Consecutive values in set theory

[IDoNotCare]

5 minutes ago, wtf said:

Ok. But the article you linked is not about Cantor's opinion about absolute infinite being God. They don't mention it, not even in passing.

Anyway we're past talking about set theory so let me know if you have any questions about the technical side. I can't really speak to the issue of computing God, except to say that I don't believe God (by any definition) could be computable; because computability has well-known limitations, whereas God (by any definition) is unlimited. 

I found an amusing Quora thread on the subject. Can God solve the Halting problem? I would say yes; and that's why God can't be computable. 

https://www.quora.com/Can-God-solve-the-Halting-Problem

You speak of God 

https://en.wikipedia.org/wiki/Boltzmann_brain

Which cannot exist yet as there's too much heat in the universe to allow the vast majority of electrons to be entangled. 

If you want to understand what entanglement is, two particles create GWs in close proximity, their GWs propagate faster than the particles like electrons can separate. So they're always entangled because the gravitational ripples were equal and opposite and shared the same initial conditions. The problem is most electrons, like the electrons in our synapses, have not had time in the universe to all entangle. If they were all entangled than we would be able to see through walls similar to how you yourself can see through to the other side of your nose when looking at it from an angle with both eyes. A qudit is a single electron, a qubit is parallel operations between two electrons as they both carry information about the atomic orbitals they propagate from to other atomic orbitals and also communicate instantly with one another about their positions and states via entanglement, a qutrit is when three communicate. Anyway in a boltzmann brain imagine all the electrons in the universe are entangled, that's a qugooglit. And from it's perspective the processing power of it's thoughts are equal in a universe than is collapsing, so there is a possibility of a mind thinking backward in time, that could create us at the end of our evolution as opposed to the beginning.  

[studiot]

17 minutes ago, wtf said:

Ok. But the article you linked is not about Cantor's opinion about absolute infinite being God. They don't mention it, not even in passing.

Anyway we're past talking about set theory so let me know if you have any questions about the technical side. I can't really speak to the issue of computing God, except to say that I don't believe God (by any definition) could be computable; because computability has well-known limitations, whereas God (by any definition) is unlimited. 

I found an amusing Quora thread on the subject. Can God solve the Halting problem? I would say yes; and that's why God can't be computable. 

https://www.quora.com/Can-God-solve-the-Halting-Problem

+1 for hanging in  there.

I have been following this discussion with interest but have insufficient knowledge to contribute anything else.

[wtf]

5 hours ago, studiot said:

+1 for hanging in  there.

I have been following this discussion with interest but have insufficient knowledge to contribute anything else.

Thank you @studiot

I enjoy having an opportunity to talk about this stuff. I am not an expert in this material by any means. Most of this is right at my mathematical periphery.

I'll take the liberty to talk a little more. 

The idea of a cumulative hierarchy of sets is more general than the von Neumann universe. I should mention that for myself personally, I'm always astonished to know that John von Neumann, who worked on the hydrogen bomb, invented mathematical game theory, worked on quantum physics and the theory of computing; also went so very deep into foundations and set theory. His intellect was incredible. 

Another cumulative hierarchy is Gödel's constructible universe. Gödel defined some sets [math]L_\alpha[/math] in the same manner as in the von Neumann construction. But where von Neumann uses powersets to go up levels; Gödel restricts [math]L_{\alpha + 1}[/math] to be exactly the collection of sets that are first-order definable with parameters; where those parameters are allowed to range over [math]L_\alpha[/math]. Then [math]L[/math] is the union of them all. 

"First-order definable with parameters" is a technical term in mathematical logic that more or less means that if you have a set, and you have a legal first-order formula of ZF, and you allow the quantifiers of that formula to range over that set; you can form another set that has been "first-order defined." Such a set satisfies constructivist urges that in order for a thing to exist, that thing must be capable of being written down or computed. The definable sets are a superset of the computable sets. Chaitin's Omega is definable but not computable. 

The resulting structure [math]L[/math] is a proper class in which:

* All the axioms of ZF are true; [* confused about this]

* The axiom of choice is true;

* And the Continuum hypothesis is true.

In other words, [math]L[/math] is a model of ZF (a proper class model, not a set) that is also a model of ZF + AC + CH. 

By Gödel's completeness theorem -- the one he did before his famous incompleteness theorem -- which says that a (first-order) theory has a model if and only if it is consistent, meaning free of contradiction; it must be the case that ZF + AC + CH is free of contradiction (if ZF is). 

This is how Gödel proved the consistency of AC and CH: by exhibiting a model of ZF in which they're both true. 

There's one point I don't follow. I always thought a model is required to be a set. So I'm not sure what lets Gödel apply his completeness theorem to a proper class. He must have proved it does apply. He was Gödel, I'm sure he figured this out.

Uh oh. I found something else I don't understand. I've always understood that these are relative consistency proofs; that is, AC and CH don't introduce any contradictions if ZF doesn't have any. But now I just said that [math]L[/math] is a model of ZF, in which case we just proved ZF is consistent. And we have NO such proof of the consistency of ZF. I have no insight into this problem at the moment.

 

 

A question that comes up is, why not just declare [math]L[/math] to be the universe of sets? If [math]V[/math] is the von Neumann universe, the question is: Is [math]V = L[/math]? If we accept [math]V = L[/math] as an axiom, we get AC and CH for free. And since virtually all of modern mathematics can in fact be done in [math]L[/math], why don't we just accept this as an axiom and be done with it?

The claim that [math]V = L[/math] is called the axiom of constructibility.

There are a number of reasons that most working set theorists do NOT believe that [math]V = L[/math]. I was hoping the Wiki article would list them but evidently not.  Found something ... Penelope Maddy discussing the question. It's on JSTOR which is a friendly academic paywall. If you register a free account you can read papers online even if you can't download them. 

https://www.jstor.org/stable/2275321?seq=1

Found a discussion on Mathoverflow on the topic. Working mathematicians kicking the question around. Some very understandable and interesting points. Such as, learning to use ZFC is relatively easy; but working in [math]ZFC + V = L[/math] is hard! You have to do some logic-level operations to prove things that are easy to prove in ZFC. So [math]V = L[/math] fails the criterion that a foundation should make it easy to do math! This is a utilitarian consideration. If one is a set-theoretic Platonist and thinks there is a "true" universe of sets, that's not much of an argument. So this goes right to the question of what a foundation for math should be. 

[The notation ZFC + V = L means the theory ZFC, plus the axiom V = L. It looks funny because it is definitely NOT an equation even though it contains an equal sign!]

https://mathoverflow.net/questions/331956/why-not-adopt-the-constructibility-axiom-v-l

ps -- Should note that in this context constructible = first-order definable and this has nothing to do with the usage in geometry of sets in the plane constructible by compass and straightedge].

 

 

 

 

Edited August 28, 2020 by wtf

[wtf]

ps - Lost my edit window but have to mention that I'm very fuzzy on what "first-order definable with parameters" actually means. According to Wiki, the parameters are not the same as the variables that range over the previous stage. One more thing I'm hazy on here. 

https://en.wikipedia.org/wiki/Constructible_universe#What_L_is

Edited August 28, 2020 by wtf

[Simmer]

Ok I need a break 

Edited August 30, 2020 by Simmer

[Simmer]

What does it mean the variable you're using to stand for elements vs the rule that tells you what those elements are?

[wtf]

4 hours ago, Simmer said:

What does it mean the variable you're using to stand for elements vs the rule that tells you what those elements are?

Can you give an example? What are you referring to? Your question is unclear.

[Simmer]

https://ncatlab.org/nlab/show/von+Neumann+hierarchy

 

This explains it quite nicely

 

but what is V and just V

what is a from Va

 

are there multiple answers? In which case what remains the same?

Edited September 13, 2020 by Simmer

[Simmer]

I’m having a hard time with how ordinal numbed come into play

I understand V it’s will. X is power. a is the coordination amongst those...

[wtf]

4 hours ago, Simmer said:

https://ncatlab.org/nlab/show/von+Neumann+hierarchy

 

This explains it quite nicely

They gave the exact same definition that's in the Wiki article I linked. But if you found this helpful that's fine. The Wiki article has a lot more explanatory context and you might find it helpful.

 

 

4 hours ago, Simmer said:

but what is V and just V

[math]V[/math] is the union of all the [math]V_\alpha[/math]'s. That is, you define [math]V_0[/math] and toss its contents into [math]V[/math]. Then you define [math]V_1[/math] and toss its contents into [math]V[/math]. You keep going like that for all the [math]V_\alpha[/math]'s. So [math]V[/math] is the collection of all sets that appear in [math]V_\alpha[/math] for some ordinal [math]\alpha[/math]. [math]V[/math] is not a set, it's a proper class.

 

 

4 hours ago, Simmer said:

what is a from Va

[math]\alpha[/math] an ordinal number. 

 

4 hours ago, Simmer said:

are there multiple answers? In which case what remains the same?

No, there is only one [math]V[/math].

 

 

3 hours ago, Simmer said:

I’m having a hard time with how ordinal numbed come into play

The ordinals index the [math]V_\alpha[/math]'s. You might find it helpful to work out for yourself the first few of them. What is [math]V_0[/math]? What is [math]V_1[/math]? What is [math]V_2[/math]? I believe the Wiki article actually shows a picture of these first few [math]V_\alpha[/math]'s.

 

 

3 hours ago, Simmer said:

I understand V it’s will. X is power. a is the coordination amongst those...

"I understand V it's will." does not parse. I could not understand what you are writing. 

What is X? 

[math]\alpha[/math] is an ordinal.

Edited September 13, 2020 by wtf

[Simmer]

How do you define a Greek letter used in set theory 

[Simmer]

Does anyone have a complete list of letters, symbols, ect used in set theory?

Edited September 14, 2020 by Simmer

[Simmer]

I don’t understand the superscript and subscript 

Edited September 14, 2020 by Simmer

[wtf]

5 hours ago, Simmer said:

How do you define a Greek letter used in set theory 

It's traditional that [math]\alpha[/math] and [math]\beta[/math] are used for ordinals, and lower-case kappa, [math]\kappa[/math], is used for a cardinal. But any paper or book you read will define the symbols they use.

 

 

 

4 hours ago, Simmer said:

Does anyone have a complete list of letters, symbols, ect used in set theory?

A Google search on the phrase, "symbols used in set theory" turned up 454,000,000 results; the first few of which are:

https://www.rapidtables.com/math/symbols/Set_Symbols.html

http://www.math.wsu.edu/faculty/martin/Math105/NoteOutlines/section0201.pdf

https://byjus.com/maths/set-theory-symbols/

https://castle.eiu.edu/~mathcs/mat2120/index/set03-2x3.pdf

https://www.onlinemath4all.com/symbols-used-in-set-theory.html

https://en.wikipedia.org/wiki/Glossary_of_set_theory

and

https://faculty.math.illinois.edu/~hildebr/347.summer19/settheory.pdf

@Simmer, It's one thing for me to make an effort to explain a little bit of advanced set theory to you. But when you ask me to do your web searches for you, that is the kind of thing that might tend to make me wonder why I persist here. Can you see that? 

 

 

3 hours ago, Simmer said:

I don’t understand the superscript and subscript 

This is the same question you asked on Aug 22 here:

 https://www.scienceforums.net/topic/122885-consecutive-values-in-set-theory/?do=findComment&comment=1151059

And I answered that same day, here: https://www.scienceforums.net/topic/122885-consecutive-values-in-set-theory/?do=findComment&comment=1151085, pointing out that the notation you asked about was defined in the very next sentence of the paper you linked.

That is, the notation [math]\Phi^{V_\alpha}[/math] denotes the sentence of set theory [math]\Phi[/math], relativized to the set  [math]V_\alpha[/math].

The core problem is that you lack the technical background to read this paper or to understand the answers to the questions you're asking. The material is beginning graduate-level set theory. Relativization means that as you build up a hierarchy of sets, at each stage you restrict quantifiers to the levels already constructed. This is a technical concept and you have to build up to it by studying exactly what they mean by a sentence of set theory, what's a model, what's a cumulative hierarchy, and so forth. 

Here is a pdf of Kunen, Set Theory: An Introduction to Independence Proofs. 

https://pdfs.semanticscholar.org/8929/ab7afdb220d582e9880b098c23082da8bc0c.pdf

You could learn a lot by starting at page 1 and working through this book at the rate of two or three days per page. It's a tough book. But I know of no other way to learn this material. As Euclid said, "There is no royal road to geometry." 

There are other grad-level set theory books out there that you can find by Googling. There's simply no other way to learn this material. Wikipedia doesn't even have a page on relativization. I found a concise definition on Proof-Wiki:

https://proofwiki.org/wiki/Definition:Relativisation

On that page they define the superscript notation. But it's not very helpful without understanding the context. The best I could find is this Wiki page on Absoluteness: https://en.wikipedia.org/wiki/Absoluteness, but it's heavy going and pretty much doesn't address the superscript notation at all.

The bottom line is you're trying to do brain surgery without knowing how to put a band-aid on a paper cut. You have to learn some of the basics.

But even so, in my earlier posts I've done my best to explain to you what relativization is. You have some formula of set theory [math]\Phi[/math], which has a bunch of quantifiers in it. Then you have some set of interest; and you relativize [math]\Phi[/math] by restricting the quantifiers to the given set. But to make sense of that description you have to know some mathematical logic and a little bit of advanced set theory; without which you will miss this explanation that's already in the paper you linked, and which I've explained several times. 

It's just a matter of prerequisites and background. 

 

 

 

Edited September 14, 2020 by wtf

[Simmer]

I understand all that

 

but is there a formula for V and for a

[Simmer]

Found V

 

V = ⋃_αV_α

 

_{now for the a}

_yeayea

[Simmer]

Does anyone know a formula to lowercase a?

[wtf]

34 minutes ago, Simmer said:

Does anyone know a formula to lowercase a?

[math]\alpha[/math] is a variable that ranges over the class of ordinal numbers. It doesn't have a formula. It's a variable that ranges over all the ordinals. It's used to index the sets [math]V_\alpha[/math].

 

  

1 hour ago, Simmer said:

 

V = ⋃_αV_α

Good. A more precise notation is

[math]\displaystyle V = \bigcup_{\alpha \in Ord} V_\alpha[/math] where [math]Ord[/math] is the class of ordinal numbers.

By the way I found the definition plus a detailed discussion of relativization in the Kunen link I gave earlier, starting on page 112. The bad news is that it's not self-contained. He refers to concepts discussed earlier in the previous 111 pages. And it's tough going. But at least it's a detailed definition. 

 

 

Edited September 17, 2020 by wtf

[Simmer]

https://en.m.wikipedia.org/wiki/Glossary_of_set_theory
 

The almost disjointness number, the least size of a maximal almost disjoint family of infinite subsets of ω
 

can you translate this to English?

Edited September 17, 2020 by Simmer

[wtf]

13 minutes ago, Simmer said:

https://en.m.wikipedia.org/wiki/Glossary_of_set_theory
 

The almost disjointness number, the least size of a maximal almost disjoint family of infinite subsets of ω
 

can you translate this to English lol?

No, "almost disjointness" is a concept from advanced set theory. It's defined on page 47 of the Kunen link I've provided you. It relates to the subject of infinitary combinatorics, although the (not very good) Wiki article doesn't mention the term. 

But 'a' in this context has NOTHING AT ALL TO DO with the variable [math]\alpha[/math] that ranges over the ordinals in the context we're discussing. You're running yourself around in circles. 

I suggest two things you could do here:

1) Work out for yourself the contents of the sets [math]V_0[/math], [math]V_1[/math], [math]V_2[/math],  and [math]V_3[/math]. You'll find this very educational.

2) Learn about ordinal numbers.

 

Edited September 17, 2020 by wtf

[Simmer]

I don’t know what it is about V0, V1, V2,  and V3. I don’t understand 

[wtf]

29 minutes ago, Simmer said:

I don’t know what it is about V0, V1, V2,  and V3. I don’t understand 

Wiki has a picture of these sets.

https://en.wikipedia.org/wiki/Von_Neumann_universe

Are you saying you don't understand the definition of the sets? Or you don't understand why I think you don't understand? I don't know if we're making any progress here.

Edited September 17, 2020 by wtf

[Simmer]

Well

V = ⋃αVα

and 

Va=⋃b<aP(Vb)

so what is a

 

(https://en.wikipedia.org/wiki/Von_Neumann_universe)

Edited September 17, 2020 by Simmer

[wtf]

9 minutes ago, Simmer said:

Well

V = ⋃αVα

and 

Va=⋃b<aP(Vb)

so what is a

 

(https://en.wikipedia.org/wiki/Von_Neumann_universe)

Where are you getting the English lower-case 'a' from? 

The [math]\alpha[/math] in [math]V_\alpha[/math] is the lower-case Greek letter alpha. In this particular context it ranges over the class of ordinal numbers, serving as an index variable. I've said this many times and that Wiki page has a picture of the first few [math]V_\alpha[/math]'s so I wonder which part is confusing you. 

Edited September 17, 2020 by wtf

[Simmer]

Sorry I was using it for alpha

what is it called, ordinalizing?

 

like beta is bounding, alpha is ordinalizing

Edited September 17, 2020 by Simmer