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A very simple upgrade for mahlo cardinals.

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Anyways I'm still waiting for more input or a refutation if 1-inaccessible quantity of an object qualifies for being the baseline tier 0 as a cardinal.
(Or maybe not baseline but being tier 0 all together, I just want to upgrade mahlo man.)
 
Cat said:
Are you talking about mahlo cardinals being inaccessible?
(Sorry I don't quite understand)
Click to expand...
Yes, and according to the Tiering System, a inaccessible cardinal (Mahlo cardinals) is being treated as being part of Tier High 1A, and not necessarily Tier 0 by default.

You will need more solid evidence of Tier 0 especially since how strict the standards are in specific cases such as this.
 
Oblivion_Of_The_Endless said:
No it isnt
Click to expand...

“High 1-A: High Outerverse level​

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.”

This is what being explicitly said on the Tiering System page itself right now.
 
HammerStrikes219 said:

“High 1-A: High Outerverse level​

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.”

This is what being explicitly said on the Tiering System page itself m.
Click to expand...
Mahlo are hyper-inaccessibles, not inaccessibles.
 
HammerStrikes219 said:
Yes, and according to the Tiering System, a inaccessible cardinal (Mahlo cardinals) is being treated as being part of Tier High 1A, and not necessarily Tier 0 by default.

You will need more solid evidence of Tier 0 especially since how strict the standards are in specific cases such as this.
Click to expand...
Ah no, the tiering system standards on High 1-A are the kappa = aleph kappa, mahlo is an inaccessible cardinal but it's also a limit of inaccessible cardinals, if you didn't read what I presented above a tier 0 cardinal becomes tier 0 when it limits and is a fixed point of kappa.

There are many ways to denote mahlo an example of this is using a k amount of hyper sequence of Hyper-inaccessible.

Or a alpha-hyper-inaccessible.
(Alpha-kappa)

Or something like a omega square root of alpha for every alpha is weaker than kappa in which you need to satisfy the conditions of things like hyper square root delta inaccessible denotes this, omega denotes all of this etc etc before making this mahlo cardinal.

Similarly definable structures of mahlo cardinal can be denoted like this but another way to achieve one is to 1st start of with a kappa-kappa(kk) which is for every kappa of a strongly inaccessible there is a cofinality of a regular kappa which creates a way to access a strongly inaccessible while being impossible to be expressed by a sum of smaller kappa's.

Which bassically gives you a strongly inaccessible to start.

Then after that you use the mahlo operation to define and iterate a certain process for strongly inaccessible and diagonalize it etc.

Case here is that you only need to be a kth to a k to be able to reach tier 0 since it's the same as being the fixed point of k to k² and more.
 
Cat said:
Ah no, the tiering system standards on High 1-A are the kappa = aleph kappa, mahlo is an inaccessible cardinal but it's also a limit of inaccessible cardinals, if you didn't read what I presented above a tier 0 cardinal becomes tier 0 when it limits and is a fixed point of kappa.
Click to expand...
That is assuming this is the case.
However, I have to mention the Wikipedia for the cardinal does have this to day.

“
The term α-Mahlo is ambiguous and different authors give inequivalent definitions. One definition is that a cardinal κ is called α-Mahlo for some ordinal α if κ is strongly inaccessible and for every ordinal β<α, the set of β-Mahlo cardinals below κ is stationary in κ. However the condition "κ is strongly inaccessible" is sometimes replaced by other conditions, such as "κ is regular" or "κ is weakly inaccessible" or "κ is Mahlo". We can define "hyper-Mahlo", "α-hyper-Mahlo", "hyper-hyper-Mahlo", "weakly α-Mahlo", "weakly hyper-Mahlo", "weakly α-hyper-Mahlo", and so on, by analogy with the definitions for inaccessibles, so for example a cardinal κ is called hyper-Mahlo if it is κ-Mahlo.

A cardinal κ is greatly Mahlo or κ+-Mahlo if and only if it is inaccessible and there is a normal (i.e. nontrivial and closed under diagonal intersections) κ-complete filter on the power set of κ that is closed under the Mahlo operation, which maps the set of ordinals S to {α
\in
S: α has uncountable cofinality and S∩α is stationary in α}

The properties of being inaccessible, Mahlo, weakly Mahlo, α-Mahlo, greatly Mahlo, etc. are preserved if we replace the universe by an inner model.

Every reflecting cardinal has strictly more consistency strength than a greatly Mahlo, but inaccessible reflecting cardinals aren't in general Mahlo -- see https://mathoverflow.net/q/212597”
 
I don't even know how you got to the conclusion of large cardinals being H1-A by wiki standards, if you consider the large cardinal hierarchy as the logic quantifying H1-A then sure but in a set theory perspective the logic defining H1-A are the sets in kappa, which is k², and other more
HammerStrikes219 said:
That is assuming this is the case.
However, I have to mention the Wikipedia for the cardinal does have this to day.

“
The term α-Mahlo is ambiguous and different authors give inequivalent definitions. One definition is that a cardinal κ is called α-Mahlo for some ordinal α if κ is strongly inaccessible and for every ordinal β<α, the set of β-Mahlo cardinals below κ is stationary in κ. However the condition "κ is strongly inaccessible" is sometimes replaced by other conditions, such as "κ is regular" or "κ is weakly inaccessible" or "κ is Mahlo". We can define "hyper-Mahlo", "α-hyper-Mahlo", "hyper-hyper-Mahlo", "weakly α-Mahlo", "weakly hyper-Mahlo", "weakly α-hyper-Mahlo", and so on, by analogy with the definitions for inaccessibles, so for example a cardinal κ is called hyper-Mahlo if it is κ-Mahlo.

A cardinal κ is greatly Mahlo or κ+-Mahlo if and only if it is inaccessible and there is a normal (i.e. nontrivial and closed under diagonal intersections) κ-complete filter on the power set of κ that is closed under the Mahlo operation, which maps the set of ordinals S to {α
\in
S: α has uncountable cofinality and S∩α is stationary in α}

The properties of being inaccessible, Mahlo, weakly Mahlo, α-Mahlo, greatly Mahlo, etc. are preserved if we replace the universe by an inner model.

Every reflecting cardinal has strictly more consistency strength than a greatly Mahlo, but inaccessible reflecting cardinals aren't in general Mahlo -- see https://mathoverflow.net/q/212597”
Click to expand...
This does not disprove the standards, a kappa is H1-A because it is a fixed points of all alephs same case for tier 1-inaccessible or mahlo being tier 0 it is stronger than the existence of the proper class of inaccessible.

Also since your probably talking about the strongly inaccessible part check the mahlo operation section
 
Cat said:
This does not disprove the standards, a kappa is H1-A because it is a fixed points of all alephs same case for tier 0 1-inaccessible or mahlo it l is stronger than the existence of the proper class of inaccessible.
Click to expand...
It doesn’t necessarily disprove the standards though, it seems to more clarified that there are larger cardinals than that of Mahlo
 
HammerStrikes219 said:
It doesn’t necessarily disprove the standards though, it seems to more clarified that there are larger cardinals than that of Mahlo
Click to expand...
There are larger cardinals than Mahlo but like mahlo being inaccessible most of those cardinals have mahloness in them, those cardinals can also have compactness and etc etc which makes things confusing.

(Also can you tell me what your trying to prove from those quotes? Since mahlo cardinals are indeed a form of inaccessible since they are created by many methods and diagonalizations with mostly inaccessible as a starter, they are also denoted by using a inaccessible just that their beyond the scope of their proper class for the most part.)
 
DontTalkDT said:
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.
Click to expand...
I dunno about you, but equalizing characters who are qualitatively beyond an infinite High 1-A hierarchy to 1-inaccessible sounds logical if High 1-A is based on inaccessible cardinals. Besides, right now it's not clear what exactly we are treating as the baseline for tier 0. We just assume Mahlo because it's the next LC property after inaccessibility, but the tiering system doesn't say, so...
 
KingPin0422 said:
I dunno about you, but equalizing characters who are qualitatively beyond an infinite High 1-A hierarchy to 1-inaccessible sounds logical if High 1-A is based on inaccessible cardinals. Besides, right now it's not clear what exactly we are treating as the baseline for tier 0. We just assume Mahlo because it's the next LC property after inaccessibility, but the tiering system doesn't say, so...
Click to expand...
Problem here is that many people believe it's mahlo cardinals probably because of the old manifold thread that upgraded gom to 0.
 
HammerStrikes219 said:
That is assuming we using the non standard Mahlo though as these are assuming they are proven tbf.
Click to expand...
That's how large cardinal works buddy, we make our own truth. (Axioms)
HammerStrikes219 said:
Also there is the regular Mahlo cardinals to being considered too.
Click to expand...
There is no regular mahlo though, aside from mahlo cardinals that are made through things like 1-mahlo or alpha-mahlo etc the only mahlo I would see is a weakly mahlo and a strongly mahlo.
 
The real big brain move is to define tier 0 by V=Ultimate L. 👀

In all seriousness, I can't follow this thread anymore, so I'm just going to step away for a while...
 
Cat said:
That's how large cardinal works buddy, we make our own truth. (Axioms)

There is no regular mahlo though, aside from mahlo cardinals that are made through things like 1-mahlo or alpha-mahlo etc the only mahlo I would see is a weakly mahlo and a strongly mahlo.
Click to expand...
From Set Theory, yes.

I will unfollow the thread since this is largely complicated and could also include Tier High 1A as fhe tiering system did only give a example of inaccessible cardinals.
 
HammerStrikes219 said:
From Set Theory, yes.

I will unfollow the thread
Click to expand...
Ok
HammerStrikes219 said:
since this is largely complicated and could also include Tier High 1A as fhe tiering system did only give a example of inaccessible cardinals.
Click to expand...
It only says inaccessible cardinal though which is safe to assume it's not even a properclass of inaccessible but just a kappa=aleph kappa.
HammerStrikes219 said:
Oh right, that also technically included other cardinals that is larger than a Mahlo Cardinal according to reflecting principles.
Click to expand...
There are many large cardinals bigger than a mahlo cardinal since without specification and only stating mahlo cardinal, it would only include a strongly mahlo cardinal and not cardinals that have mahlo-ness in them (like how mahlo has inaccessible in it) nor does it include things like 1-mahlo and etc unless if it actually states so.

(Since having mahlo-ness in you doesn't mean your a mahlo cardinal it seems, pretty confusing.

Also even a woodin is a mahlo cardinal, just that it's stronger than the properclass of strong mahlo and etc.)

I'm still really confused as to how you come to the conclusion that there are no bigger cardinals than Mahlo though....
 
Did not read the thread but kind of long but read the Td:Lr and I think plan C sounds perfect.

Also the tiering system is not based on math, we are just trying to equate mathematical structures to it.

And again incredibly high tiers requires incredibly high proof. Especially the way the fictional verse treats it.

Also your confusion comes from looking at high 1-A as fiction means one layer into High 1-A.

Yes it’s enough
Looking at 1-inaccessible as fiction is enough to grant you 2-inaccessible
 
Pain_to12 said:
Also the tiering system is not based on math, we are just trying to equate mathematical structures to it.
Click to expand...
Ah I wasn't equating it on mathematical structures.

There will be no change in the tiering system, only change here is that if a verse explicitly has something like a mahlo hyperspace then it would be many layers to 0.
 
Cat said:
Ah I wasn't equating it on mathematical structures.

There will be no change in the tiering system, only change here is that if a verse explicitly has something like a mahlo hyperspace then it would be many layers to 0.
Click to expand...
Edited the post above check it out

And you are pretty much equating on mathematical structures, if not what are you doing??

If a verse has mahlo they are already considered layers into tier 0 already depending on the numbers.
 
Pain_to12 said:
Did not read the thread but kind of long but read the Td:Lr and I think plan C sounds perfect.

Also the tiering system is not based on math, we are just trying to equate mathematical structures to it.

And again incredibly high tiers requires incredibly high proof. Especially the way the fictional verse treats it.

Also your confusion comes from looking at high 1-A as fiction means one layer into High 1-A.

Yes it’s enough
Looking at 1-inaccessible as fiction is enough to grant you 2-inaccessible
Click to expand...
I don't think so R>F here is only considered as aleph 1 snapshots if we see a H1-A as fiction it would be 2^inaccessible and not 1-inaccessible.
Pain_to12 said:
Edited the post above check it out

And you are pretty much equating on mathematical structures, if not what are you doing??
Click to expand...
As I said this thread wasn't made for the purpose of changing tier 0 standards nor equate it to strictly only mathematics, it's just purely made to lower the mathematical standards for tier 0.
 
Last edited:
Pain_to12 said:
R>F is considered based on what you are seeing as R>F
Click to expand...
As I said R>F on inaccessible amount of dimensions just makes you that of a power set of inaccessible, it doesn't mean that you view inaccessible the same way inaccessible views alephs by a simple r>f.
 
Honestly, this whole thread is kind of moot to begin with, for the simple reason that we lack an official mathematical definition of 0. It's just "you exceed the mathematical foundation of High 1-A just as it exceeds the foundation of everything below it," but unlike High 1-A which we clearly equate to inaccessible cardinals, 0 isn't exemplified by anything. Mahlo cardinals are just assumed for the sake of convenience.

We need to establish a solid baseline for tier 0 before we can even talk about changing it.
 
DontTalkDT said:
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.
Click to expand...
KingPin0422 said:
Honestly, this whole thread is kind of moot to begin with, for the simple reason that we lack an official mathematical definition of 0. It's just "you exceed the mathematical foundation of High 1-A just as it exceeds the foundation of everything below it," but unlike High 1-A which we clearly equate to inaccessible cardinals, 0 isn't exemplified by anything. Mahlo cardinals are just assumed for the sake of convenience.

We need to establish a solid baseline for tier 0 before we can even talk about changing it.
Click to expand...
@DontTalkDT @KingPin0422

I am obviously very open to if the two of you can come up with a more specific and rational mathematical definition of Tier 0 via a private discussion thread, preferably while asking @Ultima_Reality and @Agnaa for input.

I would prefer if I am kept informed about your progress though.
 
Antvasima said:
@DontTalkDT @KingPin0422

I am obviously very open to if the two of you can come up with a more specific and rational mathematical definition of Tier 0 via a private discussion thread, preferably while asking @Ultima_Reality and @Agnaa for input.

I would prefer if I am kept informed about your progress though.
Click to expand...
I'm very neutral regarding the situation for the time being.

In my eyes, large cardinals being used for High 1-A is already a bad choice, so stacking on top of that with an even higher cardinal (or whatever else) for Tier 0 will also always be a bad choice.
So for the time being (i.e. until I, if ever, talk about our High 1-A choice) I'm neutral regarding all choices for Tier 0, as they don't affect what we equalize to them anyway.
 
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