From what I understand, such predicates seem to be causing trouble. For example, the result that no set contains everything seems like too strong a result at this point.
From the book: “To specify a set, it is not enough to pronounce some magic words; it is also necessary to have at hand a set to whose elements the magic words apply”. Magic words basically mean the predicates S(x).
The book says such x’s don’t constitute a set and calls them illegal. It also mentions that class is the word to describe such x’s and that classes are irrelevant in its approach to set theory.
Perhaps when I read further I’d be able to reason better.
There is a whole interplay about intensions and extensions which might clarify but also migth confuse. If the rule is unambigious in which of the members survive to the filtered set things are clear. The possible “selections” one might call the extensions. With {a,b,c} the options are {},{a},{a,b},{a,b,c},{b},{b,c},{c}. The way they are specified one migth call the intensions. If both rules “x is small” and “x is red” give out the same {a,b} selection one might try to argue that it is two intensions picking out the same extension.
If your base set is {} then providing any wild intension can only produce a {}. If you have a meaningful intension but do not provide a base set to select from you might have requirements but you don’t have any members. “x is small and red” migth alwsays produce a narrower selection than “x is small”, but if you don’t know if you are selecting from {a,b,c}, {1,2,3} or {aleph,teth,zayin} you can’t be sure whether a given entity is a member or not. So there is no subset because there fails to be any members. So a selection rule itself can’t constitute a set.
What does “For now we will not consider such x as described by S(x):x∉x as sets.” mean?
From what I understand, such predicates seem to be causing trouble. For example, the result that no set contains everything seems like too strong a result at this point.
From the book: “To specify a set, it is not enough to pronounce some magic words; it is also necessary to have at hand a set to whose elements the magic words apply”. Magic words basically mean the predicates S(x).
The book says such x’s don’t constitute a set and calls them illegal. It also mentions that class is the word to describe such x’s and that classes are irrelevant in its approach to set theory.
Perhaps when I read further I’d be able to reason better.
There is a whole interplay about intensions and extensions which might clarify but also migth confuse. If the rule is unambigious in which of the members survive to the filtered set things are clear. The possible “selections” one might call the extensions. With {a,b,c} the options are {},{a},{a,b},{a,b,c},{b},{b,c},{c}. The way they are specified one migth call the intensions. If both rules “x is small” and “x is red” give out the same {a,b} selection one might try to argue that it is two intensions picking out the same extension.
If your base set is {} then providing any wild intension can only produce a {}. If you have a meaningful intension but do not provide a base set to select from you might have requirements but you don’t have any members. “x is small and red” migth alwsays produce a narrower selection than “x is small”, but if you don’t know if you are selecting from {a,b,c}, {1,2,3} or {aleph,teth,zayin} you can’t be sure whether a given entity is a member or not. So there is no subset because there fails to be any members. So a selection rule itself can’t constitute a set.