Cat275
He/Him- 1,428
- 789
Making this thread for fun :3 though I have heard that there was a simillar crt a while ago, to which I don't know why it got rejected but I'll try my best to upgrade mahlo and make a more accurate mathematical standard for tier 0, using this thread.
(Well the more accurate part is just my point of view of what is more accurate, Just saying.)
The problem.
Alright let's start here.
1st of quoting the tier 0 in tiering system:
• Characters who can affect objects which completely exceed the logical foundations of High 1-A, much like it exceeds the ones defining 1-A and below, meaning that all possible levels of High 1-A are exceeded
Now with H1-A:
• 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-A hierarchies.
Pretty simillar if I say so myself but the idea of tier 0 and the reason why mahlo cardinal is the standard for tier 0 is because it views inaccessible the same way it views alephs.
With this idea we can assume a cardinal k is H1-A and the framework of H1-A and 1-A are the classes and sets of inaccessible cardinals and aleph numbers since a class is the collection of all the defining set a cardinal has, this will be an important topic that will be explained below but 1st before I explain further I have heard that a 1-inaccessible cardinal is treated as 1 layer to H1-A which would be the main problem.
(Another varying problem here is mahlo being the baseline 0 in mathematics which I would propose 2 plans that solve this or really just choose the 3rd one)
Clarification + Plan A
Alright so 1st of I will quote this:
• A cardinal κ is 1-inaccessible if it is inaccessible and a limit of inaccessible cardinals. In other words, κ is 1-inaccessible if κ is the κth inaccessible cardinal, that is, if κ is a fixed point in the enumeration of all inaccessible cardinals. Equivalently, κ is 1-inaccessible if Vκ is a universe and satisfies the universe axiom
This means that:
Essentially this means that a 1-inaccessible cardinal is stronger than any sets a inaccessible cardinal has and limit of such, what I mean by that is something like this.
• 1-inaccessible > 2^inaccessible + L with the axiom of L satisfying the continuum hypothesis to achieve a bigger inaccessible cardinal and etc.
This also includes the continuous iterations of power sets and such a 1-inaccessible is greater than the power set of 2^inaccessible and etc.
(Atleast it's safe to assume so.)
Moving on we also know that a 1-inaccessible cardinal is not like a normal limit cardinal where it just limits certain sequence of such but instead it's more of a kappa version and a fixed point of such, in this case a 1-inaccessible cardinal treats the 1st set of kappa (k=aleph k) like how the 1st set of kappa treats alephs as it's the fixed point of inaccessible cardinals as well as the limit of such cardinals, sets etc and such.. It's a kth to a k.
So not only it's a limit cardinal of inaccessible it's also the fixed point of inaccessible cardinals.
Which further proves that it views inaccessible the same way inaccessible views alephs.
(This also means that 1 layer to H1-A in set theory is actually just a 2^inaccessible amount of dimensions assuming L is there to satisfy the continuum hypothesis or it could also be inaccessible + 1 if we can use the idea of GCH.)
Furthermore a 1-inaccessible cardinal is stronger than the existence of the properclass of inaccessible cardinals, which I will quote below.
(Also to note, a properclass are the collection of sets defined by a formula of it's quantifiers (logic) that ranges over sets.)
Quoting:
• 1-inaccessibillity is already consistency-wise stronger than the existence of a proper class of inaccessible cardinals, and 2-inaccessibility is stronger than the existence of a proper class of 1-inaccessible cardinals.
This quote would be good as supporting evidence for a 1-inaccessible quantity of dimensions being beyond the logical foundations of H1-A.
Anyways this hierarchy continues endlessly much like the aleph numbers, atleast excluding the idea of power sets and how much bigger they are now compared to each other.
Quoting:
• 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.
With this idea, we can assume a 1-inaccessible cardinal is enough for tier 0, and 2-inaccessible is many layers into it and this hierarchy could go for more than a countable infinite.
(I would also like to note that even a strongly or weakly inaccessible cardinal "hypothetically" has a hierarchy of itself which would probably have a quantity as much as how many power sets a aleph number has and would therefore have a one-to-one correspondance to each other, ignoring the difference of size of course.
I shall try to explain why.
• Using the continuum hypothesis and adding the axiom called L to satisfy the hypothesis we use this formula.
• 2^inaccessible/inaccessible^inaccessible cardinal to create a bigger inaccessible cardinal or to satisfy a power set for inaccessible.
• Then after that we use the continuum hypothesis again to satisfy a bigger inaccessible then again, and again and again we will use the ch until we reach the degree of omega and after that we will create a recursive sequence of omega until we reach and find a omega^omega amount of power sets on a inaccessible cardinal.
• Which would be equivalent to having a aleph 1 amount of power sets which to visualize would look like something like this.
• P(P(P(P,,,,,,,,(P(inaccessible) until you have a omega^omega amount of Power sets and this bassically continues much like how the sequence of aleph numbers work but instead it's inaccessible, alternatively you can probably also use the idea of the GCH and use things like this instead, inaccessible + 1, inaccessible + 2 etc etc all of which may possibly satisfy a power set.
I think it's safe to assume that this would create as much power set a aleph has since pretty much all aleph numbers out there all started from aleph null to power set from power set and such created a hierarchy of this namely because of power sets.
Probably the same case for inaccessible here, as long as it has a power set of course.
And yeah I indeed know a inaccessible cardinal is a limit cardinal but the main idea of let's say a strong limit cardinal is that it can't be obtained by a power set but it doesn't mean that it doesn't have a power set.
this is a important note to know how a 1-inaccessible work and how big it is as something being a kth to a k cardinal and being a fixed enumeration of all of this inaccessible cardinals and sets.)
Oh yeah just in case this part might be ask and want a bit more proof on this part:
Quoting:
• A cardinal λ is a strong limit cardinal if λ cannot be reached by repeated powerset operations.
As you can see here it doesn't say that it doesn't have a power set.
• This means that λ is nonzero and, for all κ < λ, 2κ < λ. Every strong limit cardinal is also a weak limit cardinal, because κ+ ≤ 2κ for every cardinal κ, where κ+ denotes the successor cardinal of κ.
This also further proves that you can use a power set or create a successor operation on a inaccessible or limit cardinal since a cardinal k is a fixed point of alephs and also a limit of such cardinals.
(Atleast this is of course assuming the kappa used here is the fixed point of alephs and therefore a enumeration of it, which would be kappa = aleph kappa though usually the cross sign in limit cardinals are sign of the successor operation after it.)
Generally the GCH and the aleph hierarchy already proves that a limit cardinal can have a power set since it explains that if a infinite sets cardinality lies between a infinite set and a power set, then it either has the same cardinality or a power set of such cardinality. one of the examples that proves limit cardinals having power sets was about aleph alpha + 1 = aleph alpha^aleph alpha and therefore creating a successor operation of aleph alpha in the aleph hierarchy, another good example are the epsilon numbers.
(Aleph alpha is a limit cardinal which comes before aleph lambda and etc.)
Further more you can access inaccessible cardinals by using cofinalities:
• The "union operation" always provides another way of "accessing" these cardinals (and indeed, such is the case of limit ordinals as well). Stronger notions of inaccessibility can be defined using cofinality. For a weak (respectively strong) limit cardinal κ the requirement is that cf(κ) = κ (i.e. κ be regular) so that κ cannot be expressed as a sum (union) of fewer than κ smaller cardinals. Such a cardinal is called a weakly (respectively strongly) inaccessible cardinal.
This part certainly doesn't prove nor disprove the above points but this helps on giving a clear vision on how big a inaccessible is along with 1-inaccessible.
Anyways back to the topic I have heard that an r>f on a baseline H1-A is 1 layer to it and would be 1-inaccessible in mathematics when it's clear that it's not, this would be just a random upgrade from the way r>f is treated in mathematics, since it would actually be just something that of a power set of a inaccessible in mathematics and not 1-inaccessible.
(And this itself is still a highball since r>f's are treated as aleph 1 snapshots in the wiki)
But bassically a mahlo cardinal is not a cardinal that views inaccessible cardinals the same way a inaccessible cardinal views aleph numbers, it's a lot more massive than that.
So Overall plan A is about 1-inaccessible being the standard for tier 0.
Though if you don't like the idea of 1-inaccessible being baseline 0, since mahlo will be way to massive here then we will go for plan B.
Plan B
Alright I'll start now, for plan B we'll go with hyper-inaccessible being baseline 0 in plan B since the idea here is that if the difference of the degree of inaccessibles was lowballed to something as big as the difference of alephs and not that of a limit cardinal then a hyper-inaccessible cardinal will still be a cardinal that views all proper and small classes of the degree of inaccessible cardinals like the way a normal k views alephs.
To understand the point here we shall 1st start with the fixed point of alephs or normal kappa which is the baseline High 1-A.
I will quote one right here:
• A kappa cardinal is an aleph-fixed point when kappa = aleph kappa. In this case, is the infinite cardinal. Every inaccessible cardinal is an aleph-fixed point, and a limit of such fixed points and so on. Indeed, every worldly cardinal is an aleph-fixed point and a limit of such.
this was further proved on this quote:
• A kappa cardinal being inaccessible implies the following:
kappa is an aleph fixed point and a beth fixed point.
But bassically the 1st kappa cardinal is the fixed point and enumeration of alephs cardinals and beth cardinals, this is important to the topic since this is one of the many requirments of inaccessible, and a 1-inaccessible already proves that it is a k that views k like the way the 1st k views alephs but in plan B if the idea of plan A is rejected then we would make the hyper-inaccessible cardinal the baseline 0 and the very reason is this:
• A kappa cardinal is hyperinaccessible if it is kappa-inaccessible.
This bassically means that a hyper-inaccessible is the fixed point of all the degrees of inaccessible out there, it's not like 1-inaccessible or anything where it's the fixed point of inaccessible cardinals, instead it would be literally the fixed point of the entire degree of inaccessibles but bassically with the entire hierarchy of each degree of inaccessible being inaccessible to each other a hyper-inaccessible is a kth to all of this satisfying k of n-inaccessibles.
This just means that if we lowball the degrees of inaccessible cardinal and let's say 1-inaccessible is only 1 layer to H1-A then a hyper-inaccessible still satisfies the requirement of tier 0, the main reason here is that we assume the 1st kappa is H1-A and since a hyper-inaccessible is a kappa for the whole degree of inaccessible cardinals (layers of H1-A in this case) this should be the same case here as even ignoring the idea of power sets the amount of n-inaccessibles out there would probably have the same quantity of that of the amount of aleph numbers.
(Though again a 1-inaccessible cardinal already fits the bill of a inaccessible being inaccessible to a inaccessible, but essentially this means that a hyper-inaccessible cardinal is already beyond the defining framework of the degree of inaccessible cardinals and tier H1-A even with the assumption of 1-inaccessible only being a layer to H1-A.)
Anyways quoting on how big a mahlo cardinal is:
• Every Mahlo cardinal kappa is inaccessible, and indeed hyper-inaccessible and hyper-hyper-inaccessible, up to degree kappa, and a limit of such cardinals.
And
• The rest of the proof that κ is α-hyper-inaccessible mimics the proof that it is α-inaccessible. So κ is hyper-hyper-inaccessible, etc..
• Mahlo cardinals are inaccessible, hyper-inaccessible, hyper-hyper-inaccessible, ... and so on.
And to note how big a hyper-hyper-inaccessible is:
• Continuing, kappa is hyperhyperinaccessible if kappa is kappa-hyperinaccessible.
In simple words a hyper-hyper-inaccessible is a kappa-kappa-inaccessible and the same probably applies to further hyperinaccessibles.
This means that a hyper-hyper-inaccessible may view a hyper-inaccessible as the following 2, I present here:
Furthermore whether it would be the latter or the former a Hyper-Hyper-inaccessible would still have this 2 properties I present below:
Anyways continuing (hypothetically) on how big a mahlo is, with the scans I have above assuming we have a n-hyper-inaccessible, Where each n like 2-hyper-inaccessible is equivalent to hyper-hyper-hyper-inaccessible in words then a kappa that is k-hyper-inaccessible is a kappa number that satisfies some of the conditions of being a mahlo cardinal and would also satisfy being a fixed point of all the n-hyper-inaccessible where each gap of a hyper-inaccessible is like how a 1-inaccessible is to inaccessible or may also be like how a hyper-inaccessible is to a n-inaccessibles, or you could use another way to denote and define mahlo such as alpha-hyper-inaccessible and omega squared alpha-inaccessible for every alpha is weaker than kappa or you could use the mahlo operation to diagonalize and iterate a inaccessible cardinal and make a mahlo cardinal and even more. (etc)
I'm trying to say that the cardinality of each mahlo cardinal to another mahlo cardinal varies from how it's denoted and achieved. (etc)
In this case I am using a k-hyper-inaccessible to denote one, which I consider to be the most minimal way to denote a mahlo cardinal.
Though gotta note that when I said k-hyper-inaccessible I meant something like this.
k-k-k-k-k-,,,,,,,,,,,,,,,-k-inaccessible.
Or something like this instead.
Hyper-hyper-hyper-,,,,,,,,,,,,,,-hyper-inaccessible and etc.
So Overall plan B is about Hyper-inaccessible being the standard for tier 0.
Plan C
Screw everything and just forget this thread.
Source
Alright just in case someone ask for source material, then here are some of the basic source I used here:
This, this, this, this, and this
TL;DR
Plan A- 1-inaccessible quantity of dimensions should be the baseline of tier 0.
Plan B- Hyper-inaccessible quantity of dimensions should be the baseline tier 0 instead of 1-inaccessible.
Plan C- Forget this thread.
(Well the more accurate part is just my point of view of what is more accurate, Just saying.)
The problem.
Alright let's start here.
1st of quoting the tier 0 in tiering system:
• Characters who can affect objects which completely exceed the logical foundations of High 1-A, much like it exceeds the ones defining 1-A and below, meaning that all possible levels of High 1-A are exceeded
Now with H1-A:
• 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-A hierarchies.
Pretty simillar if I say so myself but the idea of tier 0 and the reason why mahlo cardinal is the standard for tier 0 is because it views inaccessible the same way it views alephs.
With this idea we can assume a cardinal k is H1-A and the framework of H1-A and 1-A are the classes and sets of inaccessible cardinals and aleph numbers since a class is the collection of all the defining set a cardinal has, this will be an important topic that will be explained below but 1st before I explain further I have heard that a 1-inaccessible cardinal is treated as 1 layer to H1-A which would be the main problem.
(Another varying problem here is mahlo being the baseline 0 in mathematics which I would propose 2 plans that solve this or really just choose the 3rd one)
Clarification + Plan A
Alright so 1st of I will quote this:
• A cardinal κ is 1-inaccessible if it is inaccessible and a limit of inaccessible cardinals. In other words, κ is 1-inaccessible if κ is the κth inaccessible cardinal, that is, if κ is a fixed point in the enumeration of all inaccessible cardinals. Equivalently, κ is 1-inaccessible if Vκ is a universe and satisfies the universe axiom
This means that:
Essentially this means that a 1-inaccessible cardinal is stronger than any sets a inaccessible cardinal has and limit of such, what I mean by that is something like this.
• 1-inaccessible > 2^inaccessible + L with the axiom of L satisfying the continuum hypothesis to achieve a bigger inaccessible cardinal and etc.
This also includes the continuous iterations of power sets and such a 1-inaccessible is greater than the power set of 2^inaccessible and etc.
(Atleast it's safe to assume so.)
Moving on we also know that a 1-inaccessible cardinal is not like a normal limit cardinal where it just limits certain sequence of such but instead it's more of a kappa version and a fixed point of such, in this case a 1-inaccessible cardinal treats the 1st set of kappa (k=aleph k) like how the 1st set of kappa treats alephs as it's the fixed point of inaccessible cardinals as well as the limit of such cardinals, sets etc and such.. It's a kth to a k.
So not only it's a limit cardinal of inaccessible it's also the fixed point of inaccessible cardinals.
Which further proves that it views inaccessible the same way inaccessible views alephs.
(This also means that 1 layer to H1-A in set theory is actually just a 2^inaccessible amount of dimensions assuming L is there to satisfy the continuum hypothesis or it could also be inaccessible + 1 if we can use the idea of GCH.)
Furthermore a 1-inaccessible cardinal is stronger than the existence of the properclass of inaccessible cardinals, which I will quote below.
(Also to note, a properclass are the collection of sets defined by a formula of it's quantifiers (logic) that ranges over sets.)
Quoting:
• 1-inaccessibillity is already consistency-wise stronger than the existence of a proper class of inaccessible cardinals, and 2-inaccessibility is stronger than the existence of a proper class of 1-inaccessible cardinals.
This quote would be good as supporting evidence for a 1-inaccessible quantity of dimensions being beyond the logical foundations of H1-A.
Anyways this hierarchy continues endlessly much like the aleph numbers, atleast excluding the idea of power sets and how much bigger they are now compared to each other.
Quoting:
• 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.
With this idea, we can assume a 1-inaccessible cardinal is enough for tier 0, and 2-inaccessible is many layers into it and this hierarchy could go for more than a countable infinite.
(I would also like to note that even a strongly or weakly inaccessible cardinal "hypothetically" has a hierarchy of itself which would probably have a quantity as much as how many power sets a aleph number has and would therefore have a one-to-one correspondance to each other, ignoring the difference of size of course.
I shall try to explain why.
• Using the continuum hypothesis and adding the axiom called L to satisfy the hypothesis we use this formula.
• 2^inaccessible/inaccessible^inaccessible cardinal to create a bigger inaccessible cardinal or to satisfy a power set for inaccessible.
• Then after that we use the continuum hypothesis again to satisfy a bigger inaccessible then again, and again and again we will use the ch until we reach the degree of omega and after that we will create a recursive sequence of omega until we reach and find a omega^omega amount of power sets on a inaccessible cardinal.
• Which would be equivalent to having a aleph 1 amount of power sets which to visualize would look like something like this.
• P(P(P(P,,,,,,,,(P(inaccessible) until you have a omega^omega amount of Power sets and this bassically continues much like how the sequence of aleph numbers work but instead it's inaccessible, alternatively you can probably also use the idea of the GCH and use things like this instead, inaccessible + 1, inaccessible + 2 etc etc all of which may possibly satisfy a power set.
I think it's safe to assume that this would create as much power set a aleph has since pretty much all aleph numbers out there all started from aleph null to power set from power set and such created a hierarchy of this namely because of power sets.
Probably the same case for inaccessible here, as long as it has a power set of course.
And yeah I indeed know a inaccessible cardinal is a limit cardinal but the main idea of let's say a strong limit cardinal is that it can't be obtained by a power set but it doesn't mean that it doesn't have a power set.
this is a important note to know how a 1-inaccessible work and how big it is as something being a kth to a k cardinal and being a fixed enumeration of all of this inaccessible cardinals and sets.)
Oh yeah just in case this part might be ask and want a bit more proof on this part:
Quoting:
• A cardinal λ is a strong limit cardinal if λ cannot be reached by repeated powerset operations.
As you can see here it doesn't say that it doesn't have a power set.
• This means that λ is nonzero and, for all κ < λ, 2κ < λ. Every strong limit cardinal is also a weak limit cardinal, because κ+ ≤ 2κ for every cardinal κ, where κ+ denotes the successor cardinal of κ.
This also further proves that you can use a power set or create a successor operation on a inaccessible or limit cardinal since a cardinal k is a fixed point of alephs and also a limit of such cardinals.
(Atleast this is of course assuming the kappa used here is the fixed point of alephs and therefore a enumeration of it, which would be kappa = aleph kappa though usually the cross sign in limit cardinals are sign of the successor operation after it.)
Generally the GCH and the aleph hierarchy already proves that a limit cardinal can have a power set since it explains that if a infinite sets cardinality lies between a infinite set and a power set, then it either has the same cardinality or a power set of such cardinality. one of the examples that proves limit cardinals having power sets was about aleph alpha + 1 = aleph alpha^aleph alpha and therefore creating a successor operation of aleph alpha in the aleph hierarchy, another good example are the epsilon numbers.
(Aleph alpha is a limit cardinal which comes before aleph lambda and etc.)
Further more you can access inaccessible cardinals by using cofinalities:
• The "union operation" always provides another way of "accessing" these cardinals (and indeed, such is the case of limit ordinals as well). Stronger notions of inaccessibility can be defined using cofinality. For a weak (respectively strong) limit cardinal κ the requirement is that cf(κ) = κ (i.e. κ be regular) so that κ cannot be expressed as a sum (union) of fewer than κ smaller cardinals. Such a cardinal is called a weakly (respectively strongly) inaccessible cardinal.
This part certainly doesn't prove nor disprove the above points but this helps on giving a clear vision on how big a inaccessible is along with 1-inaccessible.
Anyways back to the topic I have heard that an r>f on a baseline H1-A is 1 layer to it and would be 1-inaccessible in mathematics when it's clear that it's not, this would be just a random upgrade from the way r>f is treated in mathematics, since it would actually be just something that of a power set of a inaccessible in mathematics and not 1-inaccessible.
(And this itself is still a highball since r>f's are treated as aleph 1 snapshots in the wiki)
But bassically a mahlo cardinal is not a cardinal that views inaccessible cardinals the same way a inaccessible cardinal views aleph numbers, it's a lot more massive than that.
So Overall plan A is about 1-inaccessible being the standard for tier 0.
Though if you don't like the idea of 1-inaccessible being baseline 0, since mahlo will be way to massive here then we will go for plan B.
Plan B
Alright I'll start now, for plan B we'll go with hyper-inaccessible being baseline 0 in plan B since the idea here is that if the difference of the degree of inaccessibles was lowballed to something as big as the difference of alephs and not that of a limit cardinal then a hyper-inaccessible cardinal will still be a cardinal that views all proper and small classes of the degree of inaccessible cardinals like the way a normal k views alephs.
To understand the point here we shall 1st start with the fixed point of alephs or normal kappa which is the baseline High 1-A.
I will quote one right here:
• A kappa cardinal is an aleph-fixed point when kappa = aleph kappa. In this case, is the infinite cardinal. Every inaccessible cardinal is an aleph-fixed point, and a limit of such fixed points and so on. Indeed, every worldly cardinal is an aleph-fixed point and a limit of such.
this was further proved on this quote:
• A kappa cardinal being inaccessible implies the following:
kappa is an aleph fixed point and a beth fixed point.
But bassically the 1st kappa cardinal is the fixed point and enumeration of alephs cardinals and beth cardinals, this is important to the topic since this is one of the many requirments of inaccessible, and a 1-inaccessible already proves that it is a k that views k like the way the 1st k views alephs but in plan B if the idea of plan A is rejected then we would make the hyper-inaccessible cardinal the baseline 0 and the very reason is this:
• A kappa cardinal is hyperinaccessible if it is kappa-inaccessible.
This bassically means that a hyper-inaccessible is the fixed point of all the degrees of inaccessible out there, it's not like 1-inaccessible or anything where it's the fixed point of inaccessible cardinals, instead it would be literally the fixed point of the entire degree of inaccessibles but bassically with the entire hierarchy of each degree of inaccessible being inaccessible to each other a hyper-inaccessible is a kth to all of this satisfying k of n-inaccessibles.
This just means that if we lowball the degrees of inaccessible cardinal and let's say 1-inaccessible is only 1 layer to H1-A then a hyper-inaccessible still satisfies the requirement of tier 0, the main reason here is that we assume the 1st kappa is H1-A and since a hyper-inaccessible is a kappa for the whole degree of inaccessible cardinals (layers of H1-A in this case) this should be the same case here as even ignoring the idea of power sets the amount of n-inaccessibles out there would probably have the same quantity of that of the amount of aleph numbers.
(Though again a 1-inaccessible cardinal already fits the bill of a inaccessible being inaccessible to a inaccessible, but essentially this means that a hyper-inaccessible cardinal is already beyond the defining framework of the degree of inaccessible cardinals and tier H1-A even with the assumption of 1-inaccessible only being a layer to H1-A.)
Anyways quoting on how big a mahlo cardinal is:
• Every Mahlo cardinal kappa is inaccessible, and indeed hyper-inaccessible and hyper-hyper-inaccessible, up to degree kappa, and a limit of such cardinals.
And
• The rest of the proof that κ is α-hyper-inaccessible mimics the proof that it is α-inaccessible. So κ is hyper-hyper-inaccessible, etc..
• Mahlo cardinals are inaccessible, hyper-inaccessible, hyper-hyper-inaccessible, ... and so on.
And to note how big a hyper-hyper-inaccessible is:
• Continuing, kappa is hyperhyperinaccessible if kappa is kappa-hyperinaccessible.
In simple words a hyper-hyper-inaccessible is a kappa-kappa-inaccessible and the same probably applies to further hyperinaccessibles.
This means that a hyper-hyper-inaccessible may view a hyper-inaccessible as the following 2, I present here:
Furthermore whether it would be the latter or the former a Hyper-Hyper-inaccessible would still have this 2 properties I present below:
Anyways continuing (hypothetically) on how big a mahlo is, with the scans I have above assuming we have a n-hyper-inaccessible, Where each n like 2-hyper-inaccessible is equivalent to hyper-hyper-hyper-inaccessible in words then a kappa that is k-hyper-inaccessible is a kappa number that satisfies some of the conditions of being a mahlo cardinal and would also satisfy being a fixed point of all the n-hyper-inaccessible where each gap of a hyper-inaccessible is like how a 1-inaccessible is to inaccessible or may also be like how a hyper-inaccessible is to a n-inaccessibles, or you could use another way to denote and define mahlo such as alpha-hyper-inaccessible and omega squared alpha-inaccessible for every alpha is weaker than kappa or you could use the mahlo operation to diagonalize and iterate a inaccessible cardinal and make a mahlo cardinal and even more. (etc)
I'm trying to say that the cardinality of each mahlo cardinal to another mahlo cardinal varies from how it's denoted and achieved. (etc)
In this case I am using a k-hyper-inaccessible to denote one, which I consider to be the most minimal way to denote a mahlo cardinal.
Though gotta note that when I said k-hyper-inaccessible I meant something like this.
k-k-k-k-k-,,,,,,,,,,,,,,,-k-inaccessible.
Or something like this instead.
Hyper-hyper-hyper-,,,,,,,,,,,,,,-hyper-inaccessible and etc.
So Overall plan B is about Hyper-inaccessible being the standard for tier 0.
Plan C
Screw everything and just forget this thread.
Source
Alright just in case someone ask for source material, then here are some of the basic source I used here:
This, this, this, this, and this
TL;DR
Plan A- 1-inaccessible quantity of dimensions should be the baseline of tier 0.
Plan B- Hyper-inaccessible quantity of dimensions should be the baseline tier 0 instead of 1-inaccessible.
Plan C- Forget this thread.
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