Ask about 1-A+.

[LuisGeno]

What would the equivalent of 1-A+ be denoted in set theory? I'm only aware of the notion of "infinite alefs" which, I also know is wrong, I've searched for information on. However; I'm still confused and don't understand much from there on, as far as set theory is concerned. If you can give me more detailed information to know about this, I would greatly appreciate it.

 

[ActuallySpaceMan42]

It only really matters when referring to the quantity of something like universes. Enough universes can be equivalent to 1-A+.

2-A is anything equivalent to Infinite Universes, or Aleph-0 Universes. It is countable infinity, since if given infinite time, you could theoretically count all of the universes.

Low 1-C is equivalent to Uncountably Infinite Universes, or Aleph-1 Universes. It could be understood as (2-A Amount of Universes)^(2-A Amount of Universes). The reason is simple, if N = Infinity, then N to the power of N is equivalent to R, the set of Real Numbers. Real Numbers include Natural Numbers (1, 2, 3), Whole Numbers (0, 1, 2, 3), Integers (0, -1, -2, 3), Rational Numbers (0.01, 0.001, 0.0001), etc.

The reason it is referred to as uncountable infinity is that even if given infinite time, you would not be able to count Uncountably Infinite Universes. Merely a single decimal, (0.1, 0.11, 0.111, 0.1111) would take infinite time, and then you would need to also count (0.12, 0.122, 0.1222) and so on and so forth. You simply couldn't count it even if you want to. As such, R or Aleph-1 is considered a larger infinity than Aleph-0/Infinite Universes.

So, if you keep power setting like this, you could do (Low 1-C Amount of Universes)^(Low 1-C Amount of Universes), to reach further into Low 1-C, and then do it again to reach 1-C, and then (1-C Amount of Universes)^(1-C Amount of Universes), all the way up to High 1-B.

High 1-B, is equivalent to an ℵW Universes, in your words "Infinite Alephs". And once there, you can go even further (ℵW Universes)^(ℵW Universes), reaching Low 1-A, and then doing the same to the Low 1-A Amount of Universes to reach 1-A. Repeating this process infinitely will eventually reach 1-A+.

Of course High 1-A is the cut-off point for this method, it is equivalent to an inaccessible cardinal, so no amount of addition, multiplying, or power setting will let you reach it.

 

[LuisGeno]

I see. Thank you so much. If you don't mind, I have one last question:
So, if for example in reality of universes, they were dimensions (obviously with qualitative superiority between them) and that the number of dimensions were אW, would those dimensions be 1-A+? Or did I misunderstand?

 

[LuisGeno]

By the way, אW is aleph-omega, right?

 

[ActuallySpaceMan42]

I see. Thank you so much. If you don't mind, I have one last question:
So, if for example in reality of universes, they were dimensions (obviously with qualitative superiority between them) and that the number of dimensions were אW, would those dimensions be 1-A+? Or did I misunderstand?

Yes, while High 1-B is equivalent to ℵW Universes, it is more commonly stated to be equivalent ℵ₀ Dimensions in our Tiering System.

As such, you can start the power setting loop from ℵ₀ Dimensions and go to ℵ₁ Dimensions ℵ₂ Dimensions, all the way up to אW Dimensions which would be 1-A+, just like Low 1-C to High 1-B.

By the way, אW is aleph-omega, right?

Yes, the official symbol is ℵω.

 

[LuisGeno]

Good! Again thank you very much for your help.

 

[ActuallySpaceMan42]

np

 

[Cat275]

My honest reaction.