A very simple upgrade for mahlo cardinals.

[HammerStrikes219]

Well to be more specific this isn't really gonna change the tier 0 stuff and etc and I have no such motive to do so, there is also no need to equalize such mathematics and more for this since this thread mainly made for the reason in cases of things like if a certain verse states that they have a mahlo cardinal amount of dimensions it wouldn't be treated as the baseline of tier 0 but rather many layers into it.
(Infinite layers actually thread is also made for the reason to stop people from blabbering about mahlo. )

For the record, I thinking plan C as Set Theory is considered a part of philosophy, mathematics, science, and so on.

At this point in times, it is probably best to wait till we get some clarification and/or revisions of the Tier System… again.

Sigh

 

[HammerStrikes219]

http://tph.tuwien.ac.at/~svozil/publ/set.htm

Also part of theoretical physics for the set theory

 

[Cat275]

For the record, I thinking plan C as Set Theory is considered a part of philosophy, mathematics, science, and so on.

Plan C? How does that relate to this?

At this point in times, it is probably best to wait till we get some clarification and/or revisions of the Tier System… again.

Sigh

Yeah.... Afaik even an aleph lambda can possibly qualify for H1-A, tiering system is kinda vague rn.

 

[HammerStrikes219]

Plan C? How does that relate to this?

Yeah.... Afaik even an aleph lambda can possibly qualify for H1-A, tiering system is kinda vague rn.

The main reason for this is if we planning a revision ahead of time, we need to consider the philosophy, science, and mathematical parts of the Tiering System.

The other reason is to wait and see how the ones who revised the Tiering System will have to address specific parts that has been ignored to say the least.

 

[Cat275]

The main reason for this is if we planning a revision ahead of time, we need to consider the philosophy, science, and mathematical parts of the Tiering System.

The other reason is to wait and see how the ones who revised the Tiering System will have to address specific parts that has been ignored to say the least.

Ah so you mean your suggesting me to wait?

 

[HammerStrikes219]

Ah so you mean your suggesting me to wait?

Yes, but I am under the assumption of them wanting to do some Tier 1s and Tier 0 revision in a theoretical sense.

 

[Cat275]

Well, I don't think waiting is necessary when this thread isn't even about changing the main standards or anything in tier 0 or the other part of the tiering system it's just about if a verse has mahlo cardinals with a certain quantity of an object then it's many layers to 0 instead of baseline 0.

 

[Cat275]

I guess I'll change the name of the thread a little.

 

[HammerStrikes219]

Well, I don't think waiting is necessary when this thread isn't even about changing the main standards or anything in tier 0 or the other part of the tiering system it's just about if a verse has mahlo cardinals with a certain quantity of an object then it's many layers to 0 instead of baseline 0.

Tier 0 is so complicated as in my honest opinion, there are multiple cardinals and not just the one you are using as in the set theory article uploaded from a university.

“

e cardinals​

One cannot prove in ZFC that there exists a regular limit cardinal κκ, for if κκ is such a cardinal, then LκLκ is a model of ZFC, and so ZFC would prove its own consistency, contradicting Gödel’s second incompleteness theorem. Thus, the existence of a regular limit cardinal must be postulated as a new axiom. Such a cardinal is called weakly inaccessible. If, in addition κκ is a strong limit, i.e., 2λ<κ2λ<κ, for every cardinal λ<κλ<κ, then κκ is calledstrongly inaccessible. A cardinal κκ is strongly inaccessible if and only if it is regular and VκVκ is a model of ZFC. If the GCH holds, then every weakly inaccessible cardinal is strongly inaccessible.

Large cardinals are uncountable cardinals satisfying some properties that make them very large, and whose existence cannot be proved in ZFC. The first weakly inaccessible cardinal is just the smallest of all large cardinals. Beyond inaccessible cardinals there is a rich and complex variety of large cardinals, which form a linear hierarchy in terms of consistency strength, and in many cases also in terms of outright implication. See the entry onindependence and large cardinals for more details.

To formulate the next stronger large-cardinal notion, let us say that a subset CC of an infinite cardinal κκ is closed if every limit of elements of CC is also in CC; and is unbounded if for every α<κα<κ there exists β∈Cβ∈C greater than αα. For example, the set of limit ordinals less than κκ is closed and unbounded. Also, a subset SS of κκ is called stationary if it intersects every closed unbounded subset of κκ. If κκ is regular and uncountable, then the set of all ordinals less than κκ of cofinality ωω is an example of a stationary set. A regular cardinal κκ is called Mahlo if the set of strongly inaccessible cardinals smaller than κκ is stationary. Thus, the first Mahlo cardinal is much larger than the first strongly inaccessible cardinal, as there are κκ-many strongly inaccessible cardinals smaller than κκ.

Much stronger large cardinal notions arise from considering strong reflection properties. Recall that the Reflection Principle (Section 4), which is provable in ZFC, asserts that every truesentence (i.e., every sentence that holds in VV) is true in some VαVα. A strengthening of this principle to second-order sentences yields some large cardinals. For example, κκ is strongly inaccessible if and only if every Σ11Σ11 sentence (i.e., existential second-order sentence in the language of set theory, with one additional predicate symbol) true in any structure of the form (Vκ,∈,A)(Vκ,∈,A), where A⊆VκA⊆Vκ, is true in some (Vα,∈,A∩Vα)(Vα,∈,A∩Vα), with α<κα<κ. The same type of reflection, but now for Π11Π11 sentences (i.e., universal second-order sentences), yields a much stronger large cardinal property of κκ, called weak compactness. Every weakly compact cardinal κκ is Mahlo, and the set of Mahlo cardinals smaller than κκ is stationary.”

 

[HammerStrikes219]

I have edited my response.

But generally, I tend to stay out of Tier 1s and Tier 0 as it is quite complicated

 

[Cat275]

Tier 0 is so complicated as in my honest opinion, there are multiple cardinals and not just the one you are using as in the set theory article uploaded from a universe.

I will admit the wiki scaling of tier 0 is pretty wonky and probably complicated for most people in terms of set theory and non set theory wise, I even heard a woodin cardinal is only immeasurable layers to 0 here when it is arguably stronger than a greatly mahlo cardinal (k²-mahlo) in terms of strength consistency.

(A way to measure how much axioms a cardinal has.)

 

[HammerStrikes219]

I will admit the wiki scaling of tier 0 is pretty wonky and probably complicated for most people in terms of set theory and non set theory wise, I even heard a woodin cardinal is only immeasurable layers to 0 here when it is arguably stronger than a greatly mahlo cardinal (k²-mahlo) in terms of strength consistency.

(A way to measure how much axioms a cardinal has.)

“
The most famous large cardinals, called measurable, were discovered by Stanisław Ulam in 1930 as a result of his solution to the Measure Problem. A (two-valued) measure, orultrafilter, on a cardinal κκ is a subset UU of (κ)P(κ)that has the following properties: (i) the intersection of any two elements of UU is in UU; (ii) if X∈UX∈U and YY is a subset of κκ such that X⊆YX⊆Y, then Y∈UY∈U; and (iii) for every X⊆κX⊆κ, either X∈UX∈U or κ−X∈Uκ−X∈U, but not both. A measure UU is called κκ-complete if every intersection of less than κκ elements of UU is also in UU. And a measure is called non-principal if there is no α<κα<κ that belongs to all elements of UU. A cardinal κκ is called measurable if there exists a measure on κκ that is κκ-complete and non-principal.”

There is also measurable which is considered a large cardinal higher than inaccessible cardinals.

Unfortunately you are correct that applying aspects of the set theory does complicated the matter in the Tiering System.

 

[DontTalkDT]

Well to be more specific this isn't really gonna change the tier 0 stuff and etc and I have no such motive to do so, there is also no need to equalize such mathematics and more for this since this thread mainly made for the reason in cases of things like if a certain verse states that they have a mahlo cardinal amount of dimensions it wouldn't be treated as the baseline of tier 0 but rather many layers into it.

Yeah. The thing is that, in practice, characters that transcend some infinite High 1-A hierarchy to some qualitative extent will be equalized to this if it's the baseline, which to me just seems a weird way to do it.
So... yeah, it's mostly me just having a problem with how we tier things at these levels.

 

[Cat275]

Unfortunately you are correct that applying aspects of the set theory does complicated the matter in the Tiering System.

The repeated use of ordinals make it sound confusing doesn't it? >:- )

 

[HammerStrikes219]

The repeated use of ordinals make it sound confusing doesn't it? >:- )

There is also the argument that we don’t have a true baseline on Tier 0.

In fact, now I think about it, when and where it was discussed, I don’t remember if we ever attempt to logically set a “baseline” in Tier 0 to say the least

 

[Cat275]

Yeah. The thing is that, in practice, characters that transcend some infinite High 1-A hierarchy to some qualitative extent will be equalized to this if it's the baseline, which to me just seems a weird way to do it.
So... yeah, it's mostly me just having a problem with how we tier things at these levels.

Personally I think that the tier 0 description should be revised if it can be achieved by many infinite hierarchies without even the verse stating it's actually a limit of this.

since cardinals below limit cardinals can use power sets and iterate the process as much as it wants whether it would be infinite to aleph omega to even more this iteration of successor and power set operation can be repeated no matter what.

(This is why aleph omega - bigger aleph omegas exist then a aleph alpha to aleph alpha + which denotes a successor operation of aleph alpha then a aleph lambda and more.)

 

[HammerStrikes219]

Personally I think that the tier 0 description should be revised if it can be achieved by many infinite hierarchies without even the verse stating it's actually a limit of this.

since cardinals below limit cardinals can use power sets and iterate the process as much as it wants whether it would be infinite to aleph omega to even more this iteration of successor and power set operation can be repeated no matter what.

(This is why aleph omega - bigger aleph omegas exist then a aleph alpha to aleph alpha + which denotes a successor operation of aleph alpha then a aleph lambda and more.)

I think in order to achieve Tier 0, you do have logically prove they are superior to beyond infinite hierarchies IIRC. Not just say they are necessarily equal to it.

 

[Cat275]

There is also the argument that we don’t have a true baseline on Tier 0.

That's reasonable enough but I'm just trying to set out the logic that a 1-inaccessible cardinal is more than enough for tier 0 by the logic that a tier 0 views H1-A the same way H1-A views 1-A.

 

[HammerStrikes219]

That's reasonable enough but I'm just trying to set out the logic that a 1-inaccessible cardinal is more than enough for tier 0 by the logic that a tier 0 views H1-A the same way H1-A views 1-A.

On the other hand, the logic does fall apart as we got this said on the Tiering System.

“
Characters who can affect objects 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-Ahierarchies.

A concrete example of such an object 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.”

 

[Cat275]

I think in order to achieve Tier 0, you do have logically prove they are superior to infinite hierarchy IIRC. Not just say they are necessarily equal to it.

Technically by the example of inaccessible cardinal it should be more than infinite hierarchies between infinite hierarchies, since even if you set out the logic that a verse has a aleph omega layered hierarchy this would not be good enough to reach H1-A by the tiering system examples which is why I got confused when I just joined the wiki since without the statement of a being or hierarchy being limits of this and that you could assume this would still be a iteration of successor cardinals if we use set theory logic.

 

[HammerStrikes219]

Technically by the example of inaccessible cardinal it should be more than infinite hierarchies between infinite hierarchies, since even if you set out the logic that a verse has a aleph omega layered hierarchy this would not be good enough to reach H1-A by the tiering system examples which is why I got confused when I just joined the wiki since without the statement of a being or hierarchy being limits of this and that you could assume this would still be a iteration of successor cardinals if we use set theory logic.

I tend to stay out of Tier 0 matters as fiction doesn’t always follow the set theory logic as far as I am aware and is generally something I considered more of case by case.

Also I worded that a bit weirdly since I mean character that transcend a Tier High 1A cosmology mostly. After all, you need to provide evidence and clarification for the cosmology of a fictional verse after all.

 

[Cat275]

I tend to stay out of Tier 0 matters as fiction doesn’t always follow the set theory logic as far as I am aware and is generally something I considered more of case by case.

I know that, but I was referring to the examples and description of the current tiering system.

Also I worded that a bit weirdly since I mean character that transcend a Tier High 1A cosmology mostly. After all, you need to provide evidence and clarification for the cosmology of a fictional verse after all.

Would still be part of the aleph numbers and wouldn't even be High 1-A if we actually go by the description of H1-A which is why I got confused last time since you would need a cardinal k (pressumably k=aleph k to create a fixed point of alephs) to reach such tier and if you need something more than that (weakly inaccessible perspectively strongly) then your gonna need a lot more work than that since you would need a regular kappa to be your smallest value (cofinality) on another ordinal kappa to access this cardinal.

(But yeah it's strange since this would likely be H1-A even though it's still from a aleph)

 

[HammerStrikes219]

Would still be part of the aleph numbers and wouldn't even be High 1-A if we actually go by the description of H1-A which is why I got confused last time since you would need a cardinal k to reach such tier and if you need something more than that (weakly inaccessible perspectively strongly) then your gonna need a lot more work than that since you would need a regular kappa to be your smallest value (cofinality) on another ordinal kappa to access this cardinal.

Yeah, but I was under the assumption that someone managed to prove it is a tier High 1A cosmology first, then we get characters that exceeds that cosmology in its entirety

 

[Cat275]

Yeah, but I was under the assumption that someone managed to prove it is a tier High 1A cosmology first, then we get characters that exceeds cosmology in its entirety

Alright then.

 

[Cat275]

Yeah. The thing is that, in practice, characters that transcend some infinite High 1-A hierarchy to some qualitative extent will be equalized to this if it's the baseline, which to me just seems a weird way to do it.
So... yeah, it's mostly me just having a problem with how we tier things at these levels.

Btw one question, let's assume that inaccessible-n denotes how much power sets are formed in a inaccessible. meaning that if let's say inaccessible-2 is made then in this hypothetical situation this would be the same as P(P(Inaccessible) and with this idea if we go and make a hierarchy of let's say inaccessible-omega^omega+omega layered dimension.... Would transcending this qualify for tier 0?

Since you pretty much qualitively transcend infinite layers to H1-A.
(Which if this does qualify for tier 0 then it is a bit strange that we would tier a inaccessible obtained by a power set the case same as something like a inaccessible limiting this power set and more.)

 

[HammerStrikes219]

Btw one question, let's assume that inaccessible-n denotes how much power sets are formed in a inaccessible. meaning that if let's say inaccessible-2 is made then in this hypothetical situation this would be the same as P(P(Inaccessible) and with this idea if we go and make a hierarchy of let's say inaccessible-omega + omega^omega layered dimension.... Would transcending this qualify for tier 0?

Since you pretty much qualitively transcend infinite layers to H1-A.
(Which if this does qualify for tier 0 then it is a bit strange that we would tier a inaccessible obtained by a power set the case same as something like a inaccessible limiting this power set and more.)

I think it doesn’t necessarily mean it is Tier 0 and probably stems from overthinking in that matter which does involves the question of applying set theory logic to fictional verses that only has no mentions of using it at all.

This can also just act as a High 1A thing.

 

[Cat275]

I think it doesn’t necessarily mean it is Tier 0 and probably stems from overthinking in that matter which does involves the question of applying set theory logic to fictional verses that only has no mentions of using it at all.

This can also just act as a High 1A

Sure but the case of something like a aleph omega^omega + omega is qualitively superior to a aleph omega, in this case it's this much power sets a inaccessible has formed which I'm really curious if it gets tier 0 due to it being qualitively beyond, not as much as limit cardinals but it's still qualitively beyond infinite layers.

 

[HammerStrikes219]

Sure but the case of something like a aleph omega^omega + omega is qualitively superior to a aleph omega, in this case it's this much power sets a inaccessible has formed which I'm really curious if it gets tier 0 due to it being qualitively beyond, not as much as limit cardinals but it's still qualitively beyond infinite layers.

Proving a cosmology is superior to its own structure is a odd one since the cosmology isn’t superior to itself, but contains uncountable inaccessible dimensions, universes, and so on.

I never see this get assumed on such a level mainly because there is no proof of it being the case. You get Tier High 1A since it does account for that it seems.

 

[Cat275]

Proving a cosmology is superior to its own structure is a odd one since the cosmology isn’t superior to itself

Wdym? Are you talking about the aleph omega^omega + omega part or the power sets of inaccessible part?

 

[HammerStrikes219]

Wdym? Are you talking about the aleph omega^omega + omega part or the power sets of inaccessible part?

Technically both.

Stacking alephs will technically give the same result since in the Wikipedia page for aleph numbers. https://en.m.wikipedia.org/wiki/Aleph_number
It is also considered a cardinal too.

“Aleph-omega is

where the smallest infinite ordinal is denoted ω. That is, the cardinal number

is the least upper bound of

is the first uncountable cardinal number that can be demonstrated within Zermelo–Fraenkel set theory not to be equal to the cardinality of the set of all real numbers; for any positive integer nwe can consistently assume that

and moreover it is possible to assume

is as large as we like. We are only forced to avoid setting it to certain special cardinals with cofinality

meaning there is an unbounded function from

to it (see Easton's theorem)l

 

[Oblivion_Of_The_Endless]

Technically both.

Stacking alephs will technically give the same result since in the Wikipedia page for aleph numbers. https://en.m.wikipedia.org/wiki/Aleph_number
It is also considered a cardinal too.

“Aleph-omega is

where the smallest infinite ordinal is denoted ω. That is, the cardinal number

is the least upper bound of

is the first uncountable cardinal number that can be demonstrated within Zermelo–Fraenkel set theory not to be equal to the cardinality of the set of all real numbers; for any positive integer nwe can consistently assume that

and moreover it is possible to assume

is as large as we like. We are only forced to avoid setting it to certain special cardinals with cofinality

meaning there is an unbounded function from

to it (see Easton's theorem)l

No, they are explicitly cardinals. Ordinals only apply to the enumeration you give to said alephs (Aleph-1, Aleph-2, Aleph-10, Aleph-35, Aleph-200), but Aleph-4 is still higher/bigger than Aleph-3, and so on.

 

[HammerStrikes219]

No, they are explicitly cardinals. Ordinals only apply to the enumeration you give to said alephs (Aleph-1, Aleph-2, Aleph-10, Aleph-35, Aleph-200), but Aleph-4 is still higher/bigger than Aleph-3, and so on.

I corrected it to being cardinals. If you haven’t noticed the edit to my comment

 

[HammerStrikes219]

Also cardinals can contain ordinals.

 

[Cat275]

Technically both.

Stacking alephs will technically give the same result since in the Wikipedia page for aleph numbers. https://en.m.wikipedia.org/wiki/Aleph_number
It is also considering a ordinal too.

“Aleph-omega is

where the smallest infinite ordinal is denoted ω. That is, the cardinal number

is the least upper bound of

is the first uncountable cardinal number that can be demonstrated within Zermelo–Fraenkel set theory not to be equal to the cardinality of the set of all real numbers; for any positive integer nwe can consistently assume that

and moreover it is possible to assume

is as large as we like. We are only forced to avoid setting it to certain special cardinals with cofinality

meaning there is an unbounded function from

to it (see Easton's theorem)l

Ah you seem to confuse cardinals beyond aleph omega and aleph-n-omega, 1st of we'll go with the idea of GCH

Which is this:

The generalized continuum hypothesis (GCH) states that if an infinite set’s cardinality lies between that of an infinite set S and that of the power set of S, then it either has the same cardinality as the set S or the same cardinality as the power set of S. That is, for any infinite cardinal there is no cardinal such that GCH is equivalent to:

Aleph alpha + 1=aleph alpha^aleph alpha,

Now to quote aleph alpha.

To translate the formalism, aleph number + 1 is the smallest ordinal whose cardinality is greater than the previous aleph. Aleph alpha is the limit of the sequence (aleph null,aleph 1,aleph 2, aleph 3,,,) until aleph alpha is reached when alpha is a limit ordinal.

This means that if we use the idea of GCH a aleph omega + 1 is a power set of aleph omega, furthermore a omega^omega as a ordinal creates a power set and is equivalent to aleph 1 if we use the idea of L.

(This also applies to inaccessible as quoted below:

Therefore 2-inaccessibility is weaker than 3-inaccessibility, which is weaker than 4-inaccessibility… all of which are weaker than ω-inaccessibility, which is weaker than ω+1-inaccessibility, which is weaker than ω+2-inaccessibility…… all of which are weaker than hyperinaccessibility, etc.)

 

[HammerStrikes219]

Ah you seem to confuse cardinals beyond aleph omega and aleph-n-omega, 1st of we'll go with the idea of GCH

You did mention alpha omega multiple times so you could see what cause this misunderstanding to say the least.

However, I not sure if using aleph omega and beyond it is a good example per se since Tier 1A and higher is more complicated the further you think about it.


There is also running into the problem of where to set cardinals into its proper tiering just from simply taking a step too far and doing a educated guess on it.

 

[Cat275]

You did mention alpha omega multiple times so you could see what cause this misunderstanding to say the least.

Ah sorry I thought it was common knowledge that a aleph omega + 1 could obtain a power set.

However, I not sure if using aleph omega and beyond it is a good example per se since Tier 1A and higher is more complicated the further you think about it.

I think it's fairly simple atleast if I look at it in a set theory perspective.
(Though maybe not the case for other people eh.)

There is also running into the problem of where to set cardinals into its proper tiering just from simply taking a step too far and doing a educated guess on it.

Maybe.

 

[HammerStrikes219]

Ah sorry I thought it was common knowledge that a aleph omega + 1 could obtain a power set.

I think it's fairly simple atleast if I look at it in a set theory perspective.
(Though maybe not the case for other people eh.)

Maybe.

While I am aware of the fact cardinals can contain power sets. (Something the articles I have read into mentions multiple times), I not sure if alpha omega and its power set being Tier 0 is logically sound.



Also let’s been honest, Tier High 1A is the same as proving its superiority over a Tier 1A structure in a cosmological sense IIRC.

 

[HammerStrikes219]

Also not everyone is that knowledgeable about the set theory in general tbf

 

[Cat275]

While I am aware of the fact cardinals can contain power sets. (Something the articles I have read into mentions multiple times), I not sure if alpha omega and its power set being Tier 0 is logically sound.

It does sound strange and cardinal wise it really isn't considered to be H1-A, though I will say that a aleph lambda should be possibly H1-A by the fact that it's impossible to obtain by aleph alpha+, no matter how much you iterate the successor operation.
(A kappa=aleph kappa is considered to be the baseline H1-A though even though aleph lambda seems to fit the bill... Though if we go by the idea that stacking many stronger infinite hierarchies to each would eventually create tier 0 then yeah Alephs can actually reach such tier if you logically look at it that way.)

 

[HammerStrikes219]

It does sound strange and cardinal wise it really isn't considered to be H1-A, though I will say that a aleph lambda should be possibly H1-A by the fact that it's impossible to obtain by aleph alpha+, no matter how much you iterate the successor operation.
(A kappa=aleph kappa is considered to be the baseline H1-A though even though aleph lambda seems to fit the bill... Though if we go by the idea that stacking many stronger infinite hierarchies to each would eventually create tier 0 then yeah Alephs can actually reach such tier if you logically look at it that way.)

https://vsbattles.com/threads/some-...t-tiering-system-standards-1-a-and-up.135076/

We also got this thread a few months back in April.