- #1
jordi
- 197
- 14
My main question is regarding whether the membership relation is taken as an undefined concept (as is usually hinted in set theory books) or if the membership relation can be defined within the language of first order predicate theory.
Let me describe a method to define the membership relation, and please correct me if I am wrong somewhere in my argument:
A class can be defined as the collection of objects satisfying a given property.
For me, first order predicate theory has no problems in its definitions: first order predicate theory is just a way to put order into our metalanguage (English), such that all statements are true or false. It is a somewhat "limited" language (in comparison to English) but it is useful nonetheless.
So, if x is an object (a "name") and p is a property (an "adjective"), we can define p(x) as "has the object x the property p?" and this is false or true.
Then, a class A is the collection of objects x such that p(x) is true.
We can define x ∈A as "p(x) is true", where p is the property defining A.
To sum up: we have defined ∈ just with the language of first order predicate theory (which is equivalent to classes).
Then, what are sets?
Sets are classes. But not all classes: only those classes satisfying a series of statements in first order predicate theory (which include also the membership relation; but this is not a problem, since we perfectly understand what the membership relation is: we have defined it earlier above in this post), the so called axioms of set theory.
If a class does not satisfy these axioms, it is not a set.
And then, we state that "all mathematics" can be built out of sets (not classes; well, some foundational issues use classes which are not sets, such as the class of all ordinals, but anyway).
If my explanation were right, then everything is fine for my intuition: the membership relation is just DEFINED as a property in first order predicate theory. And sets are just defined as classes (which are a relabeling of properties) which satisfy a series of "strange" axioms (the axioms of set theory). One finds these "strange" axioms are useful, and only those classes that satisfy those axioms (the sets) are necessary to build all mathematics, which is a great feature to have.
Any comments?
Let me describe a method to define the membership relation, and please correct me if I am wrong somewhere in my argument:
A class can be defined as the collection of objects satisfying a given property.
For me, first order predicate theory has no problems in its definitions: first order predicate theory is just a way to put order into our metalanguage (English), such that all statements are true or false. It is a somewhat "limited" language (in comparison to English) but it is useful nonetheless.
So, if x is an object (a "name") and p is a property (an "adjective"), we can define p(x) as "has the object x the property p?" and this is false or true.
Then, a class A is the collection of objects x such that p(x) is true.
We can define x ∈A as "p(x) is true", where p is the property defining A.
To sum up: we have defined ∈ just with the language of first order predicate theory (which is equivalent to classes).
Then, what are sets?
Sets are classes. But not all classes: only those classes satisfying a series of statements in first order predicate theory (which include also the membership relation; but this is not a problem, since we perfectly understand what the membership relation is: we have defined it earlier above in this post), the so called axioms of set theory.
If a class does not satisfy these axioms, it is not a set.
And then, we state that "all mathematics" can be built out of sets (not classes; well, some foundational issues use classes which are not sets, such as the class of all ordinals, but anyway).
If my explanation were right, then everything is fine for my intuition: the membership relation is just DEFINED as a property in first order predicate theory. And sets are just defined as classes (which are a relabeling of properties) which satisfy a series of "strange" axioms (the axioms of set theory). One finds these "strange" axioms are useful, and only those classes that satisfy those axioms (the sets) are necessary to build all mathematics, which is a great feature to have.
Any comments?