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wtf last won the day on March 6 2019

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  1. 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.
  2. 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.
  3. 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.
  4. 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.
  5. 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?
  6. 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.
  7. [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/
  8. 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.
  9. 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.
  10. 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.
  11. [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.
  12. 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.
  13. 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.
  14. Can you give an example? What are you referring to? Your question is unclear.
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