Most of the debate about the Transfinite Theory has centered on The Russell Paradox. I find this a bit odd as it seems to me that the main weakness of the theory is its misinterpretation of Galileo's Paradox. However, it is really not that surprising to see great debates centered on the Russell's work, afterall, Bertrand Russell was extremely talented at snagging the center of attention.
The Russell Paradox is the new name given to the liar's paradox. The liar's paradox has been known since antiquity. It's simplest form is: "This sentence is false." The sentence negates itself. There are many humorous versions of the paradox that include lying Cretans and confused barbers in Seville. The form of the paradox of greatest interest to this story occurs with all inclusive sets like the set of all sets.
The set of all sets is a set. This implies that it is somehow a member of itself. This idea of a set belonging to itself seems troubling to me, but we can accept it. Problems occur when we look at the set of all sets that have themselves as members (I will call this set A) and specifically the set of all sets that do not have themselves as members. I will call this set (B).
Set A is extremely interesting. The set is stable but has two possible states. If you start with the set as a member of itself, it is clear that it is a member of itself and stays that way. If it is not a member of itself, it stays in that state. The odd thing about the set is that it has two possible states.
Set B is problematic. Set B is the set of all sets that do not belong to themselves. If it is not a member of itself then it should be included in itself, if it is a member of itself, it should not be a member of itself. It is a logial impossibility.
Traditional classes on logic would spend a day laughing about different permutations of the paradox. Yhe teacher would mention that there are real paradoxes in the world, and go on to the next subject. Logic is only a tiny portion of life. We cannot derive everything from first principles, so let's move on.
As a young man, Russell was very much interesting the new foundational methods developed by Cantor and Frege. Although contemplating the liar's paradox is generally a waste of time, Russell realized that any new foundational theory would have to face the issue. This, of course, brings us to one of the most intriguing episodes in the history of mathematics.
As I mentioned, most school kids joke about then forget the liar's paradox. The only time it is a critical issue is when you happen to be creating a new foundational theory for mathematics and logic as Gottlob Frege Georg Cantor were doing at the turn of the 20th century.
Frege's main concern was in developing a symbolic logic, and developing a foundation that would unite logic and mathematics. Cantor was interesting in the definition of real numbers. Both Frege and Cantor were not the type to spend their days contemplating paradoxes.
The story goes that as Frege was going to press with his master work Grundgesetze der Arithmetik, Bertrand Russell (it is very important to emphasize that he did this humbly) sent Frege a note indicating that a portion of the work (Basic Law V) was subject to the Russell Paradox. Georg Cantor (who was a bit goofy by this time) also had an extremely hard emotional time dealing with the issue as his theory of classes was subject to the paradox.
The episode was devasting to Frege and Cantor. The story catapulted Russell to the top of intellectual community, and everyone had great fun formulating new versions of the liar's paradox.
The most amusing part of the story is that after destroying Frege, Russell began to work with Alfred North Whitehead on a similar project (The Pricipia Mathematica) that was the same ambitious program to lay the foundation for all mathematics.
To get around the paradox, Russell introduced a theory of types tries to prevent self reference by only allowing statements of a higher type reference an object.
In the The Diagonal Method on Finite Binary Strings, I suggest that the diagonal method itself is just another form of the liar's paradox. The diagonal method is essentially a trick that creates a self reference and a negation. I like to discuss the method on binary strings.
Imagine a large set of infinite binary strings. You can create a new string by taking the nth character from the nth string. If you perform a bitnot, you will create a new string that cannot be in the set.
1 0100011001... 2 1000101010... 3 1111011110... 4 0110110100... n ....
The diagonal on the above listing is 0010... The bitnot of the string is 1101... This new string cannot be in the list.
The diagonal is a powerful tool because it takes one piece of information from every single member of the set. This gives us a compact way of referring to every single item in the set.
The bitnot operation is a form of negation. It might be called a form of hypernegation. It turns all the 0s into 1s and 1s into 0s. Trying to include the diagonal product of the reals into a set of the reals creates a situation where you have a self-reference and a negation.
This is extremely clear with the set that contains only the diagonal product. The function dp(set) evaluates all of the items in the set and produces a bitnot of the diagonal. Its input is a set of numbers, it outputs a binary string that is the bitnot of the diagonal. Note, the length of the diagonal is determined by the number of members in the set.
I now try to define set A such at that A = {dp(A)}. The set has only one member, so dp(A) should evaluate to either 0 or 1. We see we have a paradox. The set cannot evaluate to 0 as the diagonal product of 0 is 1. Likewise, it cannot evaluate to 1.
The diagonal product itself is simply a clever form of the self-referential paradox.
It is interesting to note that Cantor's hierarchy of sets is quite similar to Russell's Theory of types.
Cantor claims that the diagonal method drives a dichotomy between the rationals and reals. He claims that the set of rationals is the same size as the set of integers, but that the real numbers are not.
I find this to be extremely poor logic. It seems to me that the problems with transfinite theory start with this notion that an infinite set can be the same size as a proper subset of itself. Saying that an infinite set is the same size as a proper subset of itself sets up a self reference.
Cantor's dichtomy is based on the fact that it is easier to include a negation in the set of reals (using the diagonal method) than it is to find a way to negate the set of rationals. However, if we look, we can find many tricks, like the set of namable numbers which would be considered denumerable but are subject to the diagonal product.
It seems to me that Galileo's paradox is simply a close relative to the Russell Paradox. One possible solution out of the maze is to simply dismiss the idea that an infinite set can be the same size as a proper subset. We will still be able to preserve Cantors layering of infinite sets. For that matter, accepting that infinite sets are not the same size creates an extremely rich layering of infinite sets. We would see:
Naturals < Whole Numbers < Integers < Rationals < Algebraic < Reals
Each of the layers would have their own set of attributes.