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Set theory discussion thread
- Thread starter Cat275
- Start date
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- set theory
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Alright my biggest doubt is how a verse which functions on set theory fairs up against one which doesnt?
Like lets say
A verse with an Infinite Hierarchy , with the lowest layer functioning on lets say Mahlo Cardinals
Vs
A verse which at first has an structure of Inf Hiearchies (x) , then another structure whose each layer is made up of inf 'x' where the higher layer is unreachable even with inf amounts of 'x' ..Lets say this structure is 'y'
Such a loop goes on forever , as in there is an even higher structure where each layer had inf amounts of 'y'
Then lets say a structure beyond this Loop which is absolutely unreachable by any means of editing the other structures
...
Which verse scales higher??
From what iK
Inf Hiearchy - H1B/Low1A
Beyond it - 1A
Then even inf amounts of these hiearchies stacked on top wont reach H1A
So last question
Where could the second verse scale?
Like lets say
A verse with an Infinite Hierarchy , with the lowest layer functioning on lets say Mahlo Cardinals
Vs
A verse which at first has an structure of Inf Hiearchies (x) , then another structure whose each layer is made up of inf 'x' where the higher layer is unreachable even with inf amounts of 'x' ..Lets say this structure is 'y'
Such a loop goes on forever , as in there is an even higher structure where each layer had inf amounts of 'y'
Then lets say a structure beyond this Loop which is absolutely unreachable by any means of editing the other structures
...
Which verse scales higher??
From what iK
Inf Hiearchy - H1B/Low1A
Beyond it - 1A
Then even inf amounts of these hiearchies stacked on top wont reach H1A
So last question
Where could the second verse scale?
- 709
- 189
You can't use power set with mahlo because it's not proven to exist by zfc.
Last edited:
- 709
- 189
edit text
Cat275
He/Him- 1,428
- 789
- Thread starter
- #53
Not exactly sure. Maybe 0=1? This predicts every inconsistencies in large cardinal and more if i'm not wrong.
I should check later but I'm pretty sure there was a statement of the top of large cardinals.
(Which is additional proof of 0=1.)
And how one makes large cardinals.
I should check later but I'm pretty sure there was a statement of the top of large cardinals.
(Which is additional proof of 0=1.)
And how one makes large cardinals.
- 709
- 189
this is the message of experts He told me how NBG should be used
can you check for me?
Von Neumann universe
Abstract, Von Neumann universe is a hierarchy of non-empty sets, under the condition that, every sets existing need to contain at least one element in each, that is, not greater than any other elements within the set, namely a minimal element. The universe, is therefore a class because all sets contained are well-founded, thus they all share the same characteristics. In the formal system framework of NBG, extending from ZFC, V is also a proper class in which its generalization is indicated to On. We may define the hierarchy formally as: (Vα)α∈On.
Then there is the rank of well-founded sets within the hierarchy defined inductively as the smallest ordinal, which, is needed to be greater than all the members contained within certain set. Informally put, once more, the rank of the empty set is zero and each ordinal would have its rank equal to itself. Altogether, the hierarchy would be called the cumulative hierarchy. Elaborating out some instances, once more, under the framework of ZFC and NBG, it's a collection of sets Vα indexed by the class of ordinal numbers by the definition given above; Vα is a proper class having sets with the ranks less than α. Thus there is one distinct set for all ordinals α. Furthermore, we might define Vα inductively by a transfinite recursion;
Let there be Vα, and let V0 be the empty set: V0 := ∅.
For any ordinal number β, let Vβ+1 be the power set of Vβ: Vβ+1 := P(Vβ) (therefore will have the alephs and omegas unboundedly increasing)
For any limit ordinal λ, let Vλ be the union of all the V-stages so far, or all the V-stages are existing so far: Vλ := U(Vβ): β<λ
Now that we've done with the structural, we may define its stages; the sets Vα are called stages or ranks.
The class V is defined as the union of all V-stages: V := U(Vα)α
An equivalent definition sets: Vα := U(P(Vβ)) : β<λ
For each ordinal α, where P(X) is the powerset of X. The rank of a set S is the smallest α such that S ⊆ Vα.
A visual representation elaborating the finite stages, and the idea of the hierarchy goes as:
V0 = { },
V1 = {{ }},
V2 = {{ }, {{ }}},
V3 = {{},{{}},{{}}}
Which the sequence however fashionizes through hyper-4, commonly denoted by ↑↑, which, in plain word, under the definition of exponentiation, is a right-to-left exponentiation that are iterated of the n copies of a. Precisely, n is referred as the height of the function, while a is the base.
can you check for me?
Von Neumann universe
Abstract, Von Neumann universe is a hierarchy of non-empty sets, under the condition that, every sets existing need to contain at least one element in each, that is, not greater than any other elements within the set, namely a minimal element. The universe, is therefore a class because all sets contained are well-founded, thus they all share the same characteristics. In the formal system framework of NBG, extending from ZFC, V is also a proper class in which its generalization is indicated to On. We may define the hierarchy formally as: (Vα)α∈On.
Then there is the rank of well-founded sets within the hierarchy defined inductively as the smallest ordinal, which, is needed to be greater than all the members contained within certain set. Informally put, once more, the rank of the empty set is zero and each ordinal would have its rank equal to itself. Altogether, the hierarchy would be called the cumulative hierarchy. Elaborating out some instances, once more, under the framework of ZFC and NBG, it's a collection of sets Vα indexed by the class of ordinal numbers by the definition given above; Vα is a proper class having sets with the ranks less than α. Thus there is one distinct set for all ordinals α. Furthermore, we might define Vα inductively by a transfinite recursion;
Let there be Vα, and let V0 be the empty set: V0 := ∅.
For any ordinal number β, let Vβ+1 be the power set of Vβ: Vβ+1 := P(Vβ) (therefore will have the alephs and omegas unboundedly increasing)
For any limit ordinal λ, let Vλ be the union of all the V-stages so far, or all the V-stages are existing so far: Vλ := U(Vβ): β<λ
Now that we've done with the structural, we may define its stages; the sets Vα are called stages or ranks.
The class V is defined as the union of all V-stages: V := U(Vα)α
An equivalent definition sets: Vα := U(P(Vβ)) : β<λ
For each ordinal α, where P(X) is the powerset of X. The rank of a set S is the smallest α such that S ⊆ Vα.
A visual representation elaborating the finite stages, and the idea of the hierarchy goes as:
V0 = { },
V1 = {{ }},
V2 = {{ }, {{ }}},
V3 = {{},{{}},{{}}}
Which the sequence however fashionizes through hyper-4, commonly denoted by ↑↑, which, in plain word, under the definition of exponentiation, is a right-to-left exponentiation that are iterated of the n copies of a. Precisely, n is referred as the height of the function, while a is the base.
Cat275
He/Him- 1,428
- 789
- Thread starter
- #58
Sure. I'll only comment on the way Vα/cumulitive hierarchy is used though.this is the message of experts He told me how NBG should be used
can you check for me?
Agree with this.
I also agree with this since you can see this classification on wikipedia already.
I also agree with this. The way the cumulitive hierarchy is explained here seems to be accurate as you can see this on wikipedia already.
Where did you get (asked) the explanation?
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imagine the concept of infinite,how big will be?
Cat275
He/Him- 1,428
- 789
- Thread starter
- #73
I mean aleph 0 sorry, although that's not it's highest cardinality, look at an euclidean space that is R*R, this has uncountable infinite^2 null sets as far as I know, alternitively a set that has a omega amount of null set as a cofinality has the cardinality of kappa and etc.
- 5,460
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"Reality is the sum total of everything. Reality is all that exists, has existed, or will exist. Reality lacks boundaries. Reality includes that which is possible but does not exist, for if a thing can be conceived then it is already a thing. Reality includes the absence of a thing, for nothing is, in and of itself, something. By extension, Reality includes that which isn't possible.
This seems pretty self-explanatory. Reality is the basic instance from which everything derives, or perhaps the container in which everything resides. Everything is a subset of reality: you, me, this computer, the toast I had this morning, the Yankees, cancer, love, Cryptonomicon, Madagascar, and even the bee that stung my thumb when I was a kid.
This definition seems to remove the concept
of time from the equation. Upon consideration this simplifies the definition immensely. Simply put, if something can exist (or might have once existed), then it is a part of reality. This leaves room for possibility – what might exist if the future is variable, or what will exist if it isn't. Simultaneously, all of the possibilities from times past are accounted for, as opposed to "what actually happened” being the only thing that is real. For example, it was once possible that I die in a boating accident at the age of nine. I didn't, yet that still doesn't invalidate the possibility that I could have.
The reality of something that is absent makes sense. I think you're saying that if something could exist in reality at large if it were ever conceived of, but for whatever reason isn't, then there is a “placeholder" that exists for that thing – sort of a cosmic "null set” identifier. Even if all of reality occurs without that particular thing ever existing, there's still the null state placeholder for that thing: your fire-breathing cows, cars that on Jell-o, and so on.
Coupled with the concept of infinity, there must be an infinite number of null-sets as well. In English, there's no way that reality can ever "fill up."
GUIDE TO THE TRADITIONS – PAGE 46
Is this good enough?
This seems pretty self-explanatory. Reality is the basic instance from which everything derives, or perhaps the container in which everything resides. Everything is a subset of reality: you, me, this computer, the toast I had this morning, the Yankees, cancer, love, Cryptonomicon, Madagascar, and even the bee that stung my thumb when I was a kid.
This definition seems to remove the concept
of time from the equation. Upon consideration this simplifies the definition immensely. Simply put, if something can exist (or might have once existed), then it is a part of reality. This leaves room for possibility – what might exist if the future is variable, or what will exist if it isn't. Simultaneously, all of the possibilities from times past are accounted for, as opposed to "what actually happened” being the only thing that is real. For example, it was once possible that I die in a boating accident at the age of nine. I didn't, yet that still doesn't invalidate the possibility that I could have.
The reality of something that is absent makes sense. I think you're saying that if something could exist in reality at large if it were ever conceived of, but for whatever reason isn't, then there is a “placeholder" that exists for that thing – sort of a cosmic "null set” identifier. Even if all of reality occurs without that particular thing ever existing, there's still the null state placeholder for that thing: your fire-breathing cows, cars that on Jell-o, and so on.
Coupled with the concept of infinity, there must be an infinite number of null-sets as well. In English, there's no way that reality can ever "fill up."
GUIDE TO THE TRADITIONS – PAGE 46
Is this good enough?
