- #1
bahamagreen
- 1,014
- 52
If S={a,b,c}, what does it mean that S=T when T={a,a,a,a,a,a,b,c}?
The mapping that confirms the definition of equality assumes that the duplicate symbols in a set are representative of the same entity or idea.
If S={1/2} and T={ .5 , 2/4 , ,25/,5 , 4/8 } are these sets equal? At what point does one ask about or distinguish the symbols from what they represent? Does it matter that one might define the set T as "symbols that are equal to one half"?
Likewise with names:
S={Bill, Bill, Joe, Tom}
and S=T where T={Bill, Joe, Tom}
Isn't it undefined whether one of those "Bill"s in S is merely a duplication of the "Bill" symbol naming one of three boys in a group with names Bill, Joe, and Tom, or if the duplicate symbols represent the names of two boys from a group of four boys named Bill, Joe, Tom, and the fourth named Bill as well?
The mapping of the symbols to test equality does not seem to extend to the things represented strictly by the symbols of representation. Working at just the symbol level, how does one know when it is OK to collapse symbol duplicates without confounding the objects which they represent?
I'm probably not explaining my questions and examples very well; fundamentally I'm wondering why the definition of sets does not enforce or invoke some kind of uniqueness requirement of their elements. In the name example above, a set with two "Bill"s in it does not indicate whether those two are duplicate representations of a single instance or duplicate symbols used to represent two unique discernible instances... how would one know it was OK to reduce the set to contain only one "Bill" symbol and claim it was equal to before doing so?
In modern relational data systems one uses primary keys to enforce uniqueness so that one does not attempt to employ mutually shared attributes resulting in common identifiers of different things... how is set theory avoiding that kind of requirement?
The mapping that confirms the definition of equality assumes that the duplicate symbols in a set are representative of the same entity or idea.
If S={1/2} and T={ .5 , 2/4 , ,25/,5 , 4/8 } are these sets equal? At what point does one ask about or distinguish the symbols from what they represent? Does it matter that one might define the set T as "symbols that are equal to one half"?
Likewise with names:
S={Bill, Bill, Joe, Tom}
and S=T where T={Bill, Joe, Tom}
Isn't it undefined whether one of those "Bill"s in S is merely a duplication of the "Bill" symbol naming one of three boys in a group with names Bill, Joe, and Tom, or if the duplicate symbols represent the names of two boys from a group of four boys named Bill, Joe, Tom, and the fourth named Bill as well?
The mapping of the symbols to test equality does not seem to extend to the things represented strictly by the symbols of representation. Working at just the symbol level, how does one know when it is OK to collapse symbol duplicates without confounding the objects which they represent?
I'm probably not explaining my questions and examples very well; fundamentally I'm wondering why the definition of sets does not enforce or invoke some kind of uniqueness requirement of their elements. In the name example above, a set with two "Bill"s in it does not indicate whether those two are duplicate representations of a single instance or duplicate symbols used to represent two unique discernible instances... how would one know it was OK to reduce the set to contain only one "Bill" symbol and claim it was equal to before doing so?
In modern relational data systems one uses primary keys to enforce uniqueness so that one does not attempt to employ mutually shared attributes resulting in common identifiers of different things... how is set theory avoiding that kind of requirement?