Set theory discussion thread

[Cat275]

Making this thread for people that want to talk about set theory or anything related or similar to it.

 

[Lilixa]

Oh hello Cat. I'm very very interested with this kind of thread.

Ok so basically i have several gap in my knowledge about mathematics in general which i want to cover in this thread (Hopefully)

So can we discuss about the absolute basic of set theory and mathematics in general ?

btw i want to learn set theory with these books. are these good set theory books ?

Naive Set Theory

Introduction to Set Theory, Revised and Expanded (Chapman & Hall/CRC Pure and Applied Mathematics)​

and also as we know that using mathematics simply will not grant any tier. Now i am wondering what is proper hypothetical scenario or example that using Set theory will grant any tier ?

 

[Cat275]

Oh hello Cat. I'm very very interested with this kind of thread.

Ok so basically i have several gap in my knowledge about mathematics in general which i want to cover in this thread (Hopefully)

So can we discuss about the absolute basic of set theory and mathematics in general ?

Sure, actually someone I know made this doc explaining the basics of set theory in a linguistic manner and without going to deep into it.

btw i want to learn set theory with these books. are these good set theory books ?

Naive Set Theory

Introduction to Set Theory, Revised and Expanded (Chapman & Hall/CRC Pure and Applied Mathematics)​

I may not be the best person to ask when it comes to recommendations for this sort of things.

and also as we know that using mathematics simply will not grant any tier. Now i am wondering what is proper hypothetical scenario or example that using Set theory will grant any tier ?

I'd say something like a type 4 multiverse or an absolute infinite is a good example.

 

[NHTkenshin2]

The hell is a Berkeley cardinal and how far is it into tier 0 (physically)
ELI5 pls

 

[Oblivion_Of_The_Endless]

The hell is a Berkeley cardinal and how far is it into tier 0 (physically)
ELI5 pls

Yes.

 

[Cat275]

The hell is a Berkeley cardinal and how far is it into tier 0 (physically)

Let me quote some stuff 1st

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.

And then:

A Berkeley cardinal is a cardinal κ in a model of Zermelo–Fraenkel set theory with the property that for every transitive set M that includes κ and α < κ, there is a nontrivial elementary embedding of M into M with critical point below κ and j(α) = α.[1] Berkeley cardinals are a strictly stronger cardinal axiom than Reinhardt cardinals, implying that they are not compatible with the axiom of choice.

And then here:

A weakening of being a Berkeley cardinal is that for every binary relation R on Vκ, there is a nontrivial elementary embedding of (Vκ, R) into itself. This implies that we have elementary

j1, j2, j3, ...j1: (Vκ, ∈) → (Vκ, ∈),j2: (Vκ, ∈, j1) → (Vκ, ∈, j1),j3: (Vκ, ∈, j1, j2) → (Vκ, ∈, j1, j2),

I won't go into much details but this means that a berkeley cardinal can not be used by trivialism.

And like the reinhardt cardinal (V→V) the berkely cardinal is really just a sequence of Vk's being a member of each other.

And really that's the basics of being a berkeley cardinal.

As for how far it is in tier 0 look here:

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).

So probably Omega Omega to kappa for the least berkeley cardinal.

So the least isn't that high into tier 0 as far as i know in fact it is probably not even tier 0 in size.

 

[ActuallySpaceMan42]

Welp looks like I'm reading up on set theory so I can actually engage in conversations.

 

[Cat275]

Bump.

 

[Lilixa]

i am still reading set theory books, i will give some discussion later

 

[Cat275]

Alright, I guess i'll bump this thread in like 3 days or something.

 

[Cat275]

The hell is a Berkeley cardinal and how far is it into tier 0 (physically)
ELI5 pls

I'm having a little ocd somehow so i'll go over this again with slightly more details for a bit of an overview.

1st this are the weakenings of berkeley cardinals.

A weakening of being a Berkeley cardinal is that for every binary relation R on Vκ, there is a nontrivial elementary embedding of (Vκ, R) into itself. This implies that we have elementary

j1, j2, j3, ...j1: (Vκ, ∈) → (Vκ, ∈),j2: (Vκ, ∈, j1) → (Vκ, ∈, j1),j3: (Vκ, ∈, j1, j2) → (Vκ, ∈, j1, j2),
and so on. This can be continued any finite number of times, and to the extent that the model has dependent choice, transfinitely. Thus, plausibly, this notion can be strengthened simply by asserting more dependent choice.

Now with the assumption that Vκ∈Vκ is non isomorphic and is stronger than a non trivial elementary embedding of V→V in the critical point of J and with this being required for the weakenings:

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.

Then we can assume that this^

Is greater than a strongly axiom of infinity that is equal to or formed by V→V.

Quoting on how strong V→V is:

We suggest a new approach for addressing the problem of establishing an axiomatic foundation for large cardinals. An axiom asserting the existence of a large cardinal can naturally be viewed as a strong Axiom of Infinity. However, it has not been clear on the basis of our knowledge of ω itself, or of generally agreed upon intuitions about the true nature of the mathematical universe, what the right strengthening of the Axiom of Infinity is—which large cardinals ought to be derivable? It was shown in the 1960s by Lawvere that the existence of an infinite set is equivalent to the existence of a certain kind of structure-preserving transformation from V to itself, not isomorphic to the identity. We use Lawvere's transformation, rather than ω, as a starting point for a reasonably natural sequence of strengthenings and refinements, leading to a proposed strong Axiom of Infinity. A first refinement was discussed in later work by Trnková—Blass, showing that if the preservation properties of Lawvere's transformation are strengthened to the point of requiring it to be an exact functor, such a transformation is provably equivalent to the existence of a measurable cardinal. We propose to push the preservation properties as far as possible, short of inconsistency. The resulting transformation V→V is strong enough to account for virtually all large cardinals, but is at the same time a natural generalization of an assertion about transformations V→V known to be equivalent to the Axiom of Infinity.

Meaning that V→V implies the existence of large cardinals below choice.

So going with the idea that this consequences of Vκ is a member of Vκ's are possible and not false we can create as much sequence we can with this as long as it doesn't become to big and to much for the dependant choice. so we can think of berkeley cardinals to be so big (or strong since size is arbitary) that super reinhardt/just reinhardt cardinals can not reach it.

(Atleast I think so since we use the notion of dc to reach a super reinhardt cardinal and we see that Dcλ has λ as the least fixed point above the critical point of reinhardt cardinals so im overall just assuming that we can probably create and assert a sequence of λλ/λ+λ Dc's or Dcλλ that is not isomorphic to Dcλ and even more constructive notions below Dcλλ on a short sequence of Vκ's→Vκ's→,,, as the weakenings of berkeley cardinals and extend the sequence even more.

It's also because we use kappa on a super reinhardt to form a reflection of reinhardt by the von hierarchy (we specifically use Vγ and P(Vγ)/Vγ+1 here) and we use delta-0 on a berkeley to strongly reflect a reinhardt. and by strict definition Vδ>Vκ in the rankings of the von neumann hierarchy iirc. but really this is pretty much just me assuming things on what is bigger between a super reinhardt or berkeley cardinal, although we can indeed conclude that Berkeley>Super reinhardt in strength consistency however notion of strength consistency ≠ notion of size.

So the overall conclusion on what is bigger between the two cardinals is just arbitary in my opinion.)

So overall i think defining how high berkeley is into tier 0 by layers of large cardinals below all variant of choice can't really define the actual difference here.

As for the least berkeley cardinal it really isn't tier 0 by any means, assuming that omega is it's cofinality then we can conclude that the least berkeley cardinal has the cardinality of somewhere around something like a regular kappa at best. (assuming it's singular)

For a more simple and comprehensive way to see how big/strong berkeley is compared to a reinhardt cardinal.

So essentially a berkeley cardinal can map a reinhardt cardinal.

And be stronger than it as the notion of γ < δ0 means that δ0 implies γ if and only if it is a berkeley cardinal + there exist a reinhardt cardinal.

So berkeley and many more cardinals can imply the existence of reinhardt with this described notion above.

 

[NHTkenshin2]

I'm having ocd somehow so i'll go over this again with slightly more details for a bit of an overview.

1st this are the weakenings of berkeley cardinals



Now with the assumption that Vk∈Vk is non isomorphic and is stronger than a non trivial elementary embedding of V→V in the critical point of J and the quoted words below are required for the weakenings then we can assume that this:



Is greater than a strongly axiom of infinity formed by V→V (reinhardt cardinals)

Quoting on how strong V→V is:

We suggest a new approach for addressing the problem of establishing an axiomatic foundation for large cardinals. An axiom asserting the existence of a large cardinal can naturally be viewed as a strong Axiom of Infinity. However, it has not been clear on the basis of our knowledge of ω itself, or of generally agreed upon intuitions about the true nature of the mathematical universe, what the right strengthening of the Axiom of Infinity is—which large cardinals ought to be derivable? It was shown in the 1960s by Lawvere that the existence of an infinite set is equivalent to the existence of a certain kind of structure-preserving transformation from V to itself, not isomorphic to the identity. We use Lawvere's transformation, rather than ω, as a starting point for a reasonably natural sequence of strengthenings and refinements, leading to a proposed strong Axiom of Infinity. A first refinement was discussed in later work by Trnková—Blass, showing that if the preservation properties of Lawvere's transformation are strengthened to the point of requiring it to be an exact functor, such a transformation is provably equivalent to the existence of a measurable cardinal. We propose to push the preservation properties as far as possible, short of inconsistency. The resulting transformation V→V is strong enough to account for virtually all large cardinals, but is at the same time a natural generalization of an assertion about transformations V→V known to be equivalent to the Axiom of Infinity.

Meaning that V→V implies the existence of large cardinals below choice.

So going with the idea that this consequences of Vk is a member of Vk's are possible and not false we can create as much sequence we can with this as long as it doesn't become to big for the dependant choice so we can think of berkeley cardinals to be so big (or strong since size is arbitary) that super reinhardt cardinals can not reach.
(Atleast im assuming so since we use the Dc to reach a super reinhardt cardinal.)

So i think defining how high berkeley is into tier 0 by layers of large cardinals below all variant of choice can't really define the actual difference here.

As for the least berkeley cardinal it really is not tier 0 by any means, assuming that omega is it's cofinality then we can conclude that the least berkeley cardinal has the cardinality of a regular kappa at best and if we imply that the least has the cofinality of kappa then it's cardinality only fits one of the bill of being the 1st inaccessible.

Holy hell fam yeah that explains it alot better
Thank you sm

 

[Cat275]

Bump.

Yes.

Wdym by yes?

 

[Oblivion_Of_The_Endless]

It's a joke

 

[Lilixa]

i am guess i am only participates in extension of temporal branch stories which are infinite hierarchy of transcendental nonexistence of transdual nonexistence which side byproduct of infinitesimal incomprehensible shadow of inaccessible large avatar of Icarus cardinals.

 

[Alexander]

the multiverse of type 4 would be something that would be tier 0, because it is so to speak, where all the mathematical forms are(like platonic concepts,that exist beyond and outside of reality and rule all reality itself), everything mathematical, which dictates even other cosmologies (such as 3,2,and 1) So, if there is something like infinite hierarchies, infinite divinities, and that infinity in more hierarchies, it would still be something incomparable to the type 4 multiverse.

 

[Cat275]

i am guess i am only participates in extension of temporal branch stories which are infinite hierarchy of transcendental nonexistence of transdual nonexistence which side byproduct of infinitesimal incomprehensible shadow of inaccessible large avatar of Icarus cardinals.

I'll get into that later since it's 3am here but can you rephrase that a little? Grammar is a bit sloppy sorry but im assuming we're gonna talk about icarus set later?

 

[Cat275]

the multiverse of type 4 would be something that would be tier 0, because it is so to speak, where all the mathematical forms are(like platonic concepts,that exist beyond and outside of reality and rule all reality itself), everything mathematical, which dictates even other cosmologies (such as 3,2,and 1) So, if there is something like infinite hierarchies, infinite divinities, and that infinity in more hierarchies, it would still be something incomparable to the type 4 multiverse.

Type 4 multiverse is when mathematics become so general that any TOE becomes a set of abstract mathematics, It's bassically the most simplest multiverse theory possible.

[A]n entire ensemble is often much simpler than one of its members. This principle can be stated more formally using the notion of algorithmic information content. The algorithmic information content in a number is, roughly speaking, the length of the shortest computer program that will produce that number as output. For example, consider the set of all integers. Which is simpler, the whole set or just one number? Naively, you might think that a single number is simpler, but the entire set can be generated by quite a trivial computer program, whereas a single number can be hugely long. Therefore, the whole set is actually simpler... (Similarly), the higher-level multiverses are simpler. Going from our universe to the Level I multiverse eliminates the need to specify initial conditions, upgrading to Level II eliminates the need to specify physical constants, and the Level IV multiverse eliminates the need to specify anything at all... A common feature of all four multiverse levels is that the simplest and arguably most elegant theory involves parallel universes by default. To deny the existence of those universes, one needs to complicate the theory by adding experimentally unsupported processes and ad hoc postulates: finite space, wave function collapse and ontological asymmetry.

Type 4 multiverse is simply modal realism.

Which is only L1-A (afaik) in the wiki which can go to tier 0 depending on the verse.

Anyways whats the topic 1st?

 

[Alexander]

Type 4 multiverse is when mathematics become so general that any TOE becomes a set of abstract mathematics, It's bassically the most simplest multiverse theory possible.

if it weren't for the fact that it is more complex and larger than type 3, which is high hyperversal

 

[Lilixa]

I'll get into that later since it's 3am here but can you rephrase that a little? Grammar is a bit sloppy sorry but im assuming we're gonna talk about icarus set later?

ohh sorry about that. I am still reading set theory books....

 

[NHTkenshin2]

if it weren't for the fact that it is more complex and larger than type 3, which is high hyperversal

Type 3 is low 1-C from what i heard

 

[Alexander]

Type 3 is low 1-C from what i heard

not really, the type 3 multiverse includes the hilbert space, and also the many world interpretation theory of quantum mechanics.

 

[NHTkenshin2]

not really, the type 3 multiverse includes the hilbert space, and also the many world interpretation theory of quantum mechanics.

Oh, i thought it was only many worlds

 

[Alexander]

Oh, i thought it was only many worlds

no, includes hilbert space

 

[Cat275]

ohh sorry about that. I am still reading set theory books....

I can send you a few pictures about icarus set if you'd like. (unless if you already have one)

Oh, i thought it was only many worlds

no, includes hilbert space

Tegmark argues that a Level III multiverse does not contain more possibilities in the Hubble volume than a Level I or Level II multiverse. In effect, all the different "worlds" created by "splits" in a Level III multiverse with the same physical constants can be found in some Hubble volume in a Level I multiverse. Tegmark writes that, "The only difference between Level I and Level III is where your doppelgängers reside. In Level I they live elsewhere in good old three-dimensional space. In Level III they live on another quantum branch in infinite-dimensional Hilbert space."

Here^.

 

[Cat275]

if it weren't for the fact that it is more complex and larger than type 3, which is high hyperversal

Going by this again, type 4 is more simple than type 3 because you don't feel the need to specify the sets.

It's simply more simple because of the fact that a class of sets is more simple than a set and a set is more simple than a element.

The more simple it is = the more larger the said universe is, by this notion:

[A]n entire ensemble is often much simpler than one of its members. This principle can be stated more formally using the notion of algorithmic information content. The algorithmic information content in a number is, roughly speaking, the length of the shortest computer program that will produce that number as output. For example, consider the set of all integers. Which is simpler, the whole set or just one number? Naively, you might think that a single number is simpler, but the entire set can be generated by quite a trivial computer program, whereas a single number can be hugely long. Therefore, the whole set is actually simpler... (Similarly), the higher-level multiverses are simpler. Going from our universe to the Level I multiverse eliminates the need to specify initial conditions, upgrading to Level II eliminates the need to specify physical constants, and the Level IV multiverse eliminates the need to specify anything at all... A common feature of all four multiverse levels is that the simplest and arguably most elegant theory involves parallel universes by default. To deny the existence of those universes, one needs to complicate the theory by adding experimentally unsupported processes and ad hoc postulates: finite space, wave function collapse and ontological asymmetry.

So by this definition type 4 is the simplest multiverse theory and by far not as complex as the lower types of tegmarks multiverse.

 

[NHTkenshin2]

Alright, me coming back at it with the large mathematical structures questions


What is the 0=1 axiom?

 

[Cat275]

Alright, me coming back at it with the large mathematical structures questions


What is the 0=1 axiom?

We try to achieve the strongest axiom possible with a reinhardt cardinal or V→V to account almost all large cardinals that are bound by the choice axiom. However reinhardt is still consistent to ZF so there are probably more stronger large cardinals or large cardinal properties that are implied to be inconsistent with ZF as a set. so we use this idea and put the inconsistency of 0=1 to the top of the large cardinal hierarchy to motivate and make a constructive notion of large cardinals being able to increase in strength consistency and eventually lead to an inconsistency right after.

In short 0=1 can prove everything by a inconsistent statement.

(And the inconsistency of such statement is 0 not being ≠ 1 as a more general example for this.)

 

[Oblivion_Of_The_Endless]

Alright, me coming back at it with the large mathematical structures questions


What is the 0=1 axiom?

To put in more simple words than the above. Its an axiom that assumes an inconsistency (number zero=number one) and which by virtue can prove essentially everything. Because if math breaks like that, you can also state 1 trillion = -10, Infinity = 1, Woodin Cardinal spaces = subatomic-sized, etc....

 

[Cat275]

We try to achieve the strongest axiom possible with a reinhardt cardinal or V→V to account almost all large cardinals that are bound by the choice axiom. However reinhardt is still consistent to ZF so there are probably more stronger large cardinals or large cardinal properties that are implied to be inconsistent with ZF as a set. so we use this idea and put the inconsistency of 0=1 to the top of the large cardinal hierarchy to motivate and make a constructive notion of large cardinals being able to increase in strength consistency and eventually lead to an inconsistency right after.

In short 0=1 can prove everything by a inconsistent statement.

(And the inconsistency of such statement is 0 not being ≠ 1 as a more general example for this.)

Well if this^ thing above is a bit complicated to understand for most. (atleast that's what the person above seems to be implying) then you can just think of 0=1 as the ceiling of large cardinals for a more simple and comprehensive interpretation.

And you can also check this thing, since it's related to the explanation the person said above.

 

[Alexander]

Going by this again, type 4 is more simple than type 3 because you don't feel the need to specify the sets.

It's simply more simple because of the fact that a class of sets is more simple than a set and a set is more simple than a element.

The more simple it is = the more larger the said universe is, by this notion:

The initial conditions and physical constants in the Level I, Level II and Level III multiverses can vary, but the fundamental laws that govern nature remain the same. Why stop there? Why not allow the laws themselves to vary? Welcome to the Level IV multiverse. You can think of what I'm arguing for as Platonism on steroids: that external physical reality is not only described by mathematics, but that it is mathematics. And that our physical world (our Level III multiverse) is a giant mathematical object in the Level IV multiverse of all matematical objects. I first started having ideas along these lines back in grad school around 1990, and have written several papers about it over the years.

what he is saying is that if our reality were a type 3 multiverse (high hyperversal) it would be just one more mathematical structure in the type 4 multiverse.
if I wanted to continue, the thing would be too long (I just saw a 31-page page of what the max tegmark multiverse is like and I don't have the time to live to read it all after reading 500 pages of Sofia's world)but in short,a type 4 multiverse is simpler but larger,but is more complex that in reality is,is like platonic concepts(exist outside of reality and dictated all reality)

 

[Cat275]

The initial conditions and physical constants in the Level I, Level II and Level III multiverses can vary, but the fundamental laws that govern nature remain the same. Why stop there? Why not allow the laws themselves to vary? Welcome to the Level IV multiverse. You can think of what I'm arguing for as Platonism on steroids: that external physical reality is not only described by mathematics, but that it is mathematics. And that our physical world (our Level III multiverse) is a giant mathematical object in the Level IV multiverse of all matematical objects. I first started having ideas along these lines back in grad school around 1990, and have written several papers about it over the years.

what he is saying is that if our reality were a type 3 multiverse (high hyperversal) it would be just one more mathematical structure in the type 4 multiverse.
if I wanted to continue, the thing would be too long (I just saw a 31-page page of what the max tegmark multiverse is like and I don't have the time to live to read it all after reading 500 pages of Sofia's world)but in short,a type 4 multiverse is simpler but larger

Not really sure what you're trying to prove here since I already agree with this in which I already implied in some of my previous texts.

(Simpler=Larger, Type 4 is the simplest multiverse theory simple as that.)

 

[Alexander]

Not really sure what you're trying to prove here since I already agree with this in which I already implied in some of my previous texts.

(Simpler=Larger, Type 4 is the simplest multiverse theory simple as that.)

ok,good bye*throws a ball at a door and get out*

 

[Lewis]

axiom 0=1 should use trivialism (for each p T p) philosophy to avoid paladox.

 

[Cat275]

Oh nice to see you here lewis but i already mentioned something similar right...

And you can also check this thing, since it's related to the explanation the person said above.

Here^, 0=1 is literally just an axiom used to prove that large cardinals have a limit of consistency.

 

[Lewis]

We should open a discussion about the Von Neumann–Bernays–Gödel set theory because I would like to know how to use it.

 

[Lewis]

Oh nice to see you here lewis but i already mentioned something similar right...

Here^, 0=1 is literally just an axiom used to prove that large cardinals have a limit of consistency.

Oh , i'm so sorry

 

[Cat275]

We should open a discussion about the Von Neumann–Bernays–Gödel set theory because I would like to know how to use it.

Sure, we're talking about the axiom that uses the notion of classes right? where do we start though?

Oh , i'm so sorry

Sure it's ok.

 

[Lewis]

Sure, we're talking about the axiom that uses the notion of classes right? where do we start though?

Sure it's ok.

Yes, Classes in set theory

 

[Lewis]

I found the proper class to be very strong if using axiom Vk to verify large cardinal .