- #1
Bob3141592
- 236
- 2
My efforts at self-education aren't going well. In trying to read that paper on wheels, I kept having to look up terms, which takes me to all sorts of other topics, and I lose focus. I also realized I didn't sufficiently understand the ideas I was reading about. So I kept going back and ended up with the basics of set theory. Am I on track with these ideas here?
A set is a collection of elements, and at its lowest level these elements don't even have to be of the same type. We don't know anything about these elements except that they are contained in S. Even if they are of completely different types, they must have at least one attribute in common, that attribute which qualifies them for inclusion in S. At least this should be true if there is a rational reason for any object to be in S.
Can I say that this inclusion attribute need be no more than the ability of S to name [tex]\epsilon[/tex]. Does it have to be anything more? We can name any [tex]\epsilon[/tex] by a unique nul-ary operator. There have to be as many such operators as there are elements in S. I suspect a nullary operator can't do anything more than name an element, so there should be no others. It's an identification operator - and I guess as far as operators go it's an identity. But it kind of an awkward operator, in that its not a generalization in any way. For example, we cannot label that operator with a subscript like [tex]O\sub{i}[/tex] unless we know the elements of S can themselves be sorted. Does the ability to be sorted imply at least one other attribute common to all elements of S? Does it sound right that the index of the nul-ary operator is the name (or in a sense the value) of the element? Can there be any other kind of nul-ary operators?
A set is a collection of elements, and at its lowest level these elements don't even have to be of the same type. We don't know anything about these elements except that they are contained in S. Even if they are of completely different types, they must have at least one attribute in common, that attribute which qualifies them for inclusion in S. At least this should be true if there is a rational reason for any object to be in S.
Can I say that this inclusion attribute need be no more than the ability of S to name [tex]\epsilon[/tex]. Does it have to be anything more? We can name any [tex]\epsilon[/tex] by a unique nul-ary operator. There have to be as many such operators as there are elements in S. I suspect a nullary operator can't do anything more than name an element, so there should be no others. It's an identification operator - and I guess as far as operators go it's an identity. But it kind of an awkward operator, in that its not a generalization in any way. For example, we cannot label that operator with a subscript like [tex]O\sub{i}[/tex] unless we know the elements of S can themselves be sorted. Does the ability to be sorted imply at least one other attribute common to all elements of S? Does it sound right that the index of the nul-ary operator is the name (or in a sense the value) of the element? Can there be any other kind of nul-ary operators?