them subdimensions for the sake of simplicity.

These sub dimensions are infinite in size and quantity, on top of having an Aleph Cardinal Gap/Hierarchy between them.

Meaning that an infinite set of them is basically an Aleph Omega hierarchy.

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Now let's go to the next cardinal number, Worldly Cardinals :

I'll start by the definition. A Worldly Cardinal is a cardinal - let's abbreviate it as k - such as the rank Vk is a model of ZFC Set Theory.

In the verse ZFC is the quote on quote "Axiom of Inaccessibles" I use in the verse so in other words the Worldly Cardinals exist as a further result of ZFC existing in the verse.

Now as I explained before the sub dimensions are essentially a proven Aleph Omega hierarchy of dimensions, but due to the simple fact that they're essentially... for lack of a better term pieces in the puzzle that's the normal dimensions.

This in other words makes one of the infinite set of Normal Dimensions into a Worldly Cardinal Hierarchy.

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Now let's go a step higher into the "food chain" of cardinals introducing the Inaccessible Cardinals.

Now that's kinda tricky thing to explain since these cardinals have four "sub categories" which are:

- Weakly Inaccessible

- Strongly Inaccessible

- a-Inaccessible

- Hyper-Inaccessible.

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Let's start with the simpler one which is the Weakly Inaccessible Cardinals.

For starters as Inaccessible Cardinals are any uncountable cardinals that don't come to existence from procedures such as the cardinal arithmetic (retraction, subtraction, multiplication, division etc) of smaller cardinals.