Consecutive values in set theory

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What does it mean to have  consecutive values in set theory?

How are they related

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13 minutes ago, Simmer said:

What does it mean to have  consecutive values in set theory?

How are they related

Presumably your set has an ordering relation.

Can you post an example ?

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page 11

so they’re not exactly consecutive but...

It seems to be a “to the power of V” followed by a little value “a”

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Posted (edited)
9 hours ago, Simmer said:

page 11

so they’re not exactly consecutive but...

It seems to be a “to the power of V” followed by a little value “a”

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 $\Phi^{V_\alpha}$ right there. $\Phi$ is a sentence in the language of set theory, and $\Phi^{V_\alpha}$  is $\Phi$ relativized to the set $V_\alpha$. Relativizing a sentence means restricting its quantifiers to $V_\alpha$; or if $\Phi$ is second-order, to the powerset of $V_\alpha$.

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 $V_\alpha$'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:

Edited by wtf

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Posted (edited)
7 hours ago, 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:

Honestly. That was very helpful. Thank you.

youre right though I’m in over my head

So what’s the difference between a variable above another variable and a variable below another variable

is it safe to say both O and a are sets of V?

Edited by Simmer

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Posted (edited)

Ok I’m working with Va now

the sites you posted poses the questions:

what is a separately

what is V separately

Edited by Simmer

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Posted (edited)
2 hours ago, Simmer said:

Ok I’m working with Va now

the sites you posted poses the questions:

what is a separately

what is V separately

$\alpha$ is an ordinal. $V$ is the von Neumann universe of the hereditarily well-founded sets. Before you read this paper you need to read up on these topics.

5 hours ago, Simmer said:

So what’s the difference between a variable above another variable and a variable below another variable

The notation $\Phi^{V_\alpha}$ is defined in the paper as the relativization of the sentence $\Phi$ to $V_\alpha$. There's no general principle here of upper or lower indices. It's just a notation they use for relativization of a sentence.

Can you say why you are reading this paper? You seem to lack some of the prerequisites. I could tell you what I know about it, but frankly a discussion of the reflection principle and the von Neumann universe $V$ that came from me would be far less authoritative than what you could find in a set theory text like Kunen or Jech, or even careful study of the Wiki pages I linked.

The reflection principle says that whatever is true of some universe of sets must already be true of some smaller set. The passage on page 11 is an attempt to make this precise for the case of a universe larger than $V$.

Edited by wtf

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5 minutes ago, wtf said:

α is an ordinal. V is the von Neumann universe of the hereditarily well-founded sets. Before you read this paper you need to read up on these topics.

The notation ΦVα is defined in the paper as the relativization of the sentence Φ to Vα . There's no general principle here of upper or lower indices. It's just a notation they use for relativization of a sentence.

Can you say why you are reading this paper? You seem to lack some of the prerequisites. I could tell you what I know about it, but frankly a discussion of the reflection principle and the von Neumann universe V that came from me would be far less authoritative than what you could find in a set theory text like Kunen or Jech, or even careful study of the Wiki pages I linked.

The reflection principle says that whatever is true of some universe of sets must already be true of some smaller set. The passage on page 11 is an attempt to make this precise for the case of a universe larger than V

But If I were to calculate it would Va and a and V need calculation beforehand?

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1 minute ago, Simmer said:

But If I were to calculate it would Va and a and V need calculation beforehand?

It's not an arithmetic expression. It's a definition of the entire symbol taken as a whole. There's no exponentiation or whatever you are thinking of. The entire symbol is defined as the relativization of a sentence. There is no calculation. They are defining a symbol.

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Posted (edited)

I know to use set() interpolation now

If I could find solve I’d press it

Edited by Simmer

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Posted (edited)
51 minutes ago, Simmer said:

I know to use set() interpolation now

If I could find solve I’d press it

It's a defined symbol, not a calculation.

If you could say why you're reading this paper, perhaps I could explain whatever it is you're curious about.

Edited by wtf

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Posted (edited)

# | and -> need replaced
# X, V, and a need defined
# possible mistake using "in"
#set interpolation needed for consecutive values and theur parenthasis
def x ():
all(Q) | set(Q is none) -> any(a) | set(none) Va (Q.intersection(Va))
def p ():
X -> 2,
any(x()) in X(p(x()) is 0) or all(x()) in X(p(x()) is 1)
q=""
while True:
i=32
while i < 126:
i= i + 1
a = chr(i)
p()
if 1:
q = q + a
elif 0:
q = q
if q == q + "":
break
print (q)

I still need |, V, a, and in (element of)

# X, V, and a need defined

# possible mistake using "in"

Edited by Simmer

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1 hour ago, Simmer said:

# X, V, and a need defined

If you don't know what $V$ is, that's where you should start. This last post of yours is a puzzler. What are you trying to do?

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Posted (edited)

Well right now I’m thinking about interpolating sets

what is V?

Edited by Simmer

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Posted (edited)
22 minutes ago, Simmer said:

what is V?

Ah. A good question. The proper class $V$ is the von Neumann universe of hereditarily well-founded sets. I could repeat the definition here but the def on Wiki is perfectly good. This is where you need to start. Let me know if you have any questions. The Wiki page gives the formal definition and also discusses some of the philosophical issues around the subject.

What does "interpolating sets" mean?

Edited by wtf

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Interpolating set()

means I can have both 0 and a be a set of V

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Posted (edited)
7 hours ago, Simmer said:

Interpolating set()

means I can have both 0 and a be a set of V

$0 \in V_1$, and $\alpha \in V_{\alpha+1}$ for any ordinal $\alpha$.

What are you trying to do?

Edited by wtf

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Posted (edited)

I’m trying to program V and a separately

its for a project that uses the general specification “all” to determine whether the equation itself is good or not, thus giving a response only when the response is good

Edited by Simmer

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Posted (edited)
4 hours ago, Simmer said:

I’m trying to program V and a separately

That doesn't actually make a lot of sense without more context. The sets involved are very far beyond the scope of computability; and $V$ is a proper class.

You understand that $\alpha$ is an ordinal right?  Do you know much about ordinals? There aren't even notations for all the countable ordinals, let alone all of them. You might be interested in this.

4 hours ago, Simmer said:

its for a project that uses the general specification “all” to determine whether the equation itself is good or not, thus giving a response only when the response is good

It's well known that the truth value of a "for all" statement depends on the domain. For example the statement $\forall x (x \geq 0)$ is false if we quantify over the integers; but true if we quantify over the natural numbers.

Is that the kind of thing you are interested in? If so you may find that your current interest in the von Neumann hierarchy isn't exactly what you want.

The paper you referenced is about reflection principles. Reflection principles are indeed about cutting down the scope of quantifiers by restricting the domain; but IMO advanced set theory is a bit of an overkill if you just want to study how altering the domain of a quantifier affects its truth value.

Perhaps instead of higher set theory, you are actually interested in model theory; in which they study the nature of the interpretations, or models, that can be associated with a given axiomatic theory. Models are how we assign meaning to formal symbols.

Just throwing out some ideas to see if anything is helpful. I'm curious to know what this is all about.

Edited by wtf

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Posted (edited)

I still don’t understand

Are V and a Used in certain contexts? Or do they mean specific things?

Edited by Simmer

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Posted (edited)
1 hour ago, Simmer said:

I still don’t understand

This is the beginning of understanding: to admit you don't understand.

The material we're discussing is the very beginning of advanced set theory. It's typically taught at the graduate school level although it's understandable if you know a little bit about sets.

This material is essential for anyone wanting to learn set theory and mathematical logic at the research level. For the rest of us, it's only of interest to know about the kind of things professional set theorists care about.

1 hour ago, Simmer said:

Are V and a Used in certain contexts? Or do they mean specific things?

I have pointed you to the definition of $V$, the von Neumann universe of the hereditarily well-founded sets, several times.

If you would please read this page and tell me which part is unclear to you, I can help. But at some point you have to read the Wiki page I keep pointing you to. $V$ is a specific thing in set theory.

The basic idea is that we inductively define a bunch of sets:

$V_0$ is the empty set.

$V_1$ is the powerset of $V_0$

$V_2$ is the powerset of $V_1$

and so forth. Now when we run out of natural numbers to index the sets, we jump to ordinals. The ordinals are what you get when you count 0, 1, 2, 3, 4, ... then you "keep going" after they're all done.

So for each ordinal $\alpha$, we have some set $V_\alpha$. Then $V$ is the union of all of these $V_\alpha$ sets. $V$ is "too big" to be a set, so we call it a proper class.

In effect, the $V_\alpha$'s give the "born on" date of every possible set. If you have some set, there is some $V_\alpha$ in which that set first appears. This turns out to be a useful notion when one studies higher set theory.

There are some philosophical issues. Is $V$ really all the sets that there are? The paper you linked considers reflection principles in the case in which there are sets that are not in $V$. But what are reflection principles? Well, that's the next thing to learn ... but first, you need to grapple with the definition of $V$. First things first.

Also, the article you linked is NOT about contemporary set theory; but is rather about what Cantor might have thought about these issues. But Cantor did not have the modern notions of the von Neumann universe or modern reflection principles. So the paper is part historical and part advanced set theory. What is its relevance to your interests?

If you'd like to learn this material it's not really that difficult. You do have to engage. So when I point you to the Wiki page for the von Neumann universe, you have to read that page and ask questions about the parts you don't understand.

And again, why do you care? You linked a research article that refers to these advanced concepts, for which you lack the background. No problem, the material can be explained. But why are you picking some random research article in mathematical philosophy?

A terrific popularized account of all these matters is Rudy Rucker's Infinity and the Mind, a book I highly recommend to you.

Edited by wtf

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Posted (edited)
1 hour ago, Simmer said:

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.

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.

Edited by wtf

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Cantor linked the Absolute Infinite with God,[1] and believed that it had various mathematical properties, including the reflection principle: every property of the Absolute Infinite is also held by some smaller object.[2]

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Posted (edited)
35 minutes ago, Simmer said:

Cantor linked the Absolute Infinite with God,[1] and believed that it had various mathematical properties, including the reflection principle: every property of the Absolute Infinite is also held by some smaller object.[2]

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.

Edited by wtf

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