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Simmer

Consecutive values in set theory

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16 minutes ago, Simmer said:

so I ask, what is the definition of a

[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/

Edited by wtf

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Sorry I was using it for alpha

what is it called, ordinalizing?

 

like beta is bounding, alpha is ordinalizing

Edited by Simmer

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38 minutes ago, Simmer said:

Sorry I was using it for alpha

what is it called, ordinalizing?

 

like beta is bounding, alpha is ordinalizing

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.

 

 

Edited by wtf

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What are all the qualities of a

It’s an ordinal number 

but what else

im sure one of the webpages yiou gave me has the answer 

I’m just not sure which one

Edited by Simmer

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1 hour ago, Simmer said:

What are all the qualities of a

It’s an ordinal number 

but what else

im sure one of the webpages yiou gave me has the answer 

I’m just not sure which one

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?

 

 

 

Edited by wtf

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wikipedia:

 For each ordinal \alpha, the set V_{\alpha } is defined to consist of all pure sets with rank less than \alpha..

what does this look like in set theory?

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29 minutes ago, Simmer said:

wikipedia:

 For each ordinal \alpha, the set V_{\alpha } is defined to consist of all pure sets with rank less than \alpha..

what does this look like in set theory?

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.

 

 

 

Edited by wtf

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I found the formula for a

 

a = set(Va) is set(set()) rank() < a.
 

I have to do some python stuff now

ttyl

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Hi I’m back again

 

what does the superscript and subscript mean in set theory?

i mean I understand relativization but what are they individually 

 

I know this question has already been answered I just need more clarification 

 

please use high level language 

Edited by Simmer

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1 hour ago, Simmer said:

Hi I’m back again

 

what does the superscript and subscript mean in set theory?

i mean I understand relativization but what are they individually 

 

I know this question has already been answered I just need more clarification 

 

please use high level language 

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.

 

Edited by wtf

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Well when I use .index() a syntax error occurs

same with the compliment operator ~
 

Although I’m not sure how reversing binary code means to be everything except the variable before it

 

i just wanted you to know I will be holding onto the answers you gave me. And I will (eventually) go through them thoroughly 

Edited by Simmer

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4 minutes ago, Simmer said:

same with the compliment operator ~

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?

 

 

4 minutes ago, Simmer said:

Well when I use .index() a syntax error occurs

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.

 

4 minutes ago, Simmer said:

Although I’m not sure how reversing binary code means to be everything except the variable before it

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. 

Edited by wtf

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On 8/23/2020 at 12:53 AM, wtf said:

Can you say how much of this material you understand? I can't see how what they're saying isn't clear to anyone who's made it to page 11 of this paper. I don't want to explain things you already know, but I can put some of this in context if you don't.

In particular, the question you asked is answered by the very next sentence of the exposition following Axiom 2. They define the symbol ΦVα  right there. Φ is a sentence in the language of set theory, and ΦVα   is Φ relativized to the set Vα . Relativizing a sentence means restricting its quantifiers to Vα ; or if Φ is second-order, to the powerset of Vα .

It makes me wonder if you might perhaps be in a little over your head, in which case just say so and I'll try to help. But perhaps you already know all this and you're asking a more subtle question, in which case I shouldn't try to explain what you already know.

Do you understand what V and the Vα 's are? This section of the paper is formulating a reflection principle consistent with the idea that there are sets that aren't in V ; and seeing if they can define reflection for those sets, not just the ones you get by staying within V . I can help you unpack the symbols but like I say, it's a curious question. My thought process is that if I can sort of understand what's going on with my limited knowledge of this material, anyone on page 11 should be able to.

Useful refs:

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

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

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


powerset()!!! I know that!!

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3 minutes ago, Simmer said:


powerset()!!! I know that!!

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. 

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If I were any smarter, it would probably be cuz of the progeam

 

good talk my friend. With many more to come.

Edited by Simmer

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34 minutes ago, Simmer said:

good talk my friend.

Likewise.

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Hi I’m crack with something simple

https://mathworld.wolfram.com/CantorsEquation.html

w^e=e

obviously this is the relativity you were talking about

they attribute it to ordinal and cardinal numbers

but I want to see if it works with w being an undefined set

Of course it’s not programmable without the lack of the opposite of itself included

 

so I have all the time I’m the world 

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On 10/16/2020 at 3:04 PM, Simmer said:

w^e=e

obviously this is the relativity you were talking about

 

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. 

 

On 10/16/2020 at 3:04 PM, Simmer said:

but I want to see if it works with w being an undefined set

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)

Edited by wtf

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I think it should exclude all but itself 

also do you happen to know how to do Relativizing in python?

 

i think it’s map()

 

but yeah all but itself excluded

Edited by Simmer

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16 minutes ago, Simmer said:

I think it should exclude all but itself 

 

I have no idea what this refers to or what it means.

 

 

16 minutes ago, Simmer said:

also do you happen to know how to do Relativizing in python?

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.

Edited by wtf

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4 hours ago, Simmer said:

Well I mean doing it backwards

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?

 

 

4 hours ago, Simmer said:

because I'm stupid 

 

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. 

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