Kermit1941
01-12-2008, 12:26 AM
Russel's Paradox is related to naive set theory.
This paradox is the reason mathematicians developed axiomatic set theory.
Before the discovery of Russel's paradox, it was assumed that
we could freely follow our intuition and stipulate anything we could imagine
be be collected into a set.
Under this naive presumption, we construct Russel's paradox as follows.
Define an ordinary set to be any set whose members are not sets which contain themselves.
For example, a set of spoons is an ordinary set because a spoon is not a set.
A set whose members are the sets {1,2,3},{4,5,6},{7,8,9} is also an ordinary set because { {1,2,3}, {4,5,6},{7,8,9} } is not a member of itself.
Are there any sets that are not ordinary sets?
The set of ideas is an idea.
So we naively consider the set of ideas to be an extraordinary set.
The set of sets we naively believe to be a set.
Thus we will say the set of sets is an extraordinary set.
Now we can state the paradox.
We consider the set of ordinary sets.
To make the explanation clear, let's give this set a name.
Let T be the set of ordinary sets.
if the set S is an element of T, then S is not a member of itself.
and
if the set S is not an element of T, then S is not an ordinary set, and therefore, must be a member of itself.
Now let's consider the special case where s is T.
We get the self-contradictory statements.
If T is an element of T, then T is not a member of itself.
and
If T is not an element of T, then T must be a member of itself.
It is an oversimplification to say that these last two statements are Russel's paradox.
The self-reference property was used in building Russel's Paradox.
They are not the same as Russel's Paradox.
Kermit Rose < [email protected] >
This paradox is the reason mathematicians developed axiomatic set theory.
Before the discovery of Russel's paradox, it was assumed that
we could freely follow our intuition and stipulate anything we could imagine
be be collected into a set.
Under this naive presumption, we construct Russel's paradox as follows.
Define an ordinary set to be any set whose members are not sets which contain themselves.
For example, a set of spoons is an ordinary set because a spoon is not a set.
A set whose members are the sets {1,2,3},{4,5,6},{7,8,9} is also an ordinary set because { {1,2,3}, {4,5,6},{7,8,9} } is not a member of itself.
Are there any sets that are not ordinary sets?
The set of ideas is an idea.
So we naively consider the set of ideas to be an extraordinary set.
The set of sets we naively believe to be a set.
Thus we will say the set of sets is an extraordinary set.
Now we can state the paradox.
We consider the set of ordinary sets.
To make the explanation clear, let's give this set a name.
Let T be the set of ordinary sets.
if the set S is an element of T, then S is not a member of itself.
and
if the set S is not an element of T, then S is not an ordinary set, and therefore, must be a member of itself.
Now let's consider the special case where s is T.
We get the self-contradictory statements.
If T is an element of T, then T is not a member of itself.
and
If T is not an element of T, then T must be a member of itself.
It is an oversimplification to say that these last two statements are Russel's paradox.
The self-reference property was used in building Russel's Paradox.
They are not the same as Russel's Paradox.
Kermit Rose < [email protected] >