Comprehension Schema: The Possibility of Multiple Sets with the Same Property

[robertjford80]

P(x) = x ∉ x ⊃ for any set A, there is a set B such that x ∈ B iff x ∈ A and x ∉ x

Does the above mean that different things can bear the same property. For instance, x can be bipedal means x can be an element of the set human or x can be an element of the set ostrich.

 

[mathman]

Your original statement is confusing. What do mean by "x ∉ x"?

 

[robertjford80]

It's not my statement but Jech's. See attachment.

 

[gopher_p]

In a perfect world, we'd like to say that, given any property $P$, there is a set $B$ such that $x\in B$ iff $P(x)$. This roughly says that if I know what attribute I would like the elements of my set to possesses and can describe that attribute, then I can "build" a set that contains precisely those objects which possesses that attribute. This seems like a reasonable expectation that one would have for sets, but unfortunately is a bit too greedy. Accepting this naive requirement of sets, that I can build it if I can describe it, leads to inconsistencies, the most famous of which is Russell's paradox, which is how the whole $x\not\in x$ bit applies.

The Comprehension schema is basically the way that axiomatic set theory gets around Russell's paradox. It says that given any property $P$ and any set $A$, there is a set $B\subset A$ such that $x\in B$ iff $P(x)$ and $x\in A$. It basically means that if I know what attribute I would like my set to have and am willing to limit myself to choosing my elements from a "pre-determined" set, then I can build a subset of that pre-determined set that contains precisely those elements of the pre-determined set that possesses the desired attribute.

Of course, this is a very informal take on a very formal subject. So what I've written isn't necessarily the truth, the whole truth, and nothing but the truth. It's as close to the truth as I could come up with without getting overly technical, and I think it's a pretty fair representation.

 

[robertjford80]

It says that given any property $P$ and any set $A$, there is a set $B\subset A$ such that $x\in B$ iff $P(x)$ and $x\in A$.

I would need a real life example of something that has property P and belongs to set A etc.

 

[gopher_p]

I would need a real life example of something that has property P and belongs to set A etc.

Well, to use your own example from the original post, assuming that the collections mammals, birds, and animals are all sets and that being bipedal is a property that an object might have, then the sets of bipedal mammals, bipedal birds, and bipedal animals are all guaranteed to exist given the Comprehension schema. However the collection of all bipedal things is not guaranteed to exist (as a set) ... unless of course the collection of all things is a set ... which it's not. But now we're jumping outside of the realm of the "real world", so ...

 

[robertjford80]

Well, to use your own example from the original post, assuming that the collections mammals, birds, and animals are all sets and that being bipedal is a property that an object might have, then the sets of bipedal mammals, bipedal birds, and bipedal animals are all guaranteed to exist given the Comprehension schema. However the collection of all bipedal things is not guaranteed to exist (as a set) ... unless of course the collection of all things is a set ... which it's not. But now we're jumping outside of the realm of the "real world", so ...

Thank you.