Set theory discussion thread

[Cat275]

Also why is Self Reference Engine the top of Tier 0 now? What level of math does it have?

0=1 is the math accepted in the wiki.

Their supposingly above it.

 

[Savior]

0=1 is the math accepted in the wiki.

Their supposingly above it.

0=1? Wtf?

First: isnt that the top of the cardinal hierarchy?

Second; isnt it contradictory?

 

[Cat275]

0=1? Wtf?

First: isnt that the top of the cardinal hierarchy?

Yeah. Should be equal to cantors absolute infinite.

Second; isnt it contradictory?

Not sure how being above 0=1 is contradictory in fiction, should be ok as long as it isn't a set that is above it.

 

[Savior]

Yeah. Should be equal to cantors absolute infinite.

Pretty sure not , Cantor's A.I is supposed to be beyond all Cardinals

Not sure how being above 0=1 is contradictory in fiction, should be ok as long as it isn't a set that is above it.

Well mathematicians will likely disagree but this is fiction, so yeah.

Although the justification is about a statement of the extreme limit which I don't think it mentions classes.
(Highly needed to assert a logical endpoint, atleast mathematically.)

So a layer which transcend another layer which functions on 0=1 is ok?

 

[Cat275]

Pretty sure not , Cantor's A.I is supposed to be beyond all Cardinals

It depends on your assumption.

Set of all ordinals (non-well founded) --> Set of all sets --> Class of all ordinals --> Class of all sets --> Beyond human description --> True omnipotence.

But generally the set of all ordinals or the class of all ordinals is the general notion here.

Cantor also uses the Universe of sets, NBG and the reflection principle to motivate the notion of absolute infinite even further.

So normally 0=1 = Ω.

(0=1 is one of the many sets we can use to denote absolute infinite btw)

Although we see that Ω>0=1 is true if we use higher notions of Ω.
(It's bassically adding more conditions for the reflection principle and assuming even more things for Ω.
If i remember that correctly.)

Vice versa can also happen though.
(In fact it happens alot if we only view Ω as a universal object over certain axiomatic systems)

So a layer which transcend another layer which functions on 0=1 is ok?

Depends on what you mean by layer.

But generally if you mean P(0=1) then we see 0=1=P(0=1).

 

[Savior]

It depends on your assumption.

Set of all ordinals (non-well founded) --> Set of all sets --> Class of all ordinals --> Class of all sets --> Beyond human description --> True omnipotence.

But generally the set of all ordinals or the class of all ordinals is the general notion here.

Cantor also uses the Universe of sets, NBG and the reflection principle to motivate the notion of absolute infinite even further.

So normally 0=1 = Ω.

(0=1 is one of the many sets we can use to denote absolute infinite btw)

Although we see that Ω>0=1 is true if we use higher notions of Ω.
(It's bassically adding more conditions for the reflection principle and assuming even more things for Ω.
If i remember that correctly.)

Yeah makes sense , the higher you scale Absolute Infinity, the more contradiction it creates

And by definition alot of sets can be scaled to it

Depends on what you mean by layer.

But generally if you mean P(0=1) then we see 0=1=P(0=1).

An Infinite Hierarchy of Transcending Layers (H1B Hierarchy) , if one layer functions on 0=1 , how would the next layer scale?

Also , are there any math or any other concept which could scale beyond 0=1 ?

 

[Cat275]

An Infinite Hierarchy of Transcending Layers (H1B Hierarchy) , if one layer functions on 0=1 , how would the next layer scale?

Normally by mathematics it's P(V) (if V is a set and not a class) but the problem P(V)=V arises but since this isn't really a set formation of any sort and is only on a hypothetical fictional setting then it'll probably be superior on a informal manner. (Not precisely defined)

Which just means that we can't really mathematically and precisely tell what it means by above. it could be superior by having new proposition of truths (truth value) that 0=1 does not hold (unlikely) but really it could be something else instead. we can't really precisely tell since it would most likely be defined informally.

We can only tell that it's superior in some way but not really mathematically superior (mostly refering to strength consistency and cardinality of set here) by our normal mathematical standards.
(At the moment there can be no cardinal bigger/stronger than 0=1 without falling under a lot of inconsistencies)

Although if the verse said it is mathematically stronger then we'll have to consider it bigger/consistently stronger.

Also , are there any math or any other concept which could scale beyond 0=1 ?

Not that I know of.

As far as I know Large cardinals beyond choice that is defined<HOD<Von neumann universe (This part is arbitary and it depends) = 0=1 = Universe of sets = Ultimate L conjecture etc.

On a short note 0=1 and V is one of the highest mathematical concept you can get and I doubt you can imply nor create more than V and 0=1.

(For Ω>V as a notion is fine if Ω is not a unique property of P and P does not fail.

with this and a couple more concepts, assumptions and axioms we use (such as the reflection principles, NBG, Universe of sets, Informal statements, there exist inaccessible etc) we assume that Ω holds all sets and there is a ordinary set that is infinite. (by reflection) Now with this mathematical conclusion we see that Ω is also beyond what it holds since:

1. This notion or principle of absolute here is being something way to great to be described by mathematical concepts and describing it using one will always be a limited description.
2. Apophism.

So overall Ω>V is a notion where Ω holds V but has a principle that Ω is either limited by human description or could be beyond it.
So if you consider this notion of Absolute infinite to still be a mathematical concept despite being beyond mathematical concepts and human descriptions, then you can say that Ω>V = 0=1 etc is true.

So Ω would be like an infinity inaccessible by all if Ω is beyond description.
I personally don't consider it to be a mathematical concept anymore once we reach this realm of absolute.

Do note that this notion of absolute here would still have the same cardinality as V (i think) so it's still mathematically equal since both still have the same sets and elements but just not generally equal overall.

Another notion I know that Ω>V is true is Ω being the same notion I applied or explained above but also adding the idea that Ω is linked by god and therefore omnipotent.)

But yeah as far as im aware there isn't really any official mathematical concept out there that is size and strength consistency wise greater than something like 0=1.

Athough I have no comment on other concepts above 0=1 that don't specifically involve math.

 

[Savior]

Normally by mathematics it's P(V) (if V is a set and not a class) but the problem P(V)=V arises but since this isn't really a set formation of any sort and is only on a hypothetical fictional setting then it'll probably be superior on a informal manner. (Not precisely defined)

Which just means that we can't really mathematically and precisely tell what it means by above. it could be superior by having new proposition of truths (truth value) that 0=1 does not hold (unlikely) but really it could be something else instead. we can't really precisely tell since it would most likely be defined informally.

We can only tell that it's superior in some way but not really mathematically superior (mostly refering to strength consistency and cardinality of set here) by our normal mathematical standards.
(At the moment there can be no cardinal bigger/stronger than 0=1)

Not that I know of.

As far as I know Large cardinals beyond choice that is defined<HOD<Von neumann universe(V) = 0=1 = Universe of sets = Ultimate L conjecture = V=L etc.

Although im not quite sure on the Ultimate L part but on a short note 0=1 and V is one of the highest mathematical concept you can get and I doubt you can imply nor create more than V and 0=1.

(For Ω>V as a notion is fine if Ω is not a unique property of P and P does not fail.

with this and a couple more concepts and axioms we use (such as the reflection principles, NBG, Universe of sets, Informal statements etc) we assume that Ω holds all sets and there is a ordinary set that is infinite. (by reflection) Now with this mathematical conclusion we see that Ω is also beyond what it holds since:

1. This notion or principle of absolute here is being something way to great to be described by mathematical concepts and describing it using one will always be a limited description.
2. Apophism.

So overall Ω>V is a notion where Ω holds V but has a principle that Ω is either limited by human description or could be beyond it.
So if you consider this notion of Absolute infinite to still be a mathematical concept despite being beyond mathematical concepts and human descriptions, then you can say that Ω>V=L = V = 0=1 etc is true.

So Ω would be like an infinity inaccessible by all if Ω is beyond description.
I personally don't consider it to be a mathematical concept anymore once we reach this realm of absolute.

Do note that this notion of absolute here would still have the same cardinality as V though so it's still mathematically equal since both still have the same sets and elements.

Another notion I know that Ω>V is true is Ω being the same notion I applied or explained above but also adding the idea that Ω is linked by god and therefore omnipotent.)

But yeah as far as im aware there isn't really any official mathematical concept out there that is size and strength consistency wise greater than something like 0=1.

Athough I have no comment on other concepts above 0=1 that don't specifically involve math.

Thanks alot, and i knew Absolute Inf was vague but never knew it could be to such a degree

Also i heard from one of the most famous and best powerscalers on reddit that Twin Peaks has Universe of Sets..

Its rejected by alot and accepted by alot


Also , can you recomend any books or article on the internet to learn about Ultimate L Conjecture , Universe of Sets , V=L and other Large Cardinals beyond choice?

Also how do they fair up with Berkeley and Reinhardt Cardinals?

 

[Cat275]

Thanks alot, and i knew Absolute Inf was vague but never knew it could be to such a degree

Also i heard from one of the most famous and best powerscalers on reddit that Twin Peaks has Universe of Sets..

Its rejected by alot and accepted by alot

I'm not really knowledgeable on Twin Peaks.

Also , can you recomend any books or article on the internet to learn about Ultimate L Conjecture , Universe of Sets , V=L and other Large Cardinals beyond choice?

I'll get a link for that later.

Also how do they fair up with Berkeley and Reinhardt Cardinals?

Berkeley and reinhardt cardinals are one of the many large cardinals beyond choice.

 

[Cat275]

Had a bit of problem there so I deleted my previous comment but here is the link for large cardinals beyond choice and the Ultimate L conjecture.

Wikipedia already provides external links and some books online for the universe of sets though but I'll try getting a pdf if possible once i finish school.

 

[Cat275]

Found this funny verse.

What do you think savior?

 

[NHTkenshin2]

Strongest tier 0?

 

[Savior]

Had a bit of problem there so I deleted my previous comment but here is the link for large cardinals beyond choice and the Ultimate L conjecture.

Wikipedia already provides external links and some books online for the universe of sets though but I'll try getting a pdf if possible once i finish school.

Thanks

 

[Savior]

Found this funny verse.

What do you think savior?

Is this some kind of fanfiction?

Auren's true form is The Absolute, a being so large whose sheer size reaches Absolute Infinity, completely inaccessible by any means or possibilities, whereas no levels of infinities can reach it, be them alephs or large cardinals. It is beyond any definition or rationalization, all things are aspects of itself and is one with them all. Transcends the Bleeding Edge and can significantly affect the Library

Additionally, it is above things like V=Ultimate L and Extended Modal Realism

Is this even valid? Wth

 

[Savior]

Found this funny verse.

What do you think savior?

This verse also states that Absolute Infinity to be beyond all Alephs and even Large Cardinals something which most believed till last year ,

Seems like a verse made for powerscaling

Like i dont get what Absolute Infinity does..Like if a verse states its beyond all Large Cardinals , does it become true for the verse?

While lets say if another verse far transcends just Absolute Infinity but the Sub-Math isnt as big as the other , which verse scales higher?

Are the two Absolute Infinity on different scales?

 

[Cat275]

Strongest tier 0?

Based on the wiki? Not really no.
Ultimate L conjecture is somewhere on the level of 0=1 while extended modal realism is there to add the notion of impossible things above ultimate L and I'm pretty sure self reference engine already has a chain of apophatic stuff as well as impossible and possible things.

Is this some kind of fanfiction?

No.

Is this even valid? Wth

I mean it seems to use the principle of absolute which I already explained a few message ago.

This verse also states that Absolute Infinity to be beyond all Alephs and even Large Cardinals something which most believed till last year ,

Seems like a verse made for powerscaling

Like i dont get what Absolute Infinity does..Like if a verse states its beyond all Large Cardinals , does it become true for the verse?

Yes albeit since it's a informal statement it could be only all large cardinals on a axiom and even if it's all large cardinal on one axiom we can't really say it's actually all you see.

Gödels incompleteness theorem 1 and 2 exist and with this we get that all large cardinal axiom and any system are incomplete by itself so assuming it's all large cardinal as a informal statement then there would be no axiom to satisfy nor define what it means by all and what property of all.

But bassically the main point im pointing out here is that axiom + assume this exist is different from just axiom + no assumption, by assuming more things (or by adding axioms) we see that each system becomes more complete and is different from a less complete system.
(Zfc ≠ Zfc + there exist a inaccessible cardinal as a example.)

Extra note that Worldly can't really be proven to exist in zfc in general. (without using things like reflection principle)

So absolute infinite being mathematically bigger should be fine as long as it isn't a complete framework of let's say V.

While lets say if another verse far transcends just Absolute Infinity but the Sub-Math isnt as big as the other , which verse scales higher?

Are the two Absolute Infinity on different scales?

Absolute infinite can have different scales yes.

1. It depends on the interpretation.
2. It depends on what we assume to exist.
3. It depends on what axiom we are specifically using.
4. It depends on how far the reflection hierarchy is.
And lastly 5. it depends on how complete the axiom we are using and the set hierarchy we have.

There are probably more reasons out there but this should be pretty much almost if not all most of the prominent reasons I can think of.

And by definition the class of all ordinals in NBG is different to the class of all ordinals in NBG + kappa is inaccessible.

This all plays as a key factor on what absolute infinite is bigger and smaller and how big a absolute infinite is.

Also i already told you some levels of absolute infinity.

Set of all ordinals → Set of all sets → Class of all ordinals → Class of all sets → Class of all sets + Apophatic → Class of all sets + Apophatic and omnipotence.

 

[Cat275]

Anyways where do you scale it without WOG and with WOG?

 

[NHTkenshin2]

I thought SRE is only 0=1 level?

 

[Cat275]

I thought SRE is only 0=1 level?

No? It's considered to be beyond on the wiki.

 

[Savior]

Anyways where do you scale it without WOG and with WOG?

Scale what? And whats was WOG again? (Lol i am so bad at this)

 

[Savior]

Yes albeit since it's a informal statement it could be only all large cardinals on a axiom and even if it's all large cardinal on one axiom we can't really say it's actually all you see.

Gödels incompleteness theorem 1 and 2 exist and with this we get that all large cardinal axiom and any system are incomplete by itself so assuming it's all large cardinal as a informal statement then there would be no axiom to satisfy nor define what it means by all and what property of all.

But bassically the main point im pointing out here is that axiom + assume this exist is different from just axiom + no assumption, by assuming more things (or by adding axioms) we see that each system becomes more complete and is different from a less complete system.
(Zfc ≠ Zfc + there exist a inaccessible cardinal as a example.)

Extra note that Worldly can't really be proven to exist in zfc in general. (without using things like reflection principle)

So i am guessing we cant give this verse a solid tier due to many hypotheticals?

So absolute infinite being mathematically bigger should be fine as long as it isn't a complete framework of let's say V.

Absolute infinite can have different scales yes.

1. It depends on the interpretation.
2. It depends on what we assume to exist.
3. It depends on what axiom we are specifically using.
4. It depends on how far the reflection hierarchy is.
And lastly 5. it depends on how complete the axiom we are using and the set hierarchy we have.

There are probably more reasons out there but this should be pretty much almost if not all most of the prominent reasons I can think of.

And by definition the class of all ordinals in NBG is different to the class of all ordinals in NBG + kappa is inaccessible.

This all plays as a key factor on what absolute infinite is bigger and smaller and how big a absolute infinite is.

Also i already told you some levels of absolute infinity.

Set of all ordinals → Set of all sets → Class of all ordinals → Class of all sets → Class of all sets + Apophatic → Class of all sets + Apophatic and omnipotence.

Oh yeah, you had already explained this , my bad lol

 

[Cat275]

Scale what? And whats was WOG again? (Lol i am so bad at this)

Word of god.

So i am guessing we cant give this verse a solid tier due to many hypotheticals?

We can. Standard axiom we have is zfc (1st order logic) the verse states auren is a absolute infinite unreachable by all large cardinals.
(So zfc or zfc + there exist worldly cardinals)

And so we get H1-A by standard axioms we use.
(My opinion, although I would like to know your opinion and where you scale it without WOG)

Well we have that WOG that states it's beyond V=Ultimate L so we should be able to scale it crazy high into tier 0.

 

[Oblivion_Of_The_Endless]

Would V=Ultimate L encompass only the large cardinal axioms that are currently defined, or would it also apply to the nonexistent ones?

 

[Cat275]

A complete framework of V as a universe of set does include a lot of large cardinals however it's not good at viewing the framework of it.

We mainly use L to classify the stages of each cardinals and to prove the gch and ac's consistency however it limits the nature of large cardinals and is to small to even have cardinals equal to 0#.
To solve this we use the Ultimate L and so we get V=Ultimate L which proves the stages of each large cardinals and is used so that we don't need to limit large cardinals using L.

A better way to see this is that V=Ultimate L is the same as the universe of sets or V but you can take a better look on your framework.

So in short yes it does include the non existent ones.
(Also i seem to have mistakenly said that V=L = universe of sets a few messages ago which isn't true V=L is limited through 0#.)

 

[Oblivion_Of_The_Endless]

Also, why are Reinhardt/Berkeley cardinals so ******? From what I've seen, the smallest element in a Reinhardt is already V, and then you have club sets of that Reinhardt cardinal, and then a limit of that, and so on...Is this even tierable? It seems to throw logic out of the roof. V just seems to stop making sense completely. Is there a point to this?

 

[Cat275]

I got my previous message deleted (accidentally) so I'll be reposting (kinda) but anyways I do admit it can be quite confusing so I'll keep this simple and quote the prominent ones only.

A first refinement was discussed in later work by Trnková—Blass, showing that if the preservation properties of Lawvere's transformation are strengthened to the point of requiring it to be an exact functor, such a transformation is provably equivalent to the existence of a measurable cardinal. We propose to push the preservation properties as far as possible, short of inconsistency. The resulting transformation V→V is strong enough to account for virtually all large cardinals, but is at the same time a natural generalization of an assertion about transformations V→V known to be equivalent to the Axiom of Infinity.

Now lets quote this

So essentially the proposition here is that V→V is informal.

We see that j:V→V creates a reinhardt cardinal but V→V is bound by (Vy, Vy+1) or atleast implies it's existence which falls under V so it should be fine to assume V→V is incomplete.

Again j:V→V is informal so let's see a iteration of (V, J) to investigate V→V.

We'll quote this 1st.

1960s by Lawvere that the existence of an infinite set is equivalent to the existence of a certain kind of structure-preserving transformation from V to itself, not isomorphic to the identity

Let's note this as well before we further move on.

In mathematics, the word “preserve” usually means the “preservation of properties”. Loosely speaking, whenever a mathematical construct A has some property P , after A is somehow “transformed” into A′ , the transformed object A′ also has property P.

Now quoting:

We investigate the linear iterates of Nα, Jα of (V, J) and their relationship to (V, J) forcing and definability, including that for each infinite α, every set is set-generic over Nα, but Nα is not a set-ground.

So the overall conclusion is that the preservation properties of Lawvere's transformation at it's finest will result to V→V.

Now let's quote this.

We suggest a new approach for addressing the problem of establishing an axiomatic foundation for large cardinals. An axiom asserting the existence of a large cardinal can naturally be viewed as a strong Axiom of Infinity. However, it has not been clear on the basis of our knowledge of ω itself, or of generally agreed upon intuitions about the true nature of the mathematical universe, what the right strengthening of the Axiom of Infinity is—which large cardinals ought to be derivable? It was shown in the 1960s by Lawvere that the existence of an infinite set is equivalent to the existence of a certain kind of structure-preserving transformation from V to itself, not isomorphic to the identity. We use Lawvere's transformation, rather than ω, as a starting point for a reasonably natural sequence of strengthenings and refinements, leading to a proposed strong Axiom of Infinity.

So the motivation behind V→V is to establish a foundation for large cardinals although not as big as the complete framework of V.

smallest element in a Reinhardt is already V

Now im not exactly sure why you implied that but V→V seems to be incomplete compared to V so V→V is far from V.

A weakly HOD that is close to V wipes out V→V so V→V is far from even a weakly HOD.

If you were to ask me what the smallest element I have in mind for this, then it's probably the critical point of j:V→V which is κ.

 

[Savior]

Word of god.

We can. Standard axiom we have is zfc (1st order logic) the verse states auren is a absolute infinite unreachable by all large cardinals.
(So zfc or zfc + there exist worldly cardinals)

And so we get H1-A by standard axioms we use.
(My opinion, although I would like to know your opinion and where you scale it without WOG)

Well we have that WOG that states it's beyond V=Ultimate L so we should be able to scale it crazy high into tier 0.

With WOG it is definetely too much to the point its indescribably high into 0


Without WOG , i would first like to clear a few doubts

1: i am assuming that by statements like 'unreachable by all Large Cardinals' , we'll just take the highest cardinal mentioned

So like , what is the highest one mentioned? (Worldly?)


If so, High Into H1A imo

 

[Cat275]

There's no highest large cardinal mentioned but there's still a mention of the word large cardinals so assuming the hierarchy of worldly exist should be fine.

 

[Oblivion_Of_The_Endless]

So the V from Berkeley and Reinhardt are not the complete V?

Then, would the complete V be the peak of all kinds of cardinality?

 

[Cat275]

So the V from Berkeley and Reinhardt are not the complete V?

Yes.

Then, would the complete V be the peak of all kinds of cardinality?

Yes.

 

[Andika_CL_atmadja]

can someone give their personal opinion in this thread ? https://vsbattles.com/threads/from-...hematical-set-theory-expression-of-it.141873/

thanks

 

[Cat275]

can someone give their personal opinion in this thread ? https://vsbattles.com/threads/from-...hematical-set-theory-expression-of-it.141873/

thanks

Not exactly sure what you mean by a mathematical expression of Low 1-C since tiers are not really math but im assuming you mean aleph 1? If so then it should be something like C[(a+bi)] or a+bi on a mathematical expression.

Mathematical expressions don't really express infinities often though but I guess you can do x here is the real numbers.

So x>n. or are you asking something different from what I have in mind?

 

[Andika_CL_atmadja]

Not exactly sure what you mean by a mathematical expression of Low 1-C since tiers are not really math but im assuming you mean aleph 1? If so then it should be something like C[(a+bi)] or a+bi on a mathematical expression.

Mathematical expressions don't really express infinities often though but I guess you can do x here is the real numbers.

So x>n. or are you asking something different from what I have in mind?

can you explain for each variable ? this is very similar like elementary algebra.

what is n ?

 

[Cat275]

can you explain for each variable ? this is very similar like elementary algebra.

Sure.

C denotes all a+bi.

a+b satisfies the real numbers and i satisfies the equation a+b.

 

[Cat275]

what is n ?

When I said n I was either talking about all natural numbers or numbers that are on the natural numbers i.e classified as a natural number.

 

[Andika_CL_atmadja]

When I said n I was either talking about all natural numbers or numbers that are on the natural numbers i.e classified as a natural number.

hmmm but does ">" enough to give dimensional superiority or uncountable infinity context to X relative to N ?

 

[Andika_CL_atmadja]

how about N ⊂ R or R ⊃ N ?

 

[Cat275]

how about N ⊂ R or R ⊃ N ?

Should be doable if it's a proper subset or superset.

hmmm but does ">" enough to give dimensional superiority or uncountable infinity context to X relative to N ?

X represents the set of real numbers here so X>N or R>N just means that aleph-1>aleph-0 and yeah I don't think there is a subset of aleph 1 that can be bigger than aleph-0 since all would be aleph-0. (unless if you include improper subsets)

 

[Andika_CL_atmadja]

Should be doable if it's a proper subset or superset.

X represents the set of real numbers here so X>N or R>N just means that aleph-1>aleph-0 and yeah I don't think there is a subset of aleph 1 that can be bigger than aleph-0 since all would be aleph-0. (unless if you include improper subsets)

btw cat do we assume N and R always represent low-2c timeline in here ?or we need additional context like existence quantifier ?

little bit confused since math is independent, math can be represented anything right ?

We can say X = 1 timeline
but we also can say X = infinite multiverse.
would that means X > X thus logical contradiction exist?

 

[Cat275]

btw cat do we assume N and R always represent low-2c timeline in here ?or we need additional context like existence quantifier ?

Need more context yeah.

little bit confused since math is independent, math can be represented anything right ?

I wouldn't really say anything.

We can say X = 1 timeline
but we also can say X = infinite multiverse.
would that means X > X thus logical contradiction exist?

I guess? People usually would just do Y=1 timeline and X=Infinite timelines in this case rather than both X (atleast I think so).

Oh yeah we do have real coordinate space so R³ represents 3-dimensional measures and R⁴ which represents 4-dimensional measures which seems to be more general, although temporal dimensions are not really presented on on real coordinate space since it focuses on spatial dimensions.