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Is this enough for tier-0 ?

I think it's too vague. You'd at least prove the existence of a strong axiom to support the Berkeley Cardinal, because it can't be stored in a set format that isn't ZFC compliant.
 
Lewis said:
I think it's too vague. You'd at least prove the existence of a strong axiom to support the Berkeley Cardinal, because it can't be stored in a set format that isn't ZFC compliant.
Click to expand...

Berkeley​

A cardinal κ is a Berkeley cardinal, if for any transitive set M with κ∈M and any ordinal α<κ there is an elementary embedding j:M≺M with α<crit j<κ. These cardinals are defined in the context of ZF set theory without the axiom of choice.

The Berkeley cardinals were defined by W. Hugh Woodin in about 1992 at his set-theory seminar in Berkeley, with J. D. Hamkins, A. Lewis, D. Seabold, G. Hjorth and perhaps R. Solovay in the audience, among others, issued as a challenge to refute a seemingly over-strong large cardinal axiom. Nevertheless, the existence of these cardinals remains unrefuted in ZF.

If there is a Berkeley cardinal, then there is a forcing extension that forces that the least Berkeley cardinal has cofinality ω. It seems that various strengthenings of the Berkeley property can be obtained by imposing conditions on the cofinality of κ (the larger cofinality, the stronger theory is believed to be, up to regular κ). (Bagaria, 2017)

A cardinal κ is called proto-Berkeley if for any transitive M∋κ, there is some j:M≺M with crit j<κ. More generally, a cardinal is α-proto-Berkeley if and only if for any transitive set M∋κ, there is some j:M≺M with α<crit j<κ, so that if δ≥κ, δ is also α-proto-Berkeley. The least α-proto-Berkeley cardinal is called δα.

We call κ a club Berkeley cardinal if κ is regular and for all clubs C⊆κ and all transitive sets M with κ∈M there is j∈E(M) with crit(j)∈C. (Bagaria, 2017)

We call κ a limit club Berkeley cardinal if it is a club Berkeley cardinal and a limit of Berkeley cardinals. (Bagaria, 2017)

Relations​

  • If κ is the least Berkeley cardinal, then there is γ<κ such that (Vγ,Vγ+1)⊨ZF2+“There is a Reinhardtcardinal witnessed by j and an ω-huge above κω(j)”. (Bagaria, 2017)
  • For every α, δα is Berkeley. Therefore δα is the least Berkeley cardinal above α. (Bagaria, 2017)
  • In particular, the least proto-Berkeley cardinal δ0 is also the least Berkeley cardinal. (Bagaria, 2017)
  • If κ is a limit of Berkeley cardinals, then κ is not among the δα. (Bagaria, 2017)
  • Each club Berkeley cardinal is totally Reinhardt. (Bagaria, 2017).
  • The relation between Berkeley cardinals and club Berkeley cardinals is unknown. (Bagaria, 2017)
  • If κ is a limit club Berkeley cardinal, then (Vκ,Vκ+1)⊨“There is a Berkeley cardinal that issuperReinhardt”. (Bagaria, 2017) Moreover, the class of such cardinals are stationary.

The structure of L(Vδ+1)​

If δ is a singular Berkeley cardinal, DC(cf(δ)+), and δ is a limit of cardinals themselves limits of extendible cardinals, then the structure of L(Vδ+1) is similar to the structure of L(Vλ+1) under the assumption λ is I0; i.e. there is some j:L(Vλ+1)≺L(Vλ+1). For example, Θ=ΘVδ+1L(Vδ+1), then Θ is a strong limit in L(Vδ+1), δ+ is regular and measurable in L(Vδ+1), and Θ is a limit of measurable cardinals.
Click to expand...

is this enough ?
 
Cat275 said:
It's tier 0 but not as much layers as what a berkeley cardinal could imply. For more info about their size.

Around number 3.
Click to expand...
It is not possible for me to accurately evaluate the statements made in the article you provided, as they are not clear and contain several errors and inconsistencies.

For example, the concept of a "final grand cardinal" is not a well-defined concept in set theory, and the term "super Reinhardt cardinal" is not a standard term in set theory. Additionally, the notation "oS'(X)" and "oE-η(X)" are not standard notation in set theory, and it is not clear what they are intended to represent.

It is also not clear what is meant by the statement "They are weakly extensible in Size, and are ω−large". In set theory, a cardinal is extensible if it is possible to add new subsets to the set without changing its cardinality, and a cardinal is ω-large if it is larger than the set of all countable ordinals. However, it is not clear how these concepts are relevant to the topic of Reinhardt cardinals.



By the way, is your article an "academic" article?
 
Achille said:
It is not possible for me to accurately evaluate the statements made in the article you provided, as they are not clear and contain several errors and inconsistencies.

For example, the concept of a "final grand cardinal" is not a well-defined concept in set theory, and the term "super Reinhardt cardinal" is not a standard term in set theory. Additionally, the notation "oS'(X)" and "oE-η(X)" are not standard notation in set theory, and it is not clear what they are intended to represent.


It is also not clear what is meant by the statement "They are weakly extensible in Size, and are ω−large". In set theory, a cardinal is extensible if it is possible to add new subsets to the set without changing its cardinality, and a cardinal is ω-large if it is larger than the set of all countable ordinals. However, it is not clear how these concepts are relevant to the topic of Reinhardt cardinals.
Click to expand...
1. Woodin is the one who created the concept of super reinhardt cardinals. Which I'm pretty sure is written in bargaria and harvard. (I can grab the link later if you want) And it not being a standard notation is certainly not a inconsistency.
2. Weakly extendibles is woodin koellners version of extendible where the normal extendibles are called that and a extendible in woodins version are sigma-3 inductive since it satisfies the formula sigma-2. (or pi-2)
3. The term they are this and that is quite common in set theory as trivial relationships.

Also your explanation is right but a bit off. Omega-huge are crit(j) of J:V(Lambda)->V(lambda) assuming the embeddings are nontrivial. Well it's more complex than that but that should show how off you are. Same for extendibles since you use some elementary embeddings of models.
 
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Cat275 said:
1. Woodin is the one who created the concept of super reinhardt cardinals. Which I'm pretty sure is written in bargaria and harvard. (I can grab the link later if you want) And it not being a standard notation is certainly not a inconsistency.
2. Weakly extendibles is woodin koellners version of extendible where the normal extendibles are called that and a extendible in woodins version are sigma-3 inductive since it satisfies the formula sigma-2. (or pi-2)
3. The term they are this and that is quite common in set theory as trivial relationships.

Also your explanation is right but a bit off. Omega-huge are jn targets of j:V->M. Well it's more complex than that but that should show how off you are. Same for extendibles since you use elementary embeddings of models.
Click to expand...

What I wrote and your answers seem a bit unrelated (especially the first sentence in answer 1).
What I call inconsistent is that I can't quite see the part where the Reinhardt cardinal proves to be the strongest. So I apologize for expressing myself wrong in the first place.
Also, if you're saying that the explanation I just made is "incomplete", I'll make a statement in the name of "hierarchy". Can you tell me if I'm wrong? Thanks in advance.



Reinhardt cardinals are a type of large cardinal axiom in set theory that are defined in terms of certain embeddings of the universe of set theory into itself. Specifically, a Reinhardt cardinal is an infinite cardinal κ such that there exists an embedding j: V → V (where V is the universe of set theory) such that j(κ) is a subset of κ and for all ordinals η < κ, j(η) is an element of the set oE−η(j(κ)). Here, oE−η(j(κ)) denotes the ηth level of the extender hierarchy on j(κ).

The extender hierarchy is a structure in set theory that is used to construct certain large cardinal axioms, including the Reinhardt cardinal. It is defined as follows:

• oE0(κ) = κ

• oEη+1(κ) = {j : V → V | j is an elementary embedding with critical point κ and j(κ) ⊆ oEη(κ)}

• oEη(κ) = ∪{oEα(κ) | α < η} for limit ordinals η

Intuitively, the extender hierarchy on κ is a hierarchy of embeddings j: V → V that are "more and more powerful" as η increases. At each level η, the set oEη(κ) contains all the embeddings j: V → V with critical point κ that are "at least as powerful" as all the embeddings in the lower levels of the hierarchy.

A Reinhardt cardinal is an infinite cardinal κ such that there exists an embedding j: V → V with critical point κ such that j(κ) is a subset of κ and for all ordinals η < κ, j(η) is an element of the set oE−η(j(κ)). This definition is somewhat technical, but the idea behind it is that a Reinhardt cardinal is an extremely large cardinal that is "closed off" from the rest of the universe of set theory in a certain sense. Specifically, the embedding j "shields" the cardinal κ from the rest of the universe in such a way that no ordinal below κ can "see" or "access" κ in any way.


I wrote a slightly more detailed version of the "missing" explanation you see. But if you want, I can explain it in the equations I used here.
 
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Achille said:
What I wrote and said (especially the first sentence in point 1) seems somewhat irrelevant.
What I call inconsistent is that I can't quite see the part where the Reinhardt cardinal proves to be the strongest. So I apologize for expressing myself wrong in the first place.
Also, if you're saying that the explanation I just made is "incomplete", I'll make a statement in the name of "hierarchy". Can you tell me if I'm wrong? Thanks in advance.



Reinhardt cardinals are a type of large cardinal axiom in set theory that are defined in terms of certain embeddings of the universe of set theory into itself. Specifically, a Reinhardt cardinal is an infinite cardinal κ such that there exists an embedding j: V → V (where V is the universe of set theory) such that j(κ) is a subset of κ and for all ordinals η < κ, j(η) is an element of the set oE−η(j(κ)). Here, oE−η(j(κ)) denotes the ηth level of the extender hierarchy on j(κ).

The extender hierarchy is a structure in set theory that is used to construct certain large cardinal axioms, including the Reinhardt cardinal. It is defined as follows:

• oE0(κ) = κ

• oEη+1(κ) = {j : V → V | j is an elementary embedding with critical point κ and j(κ) ⊆ oEη(κ)}

• oEη(κ) = ∪{oEα(κ) | α < η} for limit ordinals η

Intuitively, the extender hierarchy on κ is a hierarchy of embeddings j: V → V that are "more and more powerful" as η increases. At each level η, the set oEη(κ) contains all the embeddings j: V → V with critical point κ that are "at least as powerful" as all the embeddings in the lower levels of the hierarchy.

A Reinhardt cardinal is an infinite cardinal κ such that there exists an embedding j: V → V with critical point κ such that j(κ) is a subset of κ and for all ordinals η < κ, j(η) is an element of the set oE−η(j(κ)). This definition is somewhat technical, but the idea behind it is that a Reinhardt cardinal is an extremely large cardinal that is "closed off" from the rest of the universe of set theory in a certain sense. Specifically, the embedding j "shields" the cardinal κ from the rest of the universe in such a way that no ordinal below κ can "see" or "access" κ in any way.


I wrote a slightly more detailed version of the "missing" explanation you see. But if you want, I can explain it in the equations I used here.
Click to expand...
Some definitions are a bit off but seems fine overall. (Not really but you get what I mean right? I'm too lazy) I wouldn't say this is the official definition or it's actually accurate but it's definitely similar see we call a cardinal k a reinhardt iff J(k)>k meaning J=/=Id this would mean that J is not equal to the identity. Further more assume we have J:V->V then J is a class property to big for k to capture meaning k can't reach J (any set sized properties as well actually) by ordinal exponentation (your definition might be a bit wrong since you said limits of η on a reinhardt=U since we create a ultrafilter V on U and we certainly don't limit it to k in fact we are not restricted to k+1 as well and we demand a full agreement over it.) so what am I gonna point out here? well mainly the fact that assuming we have formula( n ) as a basis with k being a reinhardt then k is a infinity for upper and lower limits (sigma-1 arithmetic) another being that assume the P(J)+ZFC then the critical point must be on the ordinal exponentation (which you defined a bit similar but I would more or less define it as lambda=kn>alpha) Others I don't have much comment on since I won't be bother on analysis too much but k being inaccessible is indeed right as well since by reflecting upwards we see that j( y )=y with y being a reinhardt satisfies inaccessible properties. Also to clarify super reinhardts are trivially reinhardts but what you just defined is a reinhardt which was not what we were talking about, I can probably just send a doc about this cardinals since this cardinals was more or less used to prove consistencies of a reinhardt and etc unless if you already know them but you seem to tell me you just heard of them.
 
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amen.gif
 
Cat275 said:
Seems fine. I wouldn't say this is the official definition or it's actually accurate but it's definitely similar see we call a cardinal k a reinhardt iff J(k)>k meaning J=/=Id this would mean that J is not equal to the identity. Further more assume we have J:V->V then J is a class property to big for k to capture meaning k can't reach J (any set sized properties as well actually) by ordinal exponentation (your definition might be a bit wrong since you said limits of η on a reinhardt=U since we create a ultrafilter V on U and we certainly don't limit it to k in fact we are not restricted to k+1 as well and we demand a full agreement over it.) so what am I gonna point out here? well mainly the fact that assuming we have formula( n ) as a basis with k being a reinhardt then k is a infinity for upper and lower limits (sigma-1 arithmetic) another being that assume the P(J)+ZFC then the critical point must be on the ordinal exponentation (which you defined a bit similar but I would more or less define it as lambda=kn>alpha) Others I don't have much comment on since I won't be bother on analysis too much but k being inaccessible is indeed right as well since by reflecting upwards we see that j( y )=y with y being a reinhardt satisfies inaccessible properties. Also to clarify super reinhardts are trivially reinhardts but what you just defined is a reinhardt which was not what we were talking about, I can probably just send a doc about this cardinals since this cardinals was more or less used to prove consistencies of a reinhardt and etc unless if you already know them but you seem to tell me you just heard of them.
Click to expand...
Thanks for taking the time to reply. May I ask a few questions on this subject?

• Given an embedding j: V → V with critical point κ, how can we formally define the condition that j(κ) is a subset of κ?

• How can we express the condition that for all ordinals η < κ, j(η) is an element of the set oE−η(j(κ)) in mathematical notation?
 
Achille said:
Thanks for taking the time to reply. May I ask a few questions on this subject?

• Given an embedding j: V → V with critical point κ, how can we formally define the condition that j(κ) is a subset of κ?

• How can we express the condition that for all ordinals η < κ, j(η) is an element of the set oE−η(j(κ)) in mathematical notation?
Click to expand...
I think question 1. can be answered as "j(κ) ⊆ κ".
But I'm not entirely sure about 2. questions, I think the answer is "∀η < κ, j(η) ∈ oE−η(j(κ))" but I'm not entirely sure. What do you think?
 
Achille said:
Thanks for taking the time to reply. May I ask a few questions on this subject?

• Given an embedding j: V → V with critical point κ, how can we formally define the condition that j(κ) is a subset of κ?

• How can we express the condition that for all ordinals η < κ, j(η) is an element of the set oE−η(j(κ)) in mathematical notation?
Click to expand...
Should be the opposite since J is not equal to the identity meaning that if the critical point of j is k then J(k) is inequal but the formulae still holds so it's more or less like k is a ordinal of V and J(k) is the ordinal of the model agreed upon (which is both preserved in J) but more or less we define it like this for every ordinal alpha=j(alpha) then k is the critical point iff k<J(k) this means that k is not mapped to itself while also not ignoring the fact that it's supposed to be the smallest ordinal.

For number 2 are you asking for a formal proof with mathematical notations?
 
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Achille said:
I think question 1. can be answered as "j(κ) ⊆ κ".

But I'm not entirely sure about 2. questions, I think the answer is "∀η < κ, j(η) ∈ oE−η(j(κ))" but I'm not entirely sure. What do you think?
Click to expand...
correct notation yes. for the number one though it's probably inconsistent because it's supposed to be inequal to the identity.
 
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