What is meant by that ??

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 [itex](\forall x \in R)(\exists x)(\exists y)(z=(x,y))[/itex]

The way I understand [itex]\exists x[/itex] 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

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

Preno

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

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 [itex]x \in A[/itex] such that [itex]\varphi(x)[/itex].

mahmoud2011

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

So how mus I understand it

HallsofIvy

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

mahmoud2011

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

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

mahmoud2011

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 any one explain to me What it is meant by them logically.

sigurdW

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

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

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

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

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

sigurdW

I dont 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 dont 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 cant exhibit it.)

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