Understanding Existential Quantifiers in Set Theory

[mahmoud2011]

What is meant by that ??

This can considered as a logic question , when I say for example there exist x such that ... . Mustn't I define some set from which x belong .
In a book of set theory it defined a binary relation as following :
A set R is a binary relation if (\forall x \in R)(\exists x)(\exists y)(z=(x,y))

The way I understand \exists x is as following , as he is referring to any set x which exist , So we must consider some Set containing all sets , Such set doesn't exist . So what set must be considered , how must I understand this I know that we didn't mention the universal set if it is clear from context . Here , there is no Universal set . we want x to be arbitrary .

Thanks

 

[mathman]

It looks like the author is implicitly assuming x and y belong to some set (R?). Also the statement starts with for all x in R - shouldn't it be for all z in R?

 

[HallsofIvy]

This can considered as a logic question , when I say for example there exist x such that ... . Mustn't I define some set from which x belong .
In a book of set theory it defined a binary relation as following :
A set R is a binary relation if (\forall x \in R)(\exists x)(\exists y)(z=(x,y))

One of those "x"s should be a "z".

The way I understand \exists x is as following , as he is referring to any set x which exist , So we must consider some Set containing all sets , Such set doesn't exist . So what set must be considered , how must I understand this I know that we didn't mention the universal set if it is clear from context . Here , there is no Universal set . we want x to be arbitrary .

Thanks

 

[Preno]

This can considered as a logic question , when I say for example there exist x such that ... . Mustn't I define some set from which x belong .
In a book of set theory it defined a binary relation as following :
A set R is a binary relation if (\forall x \in R)(\exists x)(\exists y)(z=(x,y))

The way I understand \exists x is as following , as he is referring to any set x which exist , So we must consider some Set containing all sets

No. \forall x and \exists x are perfectly legitimate expressions as they are. Indeed, \forall x \in R \varphi(x) is merely shorthand for \forall x (x \in R \rightarrow \varphi(x)).

You seem to be confusing this with the axiom of specification, which says that given a set A, you can define a set B as the set of all x \in A such that \varphi(x).

 

[mahmoud2011]

This can considered as a logic question , when I say for example there exist x such that ... . Mustn't I define some set from which x belong .
In a book of set theory it defined a binary relation as following :
A set R is a binary relation if (\forall x \in R)(\exists x)(\exists y)(z=(x,y))

The way I understand \exists x is as following , as he is referring to any set x which exist , So we must consider some Set containing all sets , Such set doesn't exist . So what set must be considered , how must I understand this I know that we didn't mention the universal set if it is clear from context . Here , there is no Universal set . we want x to be arbitrary .

Thanks

sorry it must be (\forall z \in R)(\exists x)(\exists y)(z=(x,y))

So how mus I understand it

 

[HallsofIvy]

Every z in the relation consists of a pair of real numbers.

 

[mahmoud2011]

Every z in the relation consists of a pair of real numbers.

But we are dealing wit general sets in set theory so we consider "any sets" also I don't know what must be meant for saying "any sets". When I say there exist x such that ... , I understand that as "x exists according to axioms of ZFC " , But When I was reading in Logic , and when defining quantifiers it was defined with a universal set . When we write for all x such that ... (Of course when is meant any set from the context) , I understand it as if for any x we can prove to exist according ZFC we have ... . And there exist x such that ... , I understand it as that we can prove the existence of some x according to ZFC such that ... . I am not sure if my way of thinking is formal and precise or not .

 

[sigurdW]

This can considered as a logic question , when I say for example there exist x such that ... . Mustn't I define some set from which x belong .
In a book of set theory it defined a binary relation as following :
A set R is a binary relation if (\forall x \in R)(\exists x)(\exists y)(z=(x,y))

The way I understand \exists x is as following , as he is referring to any set x which exist , So we must consider some Set containing all sets , Such set doesn't exist . So what set must be considered , how must I understand this I know that we didn't mention the universal set if it is clear from context . Here , there is no Universal set . we want x to be arbitrary .

Thanks

There is no set containing all sets, but there is a class containing all sets...Mathematicians say.

 

[mahmoud2011]

There is no set containing all sets, but there is a class containing all sets...Mathematicians say.

That I am talking about , the book haven't considered classes yet ( I know some about them ) , but the question is when I say for all x such that P(x) , will it mean "if you have proved the existence of set x then P(x) holds" or what , and whwn I say there exists some x such that P(x) holds , will it mean " It can proven the existence of a set x in ZFC such that P(x)" , Here I consider the statements " It can be proven" and alike are informal . So Can anyone explain to me What it is meant by them logically.

 

[sigurdW]

That I am talking about , the book haven't considered classes yet ( I know some about them ) , but the question is when I say for all x such that P(x) , will it mean "if you have proved the existence of set x then P(x) holds" or what , and when I say there exists some x such that P(x) holds , will it mean " It can proven the existence of a set x in ZFC such that P(x)" , Here I consider the statements " It can be proven" and alike are informal . So Can anyone explain to me What it is meant by them logically.

1 When you say for all x such that P(x)...
It should mean that out of a certain set ordinarily called "the domain"
you have created a subset consisting of all objects ,x, satisfying the condition "P".

Note: You must distinguish between "x" as any object of a set and "x" considered as the set!

If x is any element of the set x then x contains itself as an element and there is a certain axiom in ZFC forbidding just that. (Since the axiom is independent of the other axioms, then you could use ZFC with the axiom replaced with its negation resulting in a set theory as consistent as ZFC is.)

2 When you say there exists some x such that P(x) holds, then you claim that your subset of the domain is not empty.

 

[mahmoud2011]

1 When you say for all x such that P(x)...
It should mean that out of a certain set ordinarily called "the domain"
you have created a subset consisting of all objects ,x, satisfying the condition "P".

Note: You must distinguish between "x" as any object of a set and "x" considered as the set!

If x is any element of the set x then x contains itself as an element and there is a certain axiom in ZFC forbidding just that. (Since the axiom is independent of the other axioms, then you could use ZFC with the axiom replaced with its negation resulting in a set theory as consistent as ZFC is.)

2 When you say there exists some x such that P(x) holds, then you claim that your subset of the domain is not empty.

I see you mean Axiom Schema of Comprehension right . So in the definition I have written at first such domain is not determined , I see that he wants to determine any set , and I can't take such domain to be the st of all sets because it doesn't exist . And I don't want to use notion of classes.

 

[sigurdW]

I see you mean Axiom Schema of Comprehension right . So in the definition I have written at first such domain is not determined , I see that he wants to determine any set , and I can't take such domain to be the st of all sets because it doesn't exist . And I don't want to use notion of classes.

Im not sure what you want to do, and how to advice you...An adventurous, perhaps stupid idea, is to take away one insignificant set from the "set of all sets" then you should have a set of almost all sets... right?

 

[mahmoud2011]

Im not sure what you want to do, and how to advice you...An adventurous, perhaps stupid idea, is to take away one insignificant set from the "set of all sets" then you should have a set of almost all sets... right?

That I am talking about . My only question what 's the meaning of there exist x such that ... .
In the definition above it and and some of axioms of set theory for example Axiom of Extensionality we begin by saying for all x and y , ... . So what " for all " Exactly mean.

 

[mahmoud2011]

I see you mean Axiom Schema of Comprehension right . So in the definition I have written at first such domain is not determined , I see that he wants to determine any set , and I can't take such domain to be the st of all sets because it doesn't exist . And I don't want to use notion of classes.

Sorry , " Axiom of regularity " , I confused only their names .

 

[sigurdW]

That I am talking about . My only question what 's the meaning of there exist x such that ... .
In the definition above it and and some of axioms of set theory for example Axiom of Extensionality we begin by saying for all x and y , ... . So what " for all " Exactly mean.

I don't remember the exact formulations of the set axioms...it was a long time since I looked at them. But it seems to me that you want to be sure that you understand the basic definitions in order to be sure that you understand the meaning of the axioms. Thats an honest strategy.
I don't mind looking at basic definitions... Do you have access to them? Some authors does not make the foundation of their theory clear and complete. Make a list of the statements you find unclear.

Are you really bothered about the meaning of "for all x..." and "there exists an x..."? The expressions are called "quantifiers" and belong to "Predicate Logic" they are perhaps assumed to already be defined and understood in the theory you are studying.

"for all x..." is called the universal quantifier and it affects a variable found in a sentence function: P(x) ... "x" is a variable so P(x) is a statement function not a statement...you get a statement if you replace x with an "individual constant" say ,a, ...then P(a) is (because of the quantifier) a true sentence.

An "interpretation" of the logical language used is some non empty set called "the Domain"
it contains all sentences all predicates and all constants:

"for all x..." now means: for each object x in the domain...

Having the universal we define the existential: "there is an x such that p(x)" means "not for all x : not p(x)"

(Note that the domain is assumed not exhibited! It might be infinite so we can't exhibit it.)

This is a very abbreviated view...does it make any sense to you?