Reading the ZF axiom of existence?

[Dale10101]

(the snapshot is from http://mplab.ucsd.edu/tutorials/settheory.pdf,
Tutorial on Axiomatic Set Theory, Javier R. Movellan)

I am not sure how to read statement (6).

As I currently understand, the statement can be thought of as consisting of two parts: the first two characters declare that there exists a set y. The remaining portion says that there exists a property, call it P(x) such that for all x, x is not an element of y.

Together, as I currently understand, the statement says that there is a set y which contains all elements that are true for P(x). Is that right please?

As I try and think about what this means my first thought is that there are two sets, set x and set y. But is that right? Set y seems to be a definite set by declaration, but what about set x, is that a definite set or a sort of dummy variable that represents all possible sets in the universal set, that is, that the unstated property of x is that it has no property, no restriction, so that it may be any set? That would make sense because P(x) would then say that no set x picked from all possible sest in the universe are contained in y therefore y must be empty.

On the other hand maybe x is a dummy variable representing all elements in y is which case P(x) says that no element of y is in y, which seems (sort of) to say that y is empty.

On the other hand (three hands?, well maybe I an alien) … y is stated to exist, but the fact that x is named seems to imply that it exists too which means perhaps the statement is about two sets which may or may not have a non-null intersection … and what would that mean … I give up, need help, the crankshaft is starting to smoke.

Ancillary related questions,

1) In P(x) , where P is any property how is the domain of x stated or implied, in general and vis-à-vis the above statement (6).

2) Sometime when you read set statements the capital “X” is used to imply a set while the lowercase “x” implies an element of “X”. I know that in set theory everything is a set (no urlements) but if you do not use the convention of a capital and lowercase letter how would you discuss the set “x” as an element of the set “X”. Again in statement (6), wouldn't it make sense to capitalize Y but leave x lowercase since at most x is a subset (proper or otherwise) element of Y?

 

[daon2]

Together, as I currently understand, the statement says that there is a set y which contains all elements that are true for P(x). Is that right please?

Going with the assumption that the use of \(\displaystyle \not \in\) implies \(\displaystyle y\) is a set, it says there is a (set) \(\displaystyle y\) such that no matter what object \(\displaystyle x\) may be, it does not belong to \(\displaystyle y\).

As I try and think about what this means my first thought is that there are two sets, set x and set y. But is that right? Set y seems to be a definite set by declaration, but what about set x, is that a definite set or a sort of dummy variable that represents all possible sets in the universal set, that is, that the unstated property of x is that it has no property, no restriction, so that it may be any set? That would make sense because P(x) would then say that no set x picked from all possible sest in the universe are contained in y therefore y must be empty.

I see no indication that \(\displaystyle x\) is necessarily a set, though no restriction it can't be either. So it is some object that belongs to some universal set \(\displaystyle U\) that you are considering (in that case you can call \(\displaystyle y\) empty relative to \(\displaystyle U\) I suppose).

On the other hand maybe x is a dummy variable representing all elements in y is which case P(x) says that no element of y is in y, which seems (sort of) to say that y is empty.

I'm not sure what you're trying to say here. \(\displaystyle x\) and \(\displaystyle y\) are assumed to have no relationship by their qualifiers alone. The P(x) you are talking about is \(\displaystyle x\not\in y\).

On the other hand (three hands?, well maybe I an alien) … y is stated to exist, but the fact that x is named seems to imply that it exists too which means perhaps the statement is about two sets which may or may not have a non-null intersection … and what would that mean … I give up, need help, the crankshaft is starting to smoke.

Stating "for all x" does not imply existence. "For all real x such that x>x". Now, if your universal set is \(\displaystyle U=\{\}\), then \(\displaystyle y\) still exists, but there are no \(\displaystyle x's\) to even consider (don't think too hard about this).

Ancillary related questions,

1) In P(x) , where P is any property how is the domain of x stated or implied, in general and vis-à-vis the above statement (6).

It isn't stated, so you assume it is as encompassing as necessary, or belonging to some arbitrary universal set \(\displaystyle U\).

2) Sometime when you read set statements the capital “X” is used to imply a set while the lowercase “x” implies an element of “X”. I know that in set theory everything is a set (no urlements) but if you do not use the convention of a capital and lowercase letter how would you discuss the set “x” as an element of the set “X”. Again in statement (6), wouldn't it make sense to capitalize Y but leave x lowercase since at most x is a subset (proper or otherwise) element of Y?

My only answer to this is to read the statements as generally as you can. The use of an inclusion symbol would tell me \(\displaystyle y\) is a set, and \(\displaystyle x\) could be anything. I agree that quantifying something clearly is important, but the author may have planned this to make the reader think. Anyway, that is my opinion on the matter.

 

[Dale10101]

Thanks

I am reading your detailed response closely. I see now that just naming a variable does not imply that it is a being used to represent a set in the capital X sense, rather than as an element of a set in a small x sense, nor that it exists at all in either case beyond being a null set. Also the general interpretation of a named variables "domain" is as broad as the context required in instances like these, that too helps. Thank you.