Complexity is partially used by the current community of mathematicians because any given member of some set is reduced to some unique name, such that its internal complexity is ignored.
By using this approach, Cardinality is the number of the unique names that belong to some set.
If set A is determined by some common property, then any set that has that property is a member of set A, such that its members are ignored and only the name of the belonged set is considered.
Some examples:
The set of all ideas, called X, is itself an idea. In that case X={X, idea_a, idea_b, idea_c, …}, where “X” in “{X, idea_a, idea_b, idea_c, …}” is only the name of set X.
Since the internal complexity of X as a member of X is ignored (only its name is considered as a member of X) we artificially avoid the non-finite regression of the internal complexity of set X, that actually prevents the completeness of X.
The set of all oranges, called Y, is not itself an orange. In that case Y={ orenge_a, orange_b, orange_c,…}, where “Y” is not one of the names that belong to set Y.
In that case there is no non-finite regression, but also in this case no member of set Y is its final member if Y is a non-finite set (this non-final state is equivalent to non-finite regression, because in both cases no level or member are final cases of the considered set), so in both cases, the term “all” cannot be related to non-finite sets.
Definition 1: “Z is the set of all sets that are not their own members.”
By definition 1, If Z is not its own member then the name “Z” must be one of the members, but then Z is not the set of all sets that are not their own members, because the name “Z” is one of the members.
Let us use Stanford's description (
http://plato.stanford.edu/entries/russell-paradox/ ):
"
Russell's paradox is the most famous of the logical or set-theoretical paradoxes. The paradox arises within naive set theory by considering the set of all sets that are not members of themselves. Such a set appears to be a member of itself if and only if it is not a member of itself, hence the paradox."
If Z's name is a member of set Z, then set Z is not "the set of all sets that are not their own members" and we cannot conclude anything about some set that includes its name as its member and about Definition 1, as wrongly appears in Stanford's description.
The problem appears only in the case that Z's name must but can't be a member of set Z, and as we show, the problem is at the definition and not in set Z because of the term "all” that appears in Definition 1.
The term “all” is not satisfied in both cases, whether the name “Z” is a member or not a member of set Z. In that case we can conclude that Definition 1 does not hold and set Z is not well-defined if the term “all” is included in the definition.
Conclusion:
The universal quantifier “for all” is valid only if any member of some set can be considered as its final member.
By using this notion the following definition holds:
Definition 1: “Z is the set of sets that are not their own members.”
This definition holds because:
Since the term “all” is avoided, then Z’s name is not necessarily one of the members of set Z, and also set Z is incomplete if it is a non-finite set (no one of its members is its final member).
Let us research the finite case by assuming that there are finitely many sets that are not their own members:
Definition 1: “Z is the set of all sets that are not their own members.”
In this case the term “all” is valid since each member of set Z is its final member.
By definition 1, If Z is not its own member then the name “Z” must be one of the finitely many members, but then Z is not the set of all finitely many sets that are not their own members, because the name “Z” is one of the members.
Definition 1 does not hold also in the finite case since the term “all” leads to self reference that causes a paradox.
Let us summarize:
In the case of infinite sets the term “all” is invalid because no member of a given non-finite set is its final member.
In the case of finite sets, definitions that are using the term “all” that leads to self reference that causes paradoxes, these definitions are invalid and are not parts of a consistent mathematical framework.
The universal set does not exist not because of Russell’s paradox, but because any infinite set is incomplete (the term “all” is invalid in the case of infinite sets).
In the case of finite sets, where the term “all” can be used (but then we do not deal with the universal set), there are definitions that lead to paradox because of the term "all", but then these definitions are not parts of a consistent mathematical framework.
So in both cases (the finite and the non-finite) Russell’s paradox has a very minor impact on Set theory.
