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Set theory discussion thread

Anyways as far as I know this is how you get a AU and it's classes using NBG.

Assume V includes an inaccessible cardinal κ let X ⊆ Vκ and let Def(X) denote the class of first order definable subsets of X with parameters.
{\displaystyle \operatorname {Def} (X):={\Bigl \{}\{x\mid x\in X{\text{ and }}(X,\in )\models \phi (x,y_{1},\ldots ,y_{n})\}:\phi {\text{ is a first-order formula and }}y_{1},\ldots ,y_{n}\in X{\Bigr \}}.}


{\displaystyle (X,\in )}
denotes the model within the domain of
X
and
\in
,
\models
denotes the satisfaction relations.
Click to expand...
So this implies that (Vκ,Vκ+1) is a model of Morse-kelley set theory and is also a model of NBG.
 
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Back at it with dumb math questions on big stuff

Absolute infinity
From what i understand, its either = ℵ1 or literally larger then any thing in math
It also seems pretty debunked
 
Not sure what the question is here but I usually scale absolute infinity statements aleph-0 without further elaboration but in the case of cantors absolute infinite it is the class of all ordinals or could also be thought of a set of all sets although it could mean more things over higher notions.

For a better way to see this cantors absolute infinite uses the universe of set and some property of reflection to define large cardinals and prove the consistency of normal ordinals being infinite.

Now with that in mind we also see that the principle of absolute is indeed beyond human descrption if the property of P does not fail. However absolute will be limited by human description if some of property P fails.
(This is one of the many conditions over absolute)

Side note:Ω as a mathematical universe i.e a set or class containing all sets (or certain mathematical objects that falls under this definition) will depend on what system it is universal in.
(The minimum and the standard is aleph-kappa)
 
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Cat said:
To put this in more simple words, absolute infinite is the unification of all infinities (class of all ordinals that are possible) and is beyond human description. (Atleast as far as i know)
Click to expand...
I think this is clearly a good conclusion.
 
Gonna add something

Not related to Set Theory but can u give a clear explanation on layers of outerverse? How do you get 1 layer into it?
 
Savior said:
Gonna add something

Not related to Set Theory but can u give a clear explanation on layers of outerverse? How do you get 1 layer into it?
Click to expand...
Common ways to get 1 layer is r>f over baseline.

Or are you asking for something more detailed?
 
I meant with an example, lets say i have a structure beyond dimensional scaling and infinite hierarchy or for wtv reason its Outerversal

If some being / another structure r>f transcends it , is it 1 layer into Outer
 
Savior said:
I meant with an example, lets say i have a structure beyond dimensional scaling and infinite hierarchy or for wtv reason its Outerversal

If some being / another structure r>f transcends it , is it 1 layer into Outer
Click to expand...
Yes it would indeed be 1 layer into outer.
 
Cat said:
Yes it would indeed be 1 layer into outer.
Click to expand...
Alright Thanks

Also another doubt i have , What is 'Tier 0 Hax' that people mention?

Also can a lower dimensional being have a higher dimensional hax similar to how 3D characters have 4/5D hax? Well 4,5D often have relation to real world physics as in Time and Space but i dont get how thag would work with high dimensional hax as those are just transcendant layers...
 
Savior said:
Alright Thanks

Also another doubt i have , What is 'Tier 0 Hax' that people mention?
Click to expand...
Probably a hax that can affect tier 0 characters or structures.
Savior said:
Also can a lower dimensional being have a higher dimensional hax similar to how 3D characters have 4/5D hax? Well 4,5D often have relation to real world physics as in Time and Space but i dont get how thag would work with high dimensional hax as those are just transcendant layers...
Click to expand...
I believe this is quite possible in fiction, this is what we call a smurf hax i believe.
(Yogiri is an example)
 
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Also dimensions are often known as realms or layers but really they ain't scientifically or mathematically that. their concept is actually more like directions.
(Width, height, length etc)
 
Would it be possible to have a Proper Class of all large cardinals properties?

And is it possible to have a Proper Class of all possible large cardinals? (That is, containing the ones that have no classification yet/have yet to exist as well). And if so, would that be the equivalent to literally all possible cardinalities? (Similar to the class of all ordinals, but with cardinals instead)
 
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Oblivion_Of_The_Endless said:
Would it be possible to have a Proper Class of all large cardinals properties?
Click to expand...
It's possible yeah.
Oblivion_Of_The_Endless said:
And is it possible to have a Proper Class of all possible large cardinals? (That is, containing the ones that have no classification yet/have yet to exist as well). And if so, would that be the equivalent to literally all possible cardinalities? (Similar to the class of all ordinals, but with cardinals instead)
Click to expand...
Class of all ordinals have the same cardinality as the class of all cardinals and yeah as long as the said cardinal is assumed to exist under the proposed system then the non classified ones falls under the said class.

A complete V or the universe of sets is a more general concept of this.
 
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Cat said:
Class of all ordinals have the same cardinality as the class of all cardinals.
Click to expand...
How can ordinals relate to cardinalities in this case? Since they are, you know, ordinals, whose cardinality shouldn't increase by adding more of them.
 
Oblivion_Of_The_Endless said:
How can ordinals relate to cardinalities in this case? Since they are, you know, ordinals, whose cardinality shouldn't increase by adding more of them.
Click to expand...
We quite use ordinals to define cardinals though? Every cardinal has an order or a preceding set to create a bigger function also the axiom schema of replacement literally helps making non-bijective sets.

Even berkeley over reinhardts use ordinal to prove it's mapping.
 
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But epsilon-null, a set of ordinal numbers that has a stupid amount of powers of omega, still has the simple cardinality of aleph-null. What is the difference?
 
I think epsilon is just a limit of aleph null according to carter's set theory.
 
I'm working on my reply, but lewis you're talking about the exponential map of epsilon null and not the other epsilon null.
(There are 2 versions as far as im aware.)
 
that satisfy the fixed point ε = ω^ω..The first epsilon number, ε0, is the limit ordinal of the set {ω, ωω, ωωω, ...}.
 
An epsilon that uses the formula: ε = ω^ω^... is not a limit ordinal.
It's more like countless new ordered pairs of ω.

But a epsilon that satisfies the formula: ε=ω^α. is indeed a fixed point of the exponential map, this version of epsilon null is a limit ordinal of the sequence of ω's if i remember correctly.
 
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Oblivion_Of_The_Endless said:
But epsilon-null, a set of ordinal numbers that has a stupid amount of powers of omega, still has the simple cardinality of aleph-null. What is the difference?
Click to expand...
The fact that omega still increases by addition?

I mean we use the axiom of replacement to prove power sets so how about we look at one that proves ω1.

So replacing each ordinals on the well ordered set of omega will result to ω1
(ω1=
\aleph_1
) so this itselfs prove that ordinals can increase by addition albeit with axioms.

Now lets go above the ordinals we know and go to a realm where bijections becomes rare.

So with this it's quite adequate that ordinals can indeed go high in rankings and is additional proof that it can increase by addition as you can use the axiom of replacement to prove higher rankings above Vω+ω and below.
(axiom of replacement mostly rely and uses ordinals.)

We also have Vβ+1 which is a great example that larger ordinal models can't be bijective even by additions since Vβ+1 is a power set of Vβ.

This also shows that some ordinals can imply a reinhardt cardinal.

We should also consider the fact that a δ that satisfies a berkeley cardinal is still a ordinal none the less as it still uses the ordinal δ to represent/denote itself.

Ordinals and order theory (with transfinite induction) in general proves that there are initial segments and induction marks which then proves there are bijections, other existing sets, bigger sets that precede smaller ones and etc.

Conclusion:
Overall while some cardinals are denoted by ordinals and while the 1st order ordinals uses different mapping methods compared to cardinals, each cardinals still has an ordered pair and in fact (from what I know) each cardinal is a ordinal but not vice versa.

Also to simply put most (if not all) of the representing symbols we see in large cardinals are ordinals.
 
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So, if the class of all ordinals has the same size as the class of all cardinals, shouldn't the Absolute Infinite be inherently Tier 0 since (last time I checked) it can be considered as a class of all ordinals?
 
Absolute infinite has many theorems, we 1st need to know what theorems are there and what kind of possibles the informal statement implies as well as what exist.

Cantors view being an inherent tier 0 is a possible since his view uses the universe of sets and reflection principle to make absolute infinite apophatic but if it's just a mathematicial universe then it really just depends on the assumed existing and consistent sets.
 
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Ok, now for the next thing.

What is Consistency Strength in simple terms? Because I've been told one time that once you reach large cardinal numbers, size becomes arbitrary and essentially inapplicable from the point of view of dimensions since you need a whole axiom to make sense out of them in relation to the aleph hierarchy (which does have degrees of infinite size and dimensions).

If the large cardinal hierarchy isn't related to size, how do we quantify them in relation to the layers of Tier 0? Because the downstreamers were stupidly high into 0 when they still scaled to Woodin cardinals. I guess self-reference engine can be another example.
 
consistency strength in simple terms is how you prove things, it gets really weird since if Y is bigger than X but X implies Y then X>Y.
 
For the relation of tier 0 it can quite depend really Woodin cardinals implies some classes that dont exist in mahlo and this becomes a subset of another and etc while also implying arbitary bigger sets.

We also have stronger strength consistency via limit cardinals and fixed points so that's also one of the ways you classify them.

0=1 also can imply bassically everything as it's the limit of consistencies for large cardinals.

Like V->V where this can map almost all large cardinals, 0=1 can map and change consistencies of large cardinals. I believe you are already aware of this yes?

The best way to see how something is bigger is by looking how much something implies by the proposed axiom.

But yeah some weird cases can happen where you have stronger strength consistency but smaller size.
 
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What the Self reference engine doesn't have is the use of class hierarchies. and there is no contradiction proof 0=1, but overlooked because it is fiction, using as" the extreme limit" to describe it.
 
Would the proper class of all sets be the same of the proper class of all cardinals/ordinals? And would the same apply to the Universe of Sets?
 
Lewis said:
What the Self reference engine doesn't have is the use of class hierarchies. and there is no contradiction proof 0=1, but overlooked because it is fiction, using as" the extreme limit" to describe it.
Click to expand...
I mean I did see no implication of Mk or NBG but yeah that's pretty much the reason for 0=1 (extreme limit or top of large cardinals) in self reference engine if I remember correctly.
 
To reach Low 1-C there would need to be Infinite Universes, with each spawning Infinite Universes, and they have to do the same and this has to have happened for eternity. (Infinity^Infinity = Infinity * Infinity * Infinity --> Infinitely)
Click to expand...
This schematic is a bit funny to me, so assuming the continuum hypothesis is true.

Is this accurate?
 
Cat said:
Probably a hax that can affect tier 0 characters or structures.

I believe this is quite possible in fiction, this is what we call a smurf hax i believe.
(Yogiri is an example)
Click to expand...
I see ..

Also can you name some characters scaling between Low complex Multiversal to Hyperversal? As in more than 4D but still Finite layers into the hierarchy
 
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