wtf

I don't know what you mean by "completeness" field. Can't even guess, I'm afraid. A commutative ring with unity in which every nonzero element is invertible is a field. Sometimes a "corpus" in French, I believe. A noncommutative ring with unity and multiplicative inverses is a division ring, called in older books a skew field. Same remark. Don't know what you mean by object or completeness field. An ordered field with the least upper bound property is complete. So the reals are complete but not the rationals, if that is what you mean. Don't know what you mean by objects, unless perhaps you mean sets. Or fields. Completeness field is a bad translation and I can't even make a guess as to what it is intended to mean. FWIW this is first-semester undergrad abstract algebra. Even the computer science majors need to know some of it, because finite fields are a big thing in CS. I agree that we're way past the original question.

If [math]x[/math] is nonzero then [math]x H[/math] is an ideal. If it's not a proper ideal then it's all of the ring, including 1. Hence every nonzero element is invertible. So a commutative ring with no proper ideals is a field. If the ring is not commutative a similar proof would show that it must be a division ring, but you'd have to check both left and right ideals for the proof. Note that our ring is required to have 1. If not, I'm not sure if this result is still true.

https://en.wikipedia.org/wiki/Field_extension#Extension_field

I truly have no idea what this refers to. If you want me to understand what you're saying, can you please supply enough context so that I know what "it" refers to? Nothing to do with intelligence. Just subject-specific education and study. I myself am right at the ragged edge of my own competence in the topics we're discussing and usually have to look things up to respond to your questions. It's just a matter of learning the material.

I have no idea what this refers to or what it means. I suppose that such a project would start by implementing model theory in Python. https://en.wikipedia.org/wiki/Model_theory https://plato.stanford.edu/entries/model-theory/ I must say, though, that this conversation reminds me of someone reading a textbook on brain surgery then flailing away at a cantaloupe with an ice pick. You clearly don't understand the subject matter; and as I'm sure you know, understanding WHAT you are trying to program is the first step in designing a program.

The notation refers to ordinal exponentiation. It's not relativization, which is something entirely different that also happens to use exponential notation. You can always tell the difference because in relativization you have a proposition as the base and some particular model as the "exponent." Totally unrelated concept to exponentiation. I don't know what an undefined set is. But [math]\omega[/math] is not a variable, it's a specific ordinal, the first infinite ordinal. It's the ordinal corresponding to the natural numbers in their usual order; that is, [math]\omega[/math] = {0, 1, 2, 3, 4, ...} as an ordered set. Oh I see what this is about. See https://www.cut-the-knot.org/WhatIs/Infinity/Ordinals.shtml The notation refers to ordinal exponentiation. The theorem is that any ordinal that satisfies the equation is inaccessible. The least ordinal that satisfies it is the fascinating ordinal [math]\epsilon_0[/math]. See https://en.wikipedia.org/wiki/Epsilon_numbers_(mathematics)

Likewise.

Ok that's a start. I appreciate your zeal and interest, but I simply have nothing to add to what I've already said. I don't see any relationship between your programming project and the higher set theory under discussion; nor can I relate any of your code fragments to any programming language I know. I truly can't be of any more assistance.

Is that a pickup artist who says, "Oh baby you're so beautiful!" to women? Just kidding. You're thinking of a complement. But why do you think that advanced set theory is built into the Python language? Have you defined the complement operator somewhere? Well how is it defined? Advanced set theory is not built in to Python, nor is it amenable to programming at all unless you are writing a proof system. Don't understand this remark. Advanced set theory isn't a programming exercise. You have to roll up your sleeves and grapple with the math, starting with working out the [math]V_\alpha[/math]'s. "Reversing a binary code, " what?? I'm afraid we are not communicating at all.

I am afraid I can't participate in this thread anymore. I don't feel that you're engaging with the math and with what I'm saying to you. I invited you to work out the definitions of the first few [math]V_\alpha[/math]'s in order to gain insight into what's going on. Doing so would clarify the definition of rank. "what does the superscript and subscript mean in set theory?" I've answered this several times already. "i mean I understand relativization but what are they individually " I've answered this several times already as well. If you understood relativization you could not ask these questions since the meanings of the symbols are part of the definition of relativization. I do invite you to read Kunen starting on page 110 (book page, not necessarily pdf page) for a thorough discussion; which I will say is quite hard going and refers to material that appears earlier in the book. It's not possible for me to continue to repeat myself. I do thank you for the conversation, because in responding to your posts I've had to review and learn a lot myself. You may not have been reading Kunen but lately I have! Only one question: If you're a programmer, how can you not understand indexing? By the way I'll leave you with a big-picture insight. Why are we going to all this trouble to define [math]V[/math]? What we're doing is building up every set that there can possibly be, one level at a time. It's a technical step in the various independence proofs such as the independence of CH and AC from ZF. And by "every set" we really mean, "Every well-founded set." The well-founded sets are the sets that don't contain themselves, or don't have membership loops, or in general don't contain any infinite back-chains of membership. They're named after von Neumann but it was Zermelo who first defined the notion.

You need to work out for yourself what are the sets: * [math]V_0[/math] * [math]V_1[/math] * [math]V_2[/math] * [math]V_3[/math] and if you are particularly industrious, [math]V_4[/math]. That will show you what's going on. You could, if you set your mind to it, convince yourself that the Wiki quote makes perfect sense in the context of these few sets; and you would thereby come to understand what's going on. If you worked out these sets you could work through that Wiki paragraph and come to understand this quote. If you are unsure of how to work these out, those are the questions you should ask. At some point when we're learning math we have to roll up our sleeves and "do the math," and this is one of those moments. There is no other way to get this material. I would like it if you'd engage with the questions I asked you in my two previous detailed posts, but I can't make you. Constantly changing the subject to a new confusion is one of the problems here. We need to pick one issue and drill it down. As it stands you give me the impression of not sufficiently engaging with the material and with what I'm saying to you; which makes this a frustrating exercise at my end.

So that I can better understand your question, answer me this. In the example of the boxes [math]B_n[/math] where [math]n[/math] is a positive integer between 1 and 15, inclusive, what are the qualities of [math]n[/math]? What do you mean by that? In the example of the houses on a street, where the odd-numbered side of the street consists of a sequence of houses [math]H_n[/math] where [math]n[/math] is an odd positive integer from, say, 101 to 501; what do you mean by asking what are the qualities of [math]n[/math]? If I get into an elevator, where we have a collection of floors [math]F_n[/math] where [math]n[/math] is a sequence of positive integers starting at 1 and anding in 15, what does it mean to ask, "What are the qualities of n?" It's just a way of counting the floors so we know where we're supposed to get off the elevator to get where we're going. It's just a number. A label. It has no inherent meaning, we just use it to distinguish one floor from another, one house from another, one box from another, one [math]V_\alpha[/math] from another. It's the same question. What do you mean "what are all the qualities of" [math]\alpha[/math]? Alpha is an ordinal number. It has all and only those qualities that pertain to ordinals. And in fact ... this is a point you seem to be having trouble with ... [math]\alpha[/math] is not a particular ordinal number. It's a variable that iterates, if you will, or "marches through," ALL of the ordinal numbers. Why are we having trouble with this? [math]\alpha[/math] is not a particular ordinal number. It's a variable that ranges over all of them. Like in high school when they taught you about the parabola with the equation [math]f(x) = x^2[/math]. It doesn't make sense to ask "What is x?", or "What are the qualities of x?" or, "What is the definition of x?" x is a variable that ranges over all the real numbers. Yes? You agree with this? I hope you'll reply specifically to this last example. You understand that when we write [math]f(x) = x^2[/math], the variable 'x' ranges over the real numbers. It's not any particular real number and it has no qualities and no definition, other than being what we might call a dummy variable in a function definition. Do you agree with this?

No source that we've referenced, not NcatLab, Wiki, Kunen, or anything else, has ever used such a word as "ordinalizing," which has no meaning. You just made it up. But if by that you mean "indexing a collection of sets by ordinals," that's exactly what it is. The key word is indexing. If you don't know what it means to index a family of sets, that's where to start. Suppose I tell you that there are a bunch of boxes in a room, labeled [math]B_1[/math], [math]B_2[/math], [math]B_3[/math], ..., [math]B_{15}[/math]. Do you understand that notation? It's the same as if I say the boxes are called [math]B_n[/math] where the variable [math]n[/math] ranges over the integers 1 through 15, inclusive. Does that make sense? If no, let's stop there and work through the notation. If yes, then the [math]V_a[/math]'s are a collection of sets indexed by a variable [math]\alpha[/math] that ranges over the ordinal numbers. The general concept is called an indexed family of sets; or in the case of the boxes, it's an indexed family of boxes. https://en.wikipedia.org/wiki/Indexed_family A very down to earth example of the concept is the houses on a street, which are indexed by their addresses: 101, 103, 105, 107, ... on the odd side of the street; and 102, 104, 106, 108, ... on the even side. Or in a high-rise building, the floors are indexed by the positive integers. Floor 1, floor 2, floor 3, etc. That's all it means. So we have a bunch of sets [math]V_0[/math], [math]V_1[/math], [math]V_2[/math], [math]V_3[/math], ... We have a set for each ordinal. Set theorists, like computer programmers, like to start counting with 0. You can think of ordinals as what you get when you count 0, 1, 2, 3, 4, ... and "keep on going" after you've run out of all the positive integers. It's a technical concept in set theory. But if you like, just to get us going here, you can think of them as the floors in an infinitely tall building or the house numbers on an infinitely long street.

[math]\alpha[/math] is a variable that ranges over the class of ordinal numbers. Please tell me exactly which parts of this you don't understand. I have repeated this exact same definition maybe six or eight times now so in order to make progress you have to tell me exactly which parts are unclear to you. There is no English letter 'a' in this discussion at all. Are you using 'a' for alpha? Or do you mean some other 'a'? For clarity, you can copy/paste greek letters and other math symbols from https://math.typeit.org/

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.

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.

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.

[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]. 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.

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. 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? 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.

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. [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. [math]\alpha[/math] an ordinal number. No, there is only one [math]V[/math]. 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. "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.

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

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

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].

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

But the article never mentions God. Searching for the string "God" brings up several references to Gödel but none to God. Here's the abstract of the article: This article is concerned with reflection principles in the context of Cantor’s conception of the set theoretic universe. We argue that within such a conception reflection principles can be formulated that confer intrinsic plausibility to strong axioms of infinity. This shows that it's a technical article about higher set theory. Higher in the sense of going beyond undergrad set theory. There's no theology here or any discussion of God at all. Why do you think this article is about God when it's a technical article about set theory? Regarding your project to program God, I find that interesting. I have a question. Don't you think that God, by any definition, must be beyond computability? I do. As the Tao Te Ching might say: The God that can be computed is not the true God.