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Normally how are cardinals portrayed in tier 0 verses?

Greatsage13th

Greatsage13th

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Because I'm pretty sure no one in their right mind would say that there are a Berkeley cardinal amount of universes lying around.
 
Epsilon_R said:
Tf is Berkeley? 😭
Click to expand...
It's a large cardinal.


Greatsage13th said:
Because I'm pretty sure no one in their right mind would say that there are a Berkeley cardinal amount of universes lying around.
Click to expand...
That's true, they wouldn't. BUT, when we introduce a large cardinal, particularly of the upper bound (above Mahlo and whatnot), we use elementary embeddings which are a class language, making it so a transitive set maps to another to define its parameters. So for example the infamous j: V→V, is built on the idea that the entire universe itself is divided into a first order formula: φ (a1, a2……….,an), which requires collection and separation schemes, meaning that any definable subclass of a set is a set, and the image of the domain set under the definable class function falls inside a set (recursion but without the axiom of choice). The importance of this, is that now every element a1,a2......,an is a class. Granting a constructible forming of the set theoretic universe.

Likewise, the reflection principle in second order language allows for a potentialist version of the universe and as such because the proposition allows for the universe to be absolutely infinite, and as such an inaccessible cardinal is reflected off something larger, a Mahlo, and a mahlo something larger, so on and so forth. So in summary, large cardinals are both related to size, and property, we define all large cardinals to be a Class, an object larger than a set, and in accordance we can apply these to the universe in particular models.
 
Guardian_Doge said:
It's a large cardinal.



That's true, they wouldn't. BUT, when we introduce a large cardinal, particularly of the upper bound (above Mahlo and whatnot), we use elementary embeddings which are a class language, making it so a transitive set maps to another to define its parameters. So for example the infamous j: V→V, is built on the idea that the entire universe itself is divided into a first order formula: φ (a1, a2……….,an), which requires collection and separation schemes, meaning that any definable subclass of a set is a set, and the image of the domain set under the definable class function falls inside a set (recursion but without the axiom of choice). The importance of this, is that now every element a1,a2......,an is a class. Granting a constructible forming of the set theoretic universe.

Likewise, the reflection principle in second order language allows for a potentialist version of the universe and as such because the proposition allows for the universe to be absolutely infinite, and as such an inaccessible cardinal is reflected off something larger, a Mahlo, and a mahlo something larger, so on and so forth. So in summary, large cardinals are both related to size, and property, we define all large cardinals to be a Class, an object larger than a set, and in accordance we can apply these to the universe in particular models.
Click to expand...
so cardinals are basically just a massive set of sets of values ?
 
imo, if someone really make feats based on single dialog that say "destroying Berkeley cardinal amount of universes" yeah the story would be sucks. But should be noted that we only observe and compile the feats regardless how bad the story is.

and also i think simply saying "destroying Berkeley cardinal amount of universes" would not grants any solid tier, probably at best only "unknown, possibly tier 0"

imho the best way to portray cardinal cosmology would be establishing the connection of mathematical theory to the cosmology first perhaps the easiest one through mathematical realism or platonism. So we know that the mathematical concept on that verse is real conceptually, narratively, and factually.
 
Greatsage13th said:
very confused. what does the set of a cardinal actually contain/encompass
Click to expand...
It depends on the elements. Cardinals are just a way to express the cardinality.

You are confusing the general word cardinal to large cardinals and cardinals to properclasses.
 
Cat275 said:
It depends on the elements. Cardinals are just a way to express the cardinality.

You are confusing the general word cardinal to large cardinals and cardinals to properclasses.
Click to expand...
what are the elements and the proper classes?
 
This will be long.

Say Set A.

A = {1,2,3}

This 3 objects are the elements.

Classes are sets atleast the small ones, the proper ones are basically quantifiers that ranges over sets and not elements.
 
Cat275 said:
This will be long.

Say Set A.

A = {1,2,3}

This 3 objects are the elements.

Classes are sets atleast the small ones, the proper ones are basically quantifiers that ranges over sets and not elements.
Click to expand...
i see, so how are quantifiers expressed?
 
Greatsage13th said:
Axioms?
Click to expand...
Partially but not really.

We more or less build up a foundation on axioms from what I know, quantifiers are just a way to quantify the open formula's in the universe of discourse.

Honestly you can probably just go learn this in wikipedia or something.
 
I'm pretty sure you use a 2nd order formula there to express that well kinda.

You express a relation when using those. Quantifiers don't usually express 2nd order so not really a good way to say it.

Just think of it as there exists X.

This is a quantifier.

Although I guess that analogy of yours can work.

Anyways I think I won't be tagging around anymore.
 
Last edited:
Cat275 said:
I'm pretty sure you use a 2nd order formula there to express that well kinda.

You express a relation when using those. Quantifiers don't usually express 2nd order so not really a good way to say it.

Just think of it as there exists X.

This is a quantifier.

Although I guess that analogy of yours can work.
Click to expand...
2nd order?
Cat275 said:
Anyways I think I won't be tagging around anymore.
Click to expand...
i see
 
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